Browse Trigonometry problems Studdy Solved!

Problem 1401

Find the value of $\sec \frac{\pi}{12} \csc \frac{5 \pi}{12}-\tan \frac{\pi}{12} \cot \frac{5 \pi}{12}$ .

See Solution

Problem 1402

Rewrite the expression $\frac{\sin ^{2} \theta+\cot ^{2} \theta+\cos ^{2} \theta}{\csc \theta}$ using basic trig identities.

See Solution

Problem 1403

Calculate $\cos 66^{\circ}$ and round your answer to four decimal places.

See Solution

Problem 1404

Find the acute angle $\theta$ where $\sin \theta = \frac{\sqrt{3}}{2}$ . What is $\theta$ in radians?

See Solution

Problem 1405

If $\sec \theta=6$ , find $\csc \theta$ , $\cot \theta$ , $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ .

See Solution

Problem 1406

Постройте график функции $y=3 \sin \left(x-\frac{\pi}{4}\right)$ .

See Solution

Problem 1407

Simplify the expression: $y=\cos ^{2} x+\sin ^{2} x+1$ .

See Solution

Problem 1408

Plot the graph of the function $y=2 \tan\left(2 x+\frac{\pi}{4}\right)-2$ .

See Solution

Problem 1409

Simplify this expression without a calculator: $\frac{\cos 45 \cdot \sin 315 + 2 \tan 120 \cdot \cos 60}{2 \sin 240 \cdot \cos 300}$

See Solution

Problem 1410

Simplify:
1. $\frac{\cos 45 \cdot \sin 315+2 \tan 120 \cdot \cos 60}{2 \sin 240 \cdot \cos 300}$
2. Prove: $\frac{\cos x}{1+\sin x}-\frac{1-\sin x}{\cos x}=0$
3. Find $x$ for $5 \sin x+3 \cos x=0$ , $0^{\circ} \leq x \leq 360$
4. Simplify: $\cos (100-x)$

See Solution

Problem 1411

Simplify the expression: $\frac{\cos (160-x)}{\tan (90+x) \sin (180+x)}$ using reduction formulas.

See Solution

Problem 1412

Find the sine of $\angle T$ .
Write your answer in simplified, rationalized form. Do not round. $\sin (T)=$ $\square$

See Solution

Problem 1413

$\tan \theta = \sin^2 \theta \tan \theta + \frac{\sin^2 \theta}{\tan \theta}$

See Solution

Problem 1414

Which set of angles has the same terminal arm as $40^{\circ}$ ? A) $400^{\circ}, 760^{\circ}, 1120^{\circ}$ B) $220^{\circ}, 400^{\circ}, 580^{\circ}$ C) $80^{\circ}, 120^{\circ}, 200^{\circ}$ D) $130^{\circ}, 220^{\circ}, 310^{\circ}$
Question 7 (1 point) $\checkmark$ Saved
In which quadrants are the sine ratios negative values? 2 and 4 3 and 4 1 and 3 None of the options 1 and 2 Question 8 (1 point) Saved

See Solution

Problem 1415

Simplify each expression: Write the formula you will use to solve the Trig function a) $\operatorname{Cos} \frac{7 \pi}{12} \operatorname{Cos} \frac{5 \pi}{12}+\operatorname{Sin} \frac{7 \pi}{12} \sin \frac{5 \pi}{12}$

See Solution

Problem 1416

Solve the equation on the interval $0 \leq \theta<2 \pi$ . $(\cot \theta-1)(\csc \theta-1)=0$
Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is $\square$ \}. (Simplify your answer. Type an exact answer, using $\pi$ as needed. Type your answer i any numbers in the expression. Use a comma to separate answers as needed.) B. There is no solution on this interval.

See Solution

Problem 1417

10. If $\sec \theta=2$ and $0 \leq \theta \leq 2 \pi$ , determine the exact value(s) of $\csc \theta$ . Include a diagram.
11. If $\sin \theta=-1$ and $\pi \leq \theta \leq 2 \pi$ , determine the exact value(s) of $\cos \theta$ and $\cot \theta$ . Include a diagram.
12. If $\cos \theta=-\frac{3}{5}$ and $\sin \theta<0$ , determine the exact value(s) of $\csc \theta$ . Include a diagram.
13. Solve for $\theta$ given $\tan \theta=-\sqrt{3}$ for $-2 \pi \leq \theta \leq 2 \pi$
14. Solve for $\theta$ given $\cot \theta=5$ for $-\pi \leq \theta \leq \pi$

See Solution

Problem 1418

Question 2 (1 point) Which graph of the following trigonometric functions has no zeros? a) $y=\sec x$ b) $y=\cos x$ c) $y=\tan x$ d) $y=\cot x$

See Solution

Problem 1419

Question 3 (1 point) The graph of $y=\csc x$ can be generated by plotting the reciprocal of each $y$ -value of the graph $y=\sin x$ . True False

See Solution

Problem 1420

ATH 0993 - Math 12 part 2 (Sep 2023) nformation ए. Flag question
2. Convert the given angle to radians: $-\mathbf{7 2 0}{ }^{\circ}$

See Solution

Problem 1421

2uestion 5 (1 point) The height, $h$ , in centimetres, of a piston moving up and down in an engine cylinder can be modelled by the function $h(t)=18 \sin (50 \pi t)+18$ , where $t$ is the time, in seconds. What is the period? a) 50 s b) 0.04 cm c) 25 s d) 0.04 s

