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Math  /  Trigonometry

QuestionFor Exercises 25-30, assume that θ\theta is an acute angle. (See Example 2)
25. If cos⁡θ=217\cos \theta=\frac{\sqrt{21}}{7}, find csc⁡θ\csc \theta.
26. If sin⁡θ=1717\sin \theta=\frac{\sqrt{17}}{17}, find cot⁡θ\cot \theta.
27. If sec⁡θ=32\sec \theta=\frac{3}{2}, find sin⁡θ\sin \theta.
28. If csc⁡θ=3\csc \theta=3, find cos⁡θ\cos \theta.
29. If tan⁡θ=159\tan \theta=\frac{\sqrt{15}}{9}, find cos⁡θ\cos \theta.
30. If cot⁡θ=32\cot \theta=\frac{\sqrt{3}}{2}, find cos⁡θ\cos \theta.
Objective 3: Determine Trigonometric Function Values for Speçal Angles For Exercise 31, use the isosceles right triangle and the 30∘−60∘−90∘30^{\circ}-60^{\circ}-90^{\circ} triangle to complete the table. (See Examples 3-4) 31. \begin{tabular}{|c|c|c|c|c|c|c|} \hlineθ\theta & sin⁡θ\sin \theta & cos⁡θ\cos \theta & tan⁡θ\tan \theta & csc⁡θ\csc \theta & sec⁡θ\sec \theta & cot⁡θ\cot \theta \\ \hline 30∘=π630^{\circ}=\frac{\pi}{6} & & & & & & \\ \hline 45∘=π445^{\circ}=\frac{\pi}{4} & & & & & −⋮-\vdots \\ \hline 60∘=π360^{\circ}=\frac{\pi}{3} & & & & & & \\ \hline \end{tabular}
32. a. Evaluate sin⁡60∘\sin 60^{\circ}. b. Evaluate sin⁡30∘+sin⁡30∘\sin 30^{\circ}+\sin 30^{\circ}.

Studdy Solution

STEP 1

1. We are given a series of exercises involving trigonometric functions of an acute angle θ\theta.
2. We need to find specific trigonometric function values based on given information.
3. We assume familiarity with trigonometric identities and relationships.

STEP 2

1. Solve Exercise 25: Find csc⁡θ\csc \theta given cos⁡θ=217\cos \theta = \frac{\sqrt{21}}{7}.
2. Solve Exercise 26: Find cot⁡θ\cot \theta given sin⁡θ=1717\sin \theta = \frac{\sqrt{17}}{17}.
3. Solve Exercise 27: Find sin⁡θ\sin \theta given sec⁡θ=32\sec \theta = \frac{3}{2}.
4. Solve Exercise 28: Find cos⁡θ\cos \theta given csc⁡θ=3\csc \theta = 3.
5. Solve Exercise 29: Find cos⁡θ\cos \theta given tan⁡θ=159\tan \theta = \frac{\sqrt{15}}{9}.
6. Solve Exercise 30: Find cos⁡θ\cos \theta given cot⁡θ=32\cot \theta = \frac{\sqrt{3}}{2}.
7. Solve Exercise 31: Complete the trigonometric table for special angles.
8. Solve Exercise 32: Evaluate specific trigonometric expressions.

STEP 3

Given cos⁡θ=217\cos \theta = \frac{\sqrt{21}}{7}, use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find sin⁡θ\sin \theta.
sin⁡2θ=1−cos⁡2θ=1−(217)2\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{\sqrt{21}}{7}\right)^2

STEP 4

Calculate sin⁡2θ\sin^2 \theta:
sin⁡2θ=1−2149=2849=47\sin^2 \theta = 1 - \frac{21}{49} = \frac{28}{49} = \frac{4}{7}
sin⁡θ=47=277\sin \theta = \sqrt{\frac{4}{7}} = \frac{2\sqrt{7}}{7}

STEP 5

Find csc⁡θ\csc \theta, which is the reciprocal of sin⁡θ\sin \theta:
csc⁡θ=1sin⁡θ=727=72\csc \theta = \frac{1}{\sin \theta} = \frac{7}{2\sqrt{7}} = \frac{\sqrt{7}}{2}

STEP 6

Given sin⁡θ=1717\sin \theta = \frac{\sqrt{17}}{17}, find cos⁡θ\cos \theta using the Pythagorean identity:
cos⁡2θ=1−sin⁡2θ=1−(1717)2\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{\sqrt{17}}{17}\right)^2

STEP 7

Calculate cos⁡2θ\cos^2 \theta:
cos⁡2θ=1−17289=272289\cos^2 \theta = 1 - \frac{17}{289} = \frac{272}{289}
cos⁡θ=272289=27217\cos \theta = \sqrt{\frac{272}{289}} = \frac{\sqrt{272}}{17}