See Solution

Problem 1422

Question 8 (1 point) Determine the amplitude of the sinusoidal function $y=-3 \sin \left[2\left(x-\frac{\pi}{3}\right)\right]+1$ . a) 2 b) -3 c) $\frac{\pi}{3}$ d) 3

See Solution

Problem 1423

Question 22 (1 point) The height, $h$ , in metres, above the ground of a car as a Ferris wheel rotates can be modelled by the function $h(t)=16 \cos \left(\frac{\pi t}{120}\right)+18$ , where $t$ is the time, in seconds. What is the radius of the Ferris wheel? a) 16 m b) 8 m c) 9 m d) 18 m

See Solution

Problem 1424

Solve for $x$ in the triangle. Round your answer to the nearest tenth. $x=$

See Solution

Problem 1425

Question 6 (1 point) Solve $\sqrt{3} \csc x+2=0$ on the interval $x \in[0,2 \pi]$ , to the nearest hundredth of a radian. a) $x \doteq 1.05, x \doteq 2.09$ b) $x \doteq 4.19, x \doteq 5.24$ C) $x=60, x=120$ d) $x=240, x=300$

See Solution

Problem 1426

Question 13 (1 point) Solve $2 \sin x+1=0$ on the interval $x \in[0,2 \pi]$ , to the nearest hundredth of a radian. a) $x=210, x=330$ b) $x \doteq 0.52, x \doteq 2.62$ C) $x=30, x=150$ d) $x \doteq 3.67, x \doteq 5.76$

See Solution

Problem 1427

A sinusoidal wave is traveling on a string with speed $107 \mathrm{~cm} / \mathrm{s}$ . The displacement of the particles of the string at $x=16 \mathrm{~cm}$ is found to vary with time according to the equation $y=(1 \mathrm{~cm}) \sin \left[0.84-\left(5.6 \mathrm{~s}^{-1}\right) t\right]$
The linear density of the string is $1.8 \mathrm{~g} / \mathrm{cm}$ . What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form $y(x, t)=y_{m} \sin (k x-\omega t),$ what are (c) $y_{m}$ , (d) $k$ , and (e) $\omega$ , and (f) the correct choice of sign in front of $\omega$ ? (g) What is the tension in the string?

See Solution

Problem 1428

A night triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find $\cos B, \tan B$ , and $\sin B$ . $\begin{array}{l} \cos B= \\ \tan B= \\ \sin B= \end{array}$ $\square$ $\square$ $\square$

See Solution

Problem 1429

A right triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find $\tan M, \sin M$ , and $\cos M$ . $\begin{array}{l} \tan M=\square \\ \sin M=\square \\ \cos M=\square \end{array}$

See Solution

Problem 1430

Select the correct answer.
In $\triangle A B C, \angle A$ is a right angle. What is the value of $y$ ? A. $7 \sin \frac{\pi}{6}$ B. $7 \cos \frac{\pi}{6}$ C. $7 \tan \frac{\pi}{6}$ D. 7

See Solution

Problem 1431

Select the correct answer.
Given a right triangle, if $\tan \theta=\frac{(3)}{(4)}$ , what is the length of the side adjacent to $\angle \theta ?$ A. 3 B. 4 C. 5 D. 75

See Solution

Problem 1432

Select the correct answer.
In a right triangle, if $\angle \theta=39^{\circ}$ and the side adjacent to $\angle \theta$ is equal to 12.0 centimeters, what is the approximate length of the opposite side? A. 7.6 centimeters B. 9.3 centimeters C. 9.7 centimeters D. 14.8 centimeters

See Solution

Problem 1433

Type the correct answer in the box. Round your answer to the nearest integer.
A train traveled a distance of 1 mile, or 5,280 feet, while climbing a hill at an angle of $5^{\circ}$ .
The vertical height that the train climbed is approximately $\square$ feet.

See Solution

Problem 1434

Solve the following equation for $\theta$ in the interval $\left[0^{\circ}, 360^{\circ}\right)$ . $4 \cos 2 \theta=1$

See Solution

Problem 1435

2) Prove the following identities $\operatorname{arcos}(\pi+\theta)=2 \cos ^{2} \theta-1$

See Solution

Problem 1436

Select the correct answer.
Use a calculator to determine the value of $\arcsin (-0.66)$ . A. $\quad 0.72$ B. 2.29 C. -0.72 D. -2.29

See Solution

Problem 1437

Select the correct answer.
What value of $x$ satisfies $\cot \left(90^{\circ}-x\right)=-\frac{\sqrt{3}}{3} ?$ A. $120^{\circ}$ B. $240^{\circ}$ C. $210^{\circ}$ D. $150^{\circ}$

See Solution

Problem 1438

If: $13 \operatorname{Sin}\left(90^{\circ}-\theta\right)-5=0$ then: $\cos \theta=\ldots \ldots \ldots$ (a) $\frac{12}{13}$ (b) $-\frac{12}{13}$ (c) $-\frac{5}{13}$ (d) $\frac{5}{13}$
بقية الأسنلة فى الصفحة التالية

See Solution

Problem 1439

(9) If : $\cos \frac{20^{\circ}+\theta}{2}=\sin \frac{40^{\circ}+\theta}{2}, 0^{\circ}<\theta<90^{\circ}$ then $\theta=$ $\qquad$ (a) $60^{\circ}$ (b) $45^{\circ}$ (c) $30^{\circ}$ (d) $20^{\circ}$ (10) If the ratio between areas of two similar triangles equals $9: 25$ and the pe the smaller triangle is 60 cm then the perimeter of the greater triangle equals (a) 60 (b) 80 (c) 100 (d) 120

See Solution

Problem 1440

4. Solve the equation $\sin (8 x)=\sin (7 x) \cos (x)$ for $x \in(0, \pi)$ .
ANS:

See Solution

Problem 1441

For the triangle shown below, use your calculator to solve for the missing sides and angles. $\theta=$ $\square$ degrees $f=$ $\square$ $e=$ $\square$ Round your answers to two decimal places. Question Help: Video 1 Video 2

See Solution

Problem 1442

$\tan \left(\tan ^{-1} \frac{52 \pi}{7}\right)$

See Solution

Problem 1443

Graph the given function. State the period, amplitude, phase shift, and vertical shift of the function. $y=-\sin \left(x+\frac{\pi}{6}\right)$

See Solution

Problem 1444

Write the expression $\cos 4 \theta \sin \theta-\sin 4 \theta \cos \theta$ as a single sine or cosine.

See Solution

Problem 1445

5. Find the phase relations for the following pairs of sinusoids: a) $u=6 \sin \left(30 t-40^{\circ}\right) V$ and $i=10 \sin (30 t-\pi / 3) \mathrm{mA}$ b) $u_{1}=-8 \sin \left(40 t-80^{\circ}\right) \mathrm{V}$ and $u_{2}=-10 \sin \left(40 t-50^{\circ}\right) \mathrm{V}$ c) $i_{1}=4 \cos \left(70 t-40^{\circ}\right) \mathrm{mA}$ and $i_{2}=-6 \cos \left(70 t+80^{\circ}\right) \mathrm{mA}$ d) $u=-4 \sin \left(45 t+5^{\circ}\right) V$ and $i=7 \cos \left(45 t+80^{\circ}\right) \mathrm{mA}$
Ans. a) $u$ leads $i$ by $20^{\circ}$ ; b) $u_{1}$ lags $u_{2}$ by $30^{\circ}$ ; c) $i_{1}$ leads $i_{2}$ by $60^{\circ}$ ; d) $u$ leads $i$ by $15^{\circ}$

See Solution

Problem 1446

4. Find the periods of a) $4+3 \sin \left(800 \pi t-15^{\circ}\right)$ ; b) $8,1 \cos ^{2} 9 \pi t$ .
Ans. a) 2,5 ms; b) 111 ms .

See Solution

Problem 1447

3. Find the sine of angle A. Give your answer as a fraction in simplest form. $\operatorname{Sin} A=$ $\qquad$ I - Choose the correct answer - $\qquad$
Clear All

See Solution

Problem 1448

Simplify. $4 \sin 2 x \cos 2 x$

See Solution

Problem 1449

Solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places $\begin{aligned} & B=78.7^{\circ}, \quad a=4.9 \\ A & =\square^{\circ} \\ c & =\square^{\circ} \\ b & =\square^{\circ} \\ c & =\square^{\circ} \end{aligned}$

See Solution

Problem 1450

(4) Solve the following equation for $0 \leq x<2 \pi$ $5 \sin x-4 \sqrt{3}=3 \sin x-5 \sqrt{3}$ (10) Let $A=121^{\circ}, C=34^{\circ}$ and $b=18$ . Use Law of Sines to solve for $c$ .

See Solution

Problem 1451

$6 \sin ^{2} x=5 \cos x-2$

See Solution

Problem 1452

A 165 -foot tall antenna has 4 guy-wires connected to the top of the antenna, and each guy-wire is anchored to the ground. A side-view of this scenario is shown.
One of the guy-wires forms an angle of $\alpha=0.3$ radians with the antenna and the opposing guy-wire forms an angle of $\beta=0.41$ radians with the antenna. a. What is the horizontal distance between anchor 1 and the base of the antenna? $165^{} \tan (0.3)$ $\qquad$ $\star$ feet $\square$ $165 \cdot \tan (0.3)=51.040481185587836$ . b. What is the horizontal distance between anchor 2 and the base of the antenna? $165^{} \tan (0.41)$ $\square$ $\otimes$ feet $\square$ $165 \cdot \tan (0.41)=71.71414875898176$ . c. What is the distance between anchor 1 and anchor 2? $\square$ feet Preview

See Solution

Problem 1453

$4 \sin (0.5 x-1)$ amplitude: period: c horizontal shift: $\square$

See Solution

Problem 1454

7. Nate is in a meadow standing exactly 185 ft from the base of a mountain. He sees someone climbing the mountain in his binoculars. His eyes are 6 ft above the ground, and is angle of elevation is $10^{\circ}$ . How far above the ground is the climber?

See Solution

Problem 1455

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $\sin \left(\frac{19 \pi}{12}\right)$

See Solution

Problem 1456

Simplify the expression: $\frac{\frac{1}{\cos \theta}-\cos \theta}{\cos \theta}$ a. $\sec ^{2} \theta$ b. $\cot ^{2} \theta$ $\frac{\frac{1}{\cos \theta}-\cos \theta}{\cos \theta}$ c. $\tan ^{2} \theta$ d. 1

See Solution

Problem 1457

8. An airplane must fly over a 120 ft tower. The plane is 400 ft away from the tower when it begins to climb. At what angle should the plane climb to make it over the tower?