STEP 8

Find cot⁡θ\cot \theta, which is the ratio of cos⁡θ\cos \theta to sin⁡θ\sin \theta:
cot⁡θ=cos⁡θsin⁡θ=272171717=27217=27217\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{272}}{17}}{\frac{\sqrt{17}}{17}} = \frac{\sqrt{272}}{\sqrt{17}} = \sqrt{\frac{272}{17}}

STEP 9

Given sec⁡θ=32\sec \theta = \frac{3}{2}, find cos⁡θ\cos \theta:
cos⁡θ=1sec⁡θ=23\cos \theta = \frac{1}{\sec \theta} = \frac{2}{3}

STEP 10

Use the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find sin⁡θ\sin \theta:
sin⁡2θ=1−(23)2=1−49=59\sin^2 \theta = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}
sin⁡θ=59=53\sin \theta = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}

STEP 11

Given csc⁡θ=3\csc \theta = 3, find sin⁡θ\sin \theta:
sin⁡θ=1csc⁡θ=13\sin \theta = \frac{1}{\csc \theta} = \frac{1}{3}

STEP 12

Use the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find cos⁡θ\cos \theta:
cos⁡2θ=1−(13)2=1−19=89\cos^2 \theta = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}
cos⁡θ=89=223\cos \theta = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}

STEP 13

Given tan⁡θ=159\tan \theta = \frac{\sqrt{15}}{9}, find sin⁡θ\sin \theta and cos⁡θ\cos \theta.
Use the identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

STEP 14

Assume sin⁡θ=15c\sin \theta = \frac{\sqrt{15}}{c} and cos⁡θ=9c\cos \theta = \frac{9}{c} for some cc.
Use the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1:
(15c)2+(9c)2=1\left(\frac{\sqrt{15}}{c}\right)^2 + \left(\frac{9}{c}\right)^2 = 1

STEP 15

Solve for cc:
15c2+81c2=1\frac{15}{c^2} + \frac{81}{c^2} = 1
96c2=1⇒c2=96⇒c=96\frac{96}{c^2} = 1 \quad \Rightarrow \quad c^2 = 96 \quad \Rightarrow \quad c = \sqrt{96}

STEP 16

Find cos⁡θ\cos \theta:
cos⁡θ=996=946=368\cos \theta = \frac{9}{\sqrt{96}} = \frac{9}{4\sqrt{6}} = \frac{3\sqrt{6}}{8}

STEP 17

Given cot⁡θ=32\cot \theta = \frac{\sqrt{3}}{2}, find cos⁡θ\cos \theta.
Use the identity cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.

STEP 18

Assume cos⁡θ=3c\cos \theta = \frac{\sqrt{3}}{c} and sin⁡θ=2c\sin \theta = \frac{2}{c} for some cc.
Use the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1:
(3c)2+(2c)2=1\left(\frac{\sqrt{3}}{c}\right)^2 + \left(\frac{2}{c}\right)^2 = 1

STEP 19

Solve for cc:
3c2+4c2=1\frac{3}{c^2} + \frac{4}{c^2} = 1
7c2=1⇒c2=7⇒c=7\frac{7}{c^2} = 1 \quad \Rightarrow \quad c^2 = 7 \quad \Rightarrow \quad c = \sqrt{7}

STEP 20

Find cos⁡θ\cos \theta:
cos⁡θ=37=217\cos \theta = \frac{\sqrt{3}}{\sqrt{7}} = \frac{\sqrt{21}}{7}

STEP 21

Complete the trigonometric table for special angles using known values for 30∘30^\circ, 45∘45^\circ, and 60∘60^\circ.

STEP 22

Evaluate sin⁡60∘\sin 60^\circ using the known value:
sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}

STEP 23

Evaluate sin⁡30∘+sin⁡30∘\sin 30^\circ + \sin 30^\circ:
sin⁡30∘=12\sin 30^\circ = \frac{1}{2}
sin⁡30∘+sin⁡30∘=12+12=1\sin 30^\circ + \sin 30^\circ = \frac{1}{2} + \frac{1}{2} = 1
The solutions to the exercises are as follows:
25. csc⁡θ=72\csc \theta = \frac{\sqrt{7}}{2}
26. cot⁡θ=27217\cot \theta = \sqrt{\frac{272}{17}}
27. sin⁡θ=53\sin \theta = \frac{\sqrt{5}}{3}
28. cos⁡θ=223\cos \theta = \frac{2\sqrt{2}}{3}
29. cos⁡θ=368\cos \theta = \frac{3\sqrt{6}}{8}
30. cos⁡θ=217\cos \theta = \frac{\sqrt{21}}{7}
31. Complete the table using known trigonometric values.
32. a. sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}
b. sin⁡30∘+sin⁡30∘=1\sin 30^\circ + \sin 30^\circ = 1

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