See Solution

Problem 1458

$\sin \left(30^{\circ}\right)=\frac{1}{2}, \tan \left(30^{\circ}\right)=\frac{\sqrt{3}}{3}$ (a) $\csc \left(30^{\circ}\right)$ $\square$ (b) $\cot \left(60^{\circ}\right)$ $\square$ (c) $\cos \left(30^{\circ}\right)$ $\square$ (d) $\cot \left(30^{\circ}\right)$ $\square$ Need Help? Read It Watch It Submit Answer

See Solution

Problem 1459

5. Prove the following trigonometric identity. (4 marks) $\begin{array}{c} \frac{\cos 2 x \sec x}{1-\tan ^{2} x}=\cos x \\ \cos ^{2} \theta-\sin ^{2} \theta\left(\frac{1}{\cos x}\right) \end{array}$

See Solution

Problem 1460

7. Simplify the following: $3 \sin (2 x) \cos (2 x)$ a. $1.5 \sin (2 x)$ b. $6 \sin (4 x)$ c. $1.5 \sin (4 x)$

See Solution

Problem 1461

Sections $4.7+4.8$
Show all work! (1) Find the exact value of each expression. state if undefined. a) $\arccos \left(\frac{1}{2}\right)$ b) $\arcsin (4)$ c) $\sin \left(\arcsin \left(-\frac{1}{2}\right)\right)$ d) $\tan \left(\arccos \left(\frac{3}{7}\right)\right) \quad \begin{array}{l}\text { sketcl this on the } \\ \text { coordinate plane. }\end{array}$ (2) Solve the problem. Use exact values (leave in terms of a trig function. Aski slope is 52 ft long and the angle A ski slope is from the ground to the summit is $42^{\circ}$ . How high is the summit? (Draw your best ski slope and mountain) "̈

See Solution

Problem 1462

An electric current, $I$ , in amps, is given by $I=\cos (w t)+\sqrt{3} \sin (w t),$ where $w \neq 0$ is a constant. What are the maximum and minimum values of $I$ ? Minimum value of $I$ : $\square$ amp
Maximum value of $I$ : $\square$ amp
Note: You can earn partial credit on this problem.

See Solution

Problem 1463

Product-to-Sum Formulas:
1. $\left.\sin (u) \cos (v)=\frac{1}{2}[\sin (u+v))+\sin (u-v)\right]$
2. $\left.\cos (u) \sin (v)=\frac{1}{2}[\sin (u+v))-\sin (u-v)\right]$
3. $\left.\cos (u) \cos (v)=\frac{1}{2}[\cos (u+v))+\cos (u-v)\right]$
4. $\left.\sin (u) \sin (v)=\frac{1}{2}[\cos (u-v))-\cos (u+v)\right]$
Rewrite the expression below using one of the given formulas. $\sin (3 x) \sin (5 x)$ Using formula number one: $\sin (3 x) \sin (5 x)=\frac{1}{2}[\cos (3 x-5 x)-\cos (3 x+5 x)]$ Using formula four: $\sin (3 x) \sin (5 x)=\frac{1}{2}[\cos (3 x+5 x)-\cos (3 x-5 x)]$ Using formula one: $\sin (3 x) \sin (5 x)=\frac{1}{2}[\cos (3 x+5 x)-\cos (3 x-5 x)]$ Using formula number four: $\sin (3 x) \sin (5 x)=\frac{1}{2}[\cos (3 x-5 x)-\cos (3 x+5 x)]$

See Solution

Problem 1464

Match the expression (I, II) with its correct formula (i, ii, iii or iv). $1 \sin (2 x)=$ $11 \cos (2 x)=$ i $\cos ^{2}(x)-\sin ^{2}(x) \quad$ ii $2 \sin \theta \cos \theta$ iii $\sin ^{2} \theta-\cos ^{2} \theta \quad$ iv $-2 \cos \theta \sin \theta$
I [Choose]
II [Choose]

See Solution

Problem 1465

Question 1 25 pts
Find the solutions of $\cos (\theta)+1=0$ when $0 \leq \theta \leq 2 \pi$ . $\theta=\pi$ $\theta=\frac{\pi}{6}$ $\theta=0,2 \pi$ $\theta=\frac{\pi}{2}$ $\theta=\frac{3 \pi}{2}$

See Solution

Problem 1466

4. Determine exact value for $2 \sin 112.5^{\circ} \cos 112.5^{\circ}$ : A. $\frac{1}{2}$ B. $-\frac{1}{2}$ C. $\frac{\sqrt{2}}{2}$ D. $-\frac{\sqrt{2}}{2}$

See Solution

Problem 1467

$-5 \sin (0.5 x+1)$ amplitude: a period: d horizontal shift: $\qquad$

See Solution

Problem 1468

6.Simplify: $\cos \left(\alpha+\frac{\pi}{2}\right)$ A. $\sin \alpha$ B. $\cos \alpha$ C. $-\sin \alpha$ D. $-\cos \alpha$

See Solution

Problem 1469

$\frac{\pi}{2} \sin \left(3\left(\frac{\pi}{2}\right)\right.$

See Solution

Problem 1470

3. At a seaport, the depth of the water, d , in meters, at time $t$ hours, during a certain day is given by: $\mathrm{d}=3.4 \sin \left(2 \pi \frac{(\mathrm{t}-7.00)}{10.6}\right)+2.8$ [4 marks] a) What is the depth of the water at $6: 30 \mathrm{pm}$ ? (Answer to the nearest hundredths). b) How long will the depth be above 4 metres during one full day of 24 hours?

See Solution

Problem 1471

9. Show that $\tan 30^{\circ}+\frac{1}{\tan 30^{\circ}}=\frac{1}{\sin 30^{\circ} \cos 30^{\circ}}$ .

See Solution

Problem 1472

Nrite the complex number in trigonometric form $r(\cos \theta+i \boldsymbol{\operatorname { s i n }} \theta)$ , with $\theta$ in the interval $\left[0^{\circ}, 360^{\circ}\right)$ . $-3+3 i$ $-3+3 i=$ $\square$ $\square$ (cos ${ }^{\circ}+i \sin$ $\square$ ${ }^{\circ}$ ) (Type the value for $r$ as an exact answer, using radicals as needed. Type the value for $\theta$ as an integ nearest tenth as needed.)

See Solution

Problem 1473

Question 1 (1 point) $\checkmark$ Saved
The graphs of the functions $y=\sin x$ and $y=\cos x$ have the same domain. True False
Question 2 (1 point) $\checkmark$ Saved
The graphs of the functions $y=\cot x$ and $y=\tan x$ have the same domain. True False
Question 3 (1 point) $\checkmark$ Saved
A solution to the trigonometric equation $\sin x+\cos x=0$ is $x=0$ . True False

See Solution

Problem 1474

Question 4 (1 point) The function $y=-3 \sin x+1$ has an amplitude of -3 . True False
Question 5 (1 point) The graph of the function $y=\sin \pi x$ has a period of 2 . True False
Question 6 (1 point) The trigonometric equation $\cos ^{2} x-\sin ^{2} x=0$ has the same solutions as the trigonometric equation $\cos 2 x=0$ . True False

See Solution

Problem 1475

Write the complex number in rectangular form. $\begin{array}{l} 12\left(\cos 150^{\circ}+i \sin 150^{\circ}\right) \\ 12\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)= \end{array}$ $\square$ (Type your answer in the form a + bi. Type an exact answer, using radicals as needed.)

See Solution

Problem 1476

Question 49 (1 point) Matthew is trying to figure out which value for $x$ is NOT a solution for $\tan x=0$ . Do you have an answer? Choose one. a) $-3 \pi$ b) 0 c) $2 \pi$ d) $\frac{\pi}{2}$

See Solution

Problem 1477

Prove $\sin \theta \tan \theta=\sec \theta-\cos \theta$

See Solution

Problem 1478

Question 3 (1 point) An equivalent trigonometric expression for $\tan (-x)$ is a) $\tan x$ b) $-\cot x$ c) $\cot x$ d) $-\tan x$

See Solution

Problem 1479

What is the exact value of $\tan \left(-\frac{\pi}{12}\right)$ ? $-2+\sqrt{3}$ $-\frac{\sqrt{3}}{3}$
$\frac{\tan \frac{7 \pi}{12}}{1-\sqrt{3}}$ $1-\sqrt{3}$

See Solution

Problem 1480

Find the angle $\alpha$ coterminal with $\theta=\frac{\pi}{14}$ in the range $-2\pi < \alpha < 0$ .

See Solution

Problem 1481

Find $\cos (\theta)$ given $\sin (\theta)<0$ and $\cot (\theta)=\frac{9}{2}$ . Provide an exact answer.

See Solution

Problem 1482

Find $\tan (\theta)$ given $\sin (\theta)<0$ and $\sec (\theta)=\frac{\sqrt{35}}{4}$ . Provide the exact answer.

See Solution

Problem 1483

Solve $\cos \left(\frac{x}{2}\right)=-\frac{\sqrt{3}}{2}$ for $x$ in $[0,2\pi)$ . Provide exact radian solutions.

See Solution

Problem 1484

Solve $\cos \left(\frac{x}{2}\right)=\frac{1}{2}$ for $x$ in $[0,2 \pi)$ , and give the answer in exact radians.

See Solution

Problem 1485

Find the value of $f(x)=5 \sin^{-1}(\sin(x)) + 3 \cos^{-1}(\sin(4x))$ at $x=\frac{\pi}{3}$ .

See Solution

Problem 1486

Find the value of $\sin ^{-1}\left[\sin \left(-\frac{7 \pi}{6}\right)\right]$ without a calculator.

See Solution

Problem 1487

Simplify $\cot \left(\sin ^{-1}(x)\right)$ using a triangle or trigonometric identity, assuming $x > 0$ .

See Solution

Problem 1488

Find the value of the trigonometric expression: $\tan \left[\sec ^{-1}(-5)\right]$ .

See Solution

Problem 1489

Find the value of $\cos \left(\cot ^{-1}(10)\right)$ using trigonometric identities.

See Solution

Problem 1490

Gina and Hone are on opposite sides of a tower. Gina is 20 m away with an angle of elevation of $53^{\circ}$ . Hone's angle is $37^{\circ}$ . Find Hone's distance from the tower.

See Solution

Problem 1491

Peter sees a flag-pole $6 \mathrm{~m}$ away. Angle of depression is $65^{\circ}$ , and elevation is $47^{\circ}$ . Find the height.

See Solution

Problem 1492

Find the family of special angles for $\theta = \frac{14 \pi}{3}$ and its least nonnegative coterminal angle $\theta_{c}$ .

See Solution

Problem 1493

Find the six trigonometric functions of the angle $\theta$ with point $(-24,7)$ on its terminal side. $\sin \theta =$

See Solution

Problem 1494

Identify the family of special angles for $-\frac{43 \pi}{6}$ and find the least nonnegative coterminal angle $\theta_{c}$ .

See Solution

Problem 1495

Find the exact value of $\sin \left(-\frac{9 \pi}{2}\right)$ and answer parts a, b, c, and d. Where is the terminal side of $\theta=-\frac{9 \pi}{2}$ ?

See Solution

Problem 1496

Find $\cot(-12\pi)$ exactly. Answer parts a, b, c, and d: a. Where is the terminal side of $\theta = -12\pi$ ?

See Solution

Problem 1497

Find $\cot (-12 \pi)$ without a calculator. a. Where is the terminal side of $\theta=-12 \pi$ ? b. Give the coordinates on the terminal side with $r=1$ .

See Solution

Problem 1498

Find $\cot (-12 \pi)$ by answering: a) Where is $\theta=-12 \pi$ ? b) Coordinates for $r=1$ . c) $\cot \theta$ definition?

See Solution

Problem 1499

Find $\cot(-12\pi)$ : a. Where is $\theta=-12\pi$ ? b. Coordinates on terminal side? c. What is $\cot \theta$ ?

See Solution

Problem 1500

Find the value of $\tan(-7 \pi)$ and answer: a. Where is the terminal side of $\theta=-7 \pi$ ?

See Solution

Trigonometry

Problem 1401

Find the value of sec⁡π12csc⁡5π12−tan⁡π12cot⁡5π12\sec \frac{\pi}{12} \csc \frac{5 \pi}{12}-\tan \frac{\pi}{12} \cot \frac{5 \pi}{12}sec12π​csc125π​−tan12π​cot125π​.

Problem 1402

Rewrite the expression sin⁡2θ+cot⁡2θ+cos⁡2θcsc⁡θ\frac{\sin ^{2} \theta+\cot ^{2} \theta+\cos ^{2} \theta}{\csc \theta}cscθsin2θ+cot2θ+cos2θ​ using basic trig identities.

Problem 1403

Calculate cos⁡66∘\cos 66^{\circ}cos66∘ and round your answer to four decimal places.

Problem 1404

Find the acute angle θ\thetaθ where sin⁡θ=32\sin \theta = \frac{\sqrt{3}}{2}sinθ=23​​. What is θ\thetaθ in radians?

Problem 1405

If sec⁡θ=6\sec \theta=6secθ=6, find csc⁡θ\csc \thetacscθ, cot⁡θ\cot \thetacotθ, sin⁡θ\sin \thetasinθ, cos⁡θ\cos \thetacosθ, and tan⁡θ\tan \thetatanθ.

Problem 1406

Постройте график функции y=3sin⁡(x−π4)y=3 \sin \left(x-\frac{\pi}{4}\right)y=3sin(x−4π​).

Problem 1407

Simplify the expression: y=cos⁡2x+sin⁡2x+1y=\cos ^{2} x+\sin ^{2} x+1y=cos2x+sin2x+1.

Problem 1408

Plot the graph of the function y=2tan⁡(2x+π4)−2y=2 \tan\left(2 x+\frac{\pi}{4}\right)-2y=2tan(2x+4π​)−2.

Problem 1409

Simplify this expression without a calculator: cos⁡45⋅sin⁡315+2tan⁡120⋅cos⁡602sin⁡240⋅cos⁡300\frac{\cos 45 \cdot \sin 315 + 2 \tan 120 \cdot \cos 60}{2 \sin 240 \cdot \cos 300}2sin240⋅cos300cos45⋅sin315+2tan120⋅cos60​

Problem 1410

Problem 1411

Simplify the expression: cos⁡(160−x)tan⁡(90+x)sin⁡(180+x)\frac{\cos (160-x)}{\tan (90+x) \sin (180+x)}tan(90+x)sin(180+x)cos(160−x)​ using reduction formulas.

Problem 1412

Find the sine of ∠T\angle T∠T.Write your answer in simplified, rationalized form. Do not round. sin⁡(T)=\sin (T)=sin(T)= □\square□

Problem 1413

tan⁡θ=sin⁡2θtan⁡θ+sin⁡2θtan⁡θ\tan \theta = \sin^2 \theta \tan \theta + \frac{\sin^2 \theta}{\tan \theta}tanθ=sin2θtanθ+tanθsin2θ​

Problem 1414

Problem 1415

Problem 1416

Problem 1417

Problem 1418

Question 2 (1 point) Which graph of the following trigonometric functions has no zeros? a) y=sec⁡xy=\sec xy=secx b) y=cos⁡xy=\cos xy=cosx c) y=tan⁡xy=\tan xy=tanx d) y=cot⁡xy=\cot xy=cotx

Problem 1419

Question 3 (1 point) The graph of y=csc⁡xy=\csc xy=cscx can be generated by plotting the reciprocal of each yyy-value of the graph y=sin⁡xy=\sin xy=sinx. True False

Problem 1420

ATH 0993 - Math 12 part 2 (Sep 2023) nformation ए. Flag question2. Convert the given angle to radians: −720∘-\mathbf{7 2 0}{ }^{\circ}−720∘

Problem 1421

Problem 1422

Question 8 (1 point) Determine the amplitude of the sinusoidal function y=−3sin⁡[2(x−π3)]+1y=-3 \sin \left[2\left(x-\frac{\pi}{3}\right)\right]+1y=−3sin[2(x−3π​)]+1. a) 2 b) -3 c) π3\frac{\pi}{3}3π​ d) 3

Problem 1423

Problem 1424

Solve for xxx in the triangle. Round your answer to the nearest tenth. x=x=x=

Problem 1425

Problem 1426

Problem 1427

Problem 1428

A night triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find cos⁡B,tan⁡B\cos B, \tan BcosB,tanB, and sin⁡B\sin BsinB. cos⁡B=tan⁡B=sin⁡B=\begin{array}{l} \cos B= \\ \tan B= \\ \sin B= \end{array}cosB=tanB=sinB=​ □\square□ □\square□ □\square□

Problem 1429

A right triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find tan⁡M,sin⁡M\tan M, \sin MtanM,sinM, and cos⁡M\cos McosM. tan⁡M=□sin⁡M=□cos⁡M=□\begin{array}{l} \tan M=\square \\ \sin M=\square \\ \cos M=\square \end{array}tanM=□sinM=□cosM=□​

Problem 1430

Select the correct answer.In △ABC,∠A\triangle A B C, \angle A△ABC,∠A is a right angle. What is the value of yyy ? A. 7sin⁡π67 \sin \frac{\pi}{6}7sin6π​ B. 7cos⁡π67 \cos \frac{\pi}{6}7cos6π​ C. 7tan⁡π67 \tan \frac{\pi}{6}7tan6π​ D. 7

Problem 1431

Select the correct answer.Given a right triangle, if tan⁡θ=(3)(4)\tan \theta=\frac{(3)}{(4)}tanθ=(4)(3)​, what is the length of the side adjacent to ∠θ?\angle \theta ?∠θ? A. 3 B. 4 C. 5 D. 75

Problem 1432

Problem 1433

Type the correct answer in the box. Round your answer to the nearest integer.A train traveled a distance of 1 mile, or 5,280 feet, while climbing a hill at an angle of 5∘5^{\circ}5∘.The vertical height that the train climbed is approximately □\square□ feet.

Problem 1434

Solve the following equation for θ\thetaθ in the interval [0∘,360∘)\left[0^{\circ}, 360^{\circ}\right)[0∘,360∘). 4cos⁡2θ=14 \cos 2 \theta=14cos2θ=1

Problem 1435

2) Prove the following identities arcos⁡(π+θ)=2cos⁡2θ−1\operatorname{arcos}(\pi+\theta)=2 \cos ^{2} \theta-1arcos(π+θ)=2cos2θ−1

Problem 1436

Select the correct answer.Use a calculator to determine the value of arcsin⁡(−0.66)\arcsin (-0.66)arcsin(−0.66). A. 0.72\quad 0.720.72 B. 2.29 C. -0.72 D. -2.29

Problem 1437

Select the correct answer.What value of xxx satisfies cot⁡(90∘−x)=−33?\cot \left(90^{\circ}-x\right)=-\frac{\sqrt{3}}{3} ?cot(90∘−x)=−33​​? A. 120∘120^{\circ}120∘ B. 240∘240^{\circ}240∘ C. 210∘210^{\circ}210∘ D. 150∘150^{\circ}150∘

Problem 1438

Problem 1439

Problem 1440

4. Solve the equation sin⁡(8x)=sin⁡(7x)cos⁡(x)\sin (8 x)=\sin (7 x) \cos (x)sin(8x)=sin(7x)cos(x) for x∈(0,π)x \in(0, \pi)x∈(0,π).ANS:

Problem 1441

For the triangle shown below, use your calculator to solve for the missing sides and angles. θ=\theta=θ= □\square□ degrees f=f=f= □\square□ e=e=e= □\square□ Round your answers to two decimal places. Question Help: Video 1 Video 2

Problem 1442

tan⁡(tan⁡−152π7)\tan \left(\tan ^{-1} \frac{52 \pi}{7}\right)tan(tan−1752π​)

Problem 1443

Graph the given function. State the period, amplitude, phase shift, and vertical shift of the function. y=−sin⁡(x+π6)y=-\sin \left(x+\frac{\pi}{6}\right)y=−sin(x+6π​)

Problem 1444

Write the expression cos⁡4θsin⁡θ−sin⁡4θcos⁡θ\cos 4 \theta \sin \theta-\sin 4 \theta \cos \thetacos4θsinθ−sin4θcosθ as a single sine or cosine.

Problem 1445

Problem 1446

4. Find the periods of a) 4+3sin⁡(800πt−15∘)4+3 \sin \left(800 \pi t-15^{\circ}\right)4+3sin(800πt−15∘); b) 8,1cos⁡29πt8,1 \cos ^{2} 9 \pi t8,1cos29πt.Ans. a) 2,5 ms; b) 111 ms .

Problem 1447

Find the value of $\sec \frac{\pi}{12} \csc \frac{5 \pi}{12}-\tan \frac{\pi}{12} \cot \frac{5 \pi}{12}$ .

Rewrite the expression $\frac{\sin ^{2} \theta+\cot ^{2} \theta+\cos ^{2} \theta}{\csc \theta}$ using basic trig identities.

Calculate $\cos 66^{\circ}$ and round your answer to four decimal places.

Find the acute angle $\theta$ where $\sin \theta = \frac{\sqrt{3}}{2}$ . What is $\theta$ in radians?

If $\sec \theta=6$ , find $\csc \theta$ , $\cot \theta$ , $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ .

Постройте график функции $y=3 \sin \left(x-\frac{\pi}{4}\right)$ .

Simplify the expression: $y=\cos ^{2} x+\sin ^{2} x+1$ .

Plot the graph of the function $y=2 \tan\left(2 x+\frac{\pi}{4}\right)-2$ .

Simplify this expression without a calculator: $\frac{\cos 45 \cdot \sin 315 + 2 \tan 120 \cdot \cos 60}{2 \sin 240 \cdot \cos 300}$

Simplify the expression: $\frac{\cos (160-x)}{\tan (90+x) \sin (180+x)}$ using reduction formulas.

Find the sine of $\angle T$ .
Write your answer in simplified, rationalized form. Do not round. $\sin (T)=$ $\square$

$\tan \theta = \sin^2 \theta \tan \theta + \frac{\sin^2 \theta}{\tan \theta}$

Question 2 (1 point) Which graph of the following trigonometric functions has no zeros? a) $y=\sec x$ b) $y=\cos x$ c) $y=\tan x$ d) $y=\cot x$

Question 3 (1 point) The graph of $y=\csc x$ can be generated by plotting the reciprocal of each $y$ -value of the graph $y=\sin x$ . True False

ATH 0993 - Math 12 part 2 (Sep 2023) nformation ए. Flag question
2. Convert the given angle to radians: $-\mathbf{7 2 0}{ }^{\circ}$

Question 8 (1 point) Determine the amplitude of the sinusoidal function $y=-3 \sin \left[2\left(x-\frac{\pi}{3}\right)\right]+1$ . a) 2 b) -3 c) $\frac{\pi}{3}$ d) 3

Solve for $x$ in the triangle. Round your answer to the nearest tenth. $x=$

A night triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find $\cos B, \tan B$ , and $\sin B$ . $\begin{array}{l} \cos B= \\ \tan B= \\ \sin B= \end{array}$ $\square$ $\square$ $\square$

A right triangle has side lengths 5,12 , and 13 as shown below. Use these lengths to find $\tan M, \sin M$ , and $\cos M$ . $\begin{array}{l} \tan M=\square \\ \sin M=\square \\ \cos M=\square \end{array}$

Select the correct answer.
In $\triangle A B C, \angle A$ is a right angle. What is the value of $y$ ? A. $7 \sin \frac{\pi}{6}$ B. $7 \cos \frac{\pi}{6}$ C. $7 \tan \frac{\pi}{6}$ D. 7

Select the correct answer.
Given a right triangle, if $\tan \theta=\frac{(3)}{(4)}$ , what is the length of the side adjacent to $\angle \theta ?$ A. 3 B. 4 C. 5 D. 75

Type the correct answer in the box. Round your answer to the nearest integer.
A train traveled a distance of 1 mile, or 5,280 feet, while climbing a hill at an angle of $5^{\circ}$ .
The vertical height that the train climbed is approximately $\square$ feet.

Solve the following equation for $\theta$ in the interval $\left[0^{\circ}, 360^{\circ}\right)$ . $4 \cos 2 \theta=1$

2) Prove the following identities $\operatorname{arcos}(\pi+\theta)=2 \cos ^{2} \theta-1$

Select the correct answer.
Use a calculator to determine the value of $\arcsin (-0.66)$ . A. $\quad 0.72$ B. 2.29 C. -0.72 D. -2.29

Select the correct answer.
What value of $x$ satisfies $\cot \left(90^{\circ}-x\right)=-\frac{\sqrt{3}}{3} ?$ A. $120^{\circ}$ B. $240^{\circ}$ C. $210^{\circ}$ D. $150^{\circ}$

4. Solve the equation $\sin (8 x)=\sin (7 x) \cos (x)$ for $x \in(0, \pi)$ .
ANS:

For the triangle shown below, use your calculator to solve for the missing sides and angles. $\theta=$ $\square$ degrees $f=$ $\square$ $e=$ $\square$ Round your answers to two decimal places. Question Help: Video 1 Video 2

$\tan \left(\tan ^{-1} \frac{52 \pi}{7}\right)$

Graph the given function. State the period, amplitude, phase shift, and vertical shift of the function. $y=-\sin \left(x+\frac{\pi}{6}\right)$

Write the expression $\cos 4 \theta \sin \theta-\sin 4 \theta \cos \theta$ as a single sine or cosine.

4. Find the periods of a) $4+3 \sin \left(800 \pi t-15^{\circ}\right)$ ; b) $8,1 \cos ^{2} 9 \pi t$ .
Ans. a) 2,5 ms; b) 111 ms .

3. Find the sine of angle A. Give your answer as a fraction in simplest form. $\operatorname{Sin} A=$ $\qquad$ I - Choose the correct answer - $\qquad$
Clear All

Simplify. $4 \sin 2 x \cos 2 x$

(4) Solve the following equation for $0 \leq x<2 \pi$ $5 \sin x-4 \sqrt{3}=3 \sin x-5 \sqrt{3}$ (10) Let $A=121^{\circ}, C=34^{\circ}$ and $b=18$ . Use Law of Sines to solve for $c$ .

$6 \sin ^{2} x=5 \cos x-2$

$4 \sin (0.5 x-1)$ amplitude: period: c horizontal shift: $\square$

7. Nate is in a meadow standing exactly 185 ft from the base of a mountain. He sees someone climbing the mountain in his binoculars. His eyes are 6 ft above the ground, and is angle of elevation is $10^{\circ}$ . How far above the ground is the climber?

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $\sin \left(\frac{19 \pi}{12}\right)$

7. Simplify the following: $3 \sin (2 x) \cos (2 x)$ a. $1.5 \sin (2 x)$ b. $6 \sin (4 x)$ c. $1.5 \sin (4 x)$

4. Determine exact value for $2 \sin 112.5^{\circ} \cos 112.5^{\circ}$ : A. $\frac{1}{2}$ B. $-\frac{1}{2}$ C. $\frac{\sqrt{2}}{2}$ D. $-\frac{\sqrt{2}}{2}$

$-5 \sin (0.5 x+1)$ amplitude: a period: d horizontal shift: $\qquad$

6.Simplify: $\cos \left(\alpha+\frac{\pi}{2}\right)$ A. $\sin \alpha$ B. $\cos \alpha$ C. $-\sin \alpha$ D. $-\cos \alpha$

$\frac{\pi}{2} \sin \left(3\left(\frac{\pi}{2}\right)\right.$

9. Show that $\tan 30^{\circ}+\frac{1}{\tan 30^{\circ}}=\frac{1}{\sin 30^{\circ} \cos 30^{\circ}}$ .