QuestionIn a right triangle, if $a=2$ (opposite angle $A$ ) and $b=7$ (hypotenuse), find $\sin(A)$ , $\cos(A)$ , $\tan(A)$ , $\sec(A)$ , $\csc(A)$ , $\cot(A)$ .

Studdy Solution

STEP 1

Assumptions1. The triangle is a right triangle. $a$ is the length of the side opposite angle $A$
3. $b$ is the length of the hypotenuse4. $a=$
5. $b=7$

STEP 2

First, we need to find the length of the side adjacent to angle $A$ . We can use the Pythagorean theorem for this, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Let's denote the length of the side adjacent to angle $A$ as $c$ .
$c^2 = b^2 - a^2$

STEP 3

Now, plug in the given values for $a$ and $b$ to calculate $c^2$ .
$c^2 =7^2 -2^2$

STEP 4

Calculate $c^2$ .
$c^2 =49 -4 =45$

STEP 5

Take the square root of both sides to solve for $c$ .
$c = \sqrt{45}$

STEP 6

implify the square root.
$c =3\sqrt{5}$

STEP 7

Now that we have all the side lengths, we can calculate the trigonometric functions. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
$\sin(A) = \frac{a}{b}$

STEP 8

Plug in the values for $a$ and $b$ to calculate $\sin(A)$ .
$\sin(A) = \frac{2}{7}$

STEP 9

The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
$\cos(A) = \frac{c}{b}$

STEP 10

Plug in the values for $c$ and $b$ to calculate $\cos(A)$ .
$\cos(A) = \frac{3\sqrt{5}}{7}$

STEP 11

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
$\tan(A) = \frac{a}{c}$

STEP 12

Plug in the values for $a$ and $c$ to calculate $\tan(A)$ .
$\tan(A) = \frac{2}{\sqrt{5}}$

STEP 13

implify $\tan(A)$ by rationalizing the denominator.
$\tan(A) = \frac{2\sqrt{5}}{15}$

STEP 14

The secant of an angle in a right triangle is defined as the reciprocal of the cosine of the angle.
$\sec(A) = \frac{}{\cos(A)}$

STEP 15

Plug in the value for $\cos(A)$ to calculate $\sec(A)$ .
$\sec(A) = \frac{}{\frac{3\sqrt{5}}{7}}$

STEP 16

implify $\sec(A)$ by multiplying the numerator and denominator by $$.
$\sec(A) = \frac{}{3\sqrt{5}}$

STEP 17

Rationalize the denominator to simplify $\sec(A)$ .
$\sec(A) = \frac{7\sqrt{5}}{15}$

STEP 18

The cosecant of an angle in a right triangle is defined as the reciprocal of the sine of the angle.
$\csc(A) = \frac{}{\sin(A)}$

STEP 19

Plug in the value for $\sin(A)$ to calculate $\csc(A)$ .
$\csc(A) = \frac{1}{\frac{}{7}}$

STEP 20

implify $\csc(A)$ by multiplying the numerator and denominator by $7$ .
$\csc(A) = \frac{7}{}$

STEP 21

The cotangent of an angle in a right triangle is defined as the reciprocal of the tangent of the angle.
$\cot(A) = \frac{1}{\tan(A)}$

STEP 22

Plug in the value for $\tan(A)$ to calculate $\cot(A)$ .
$\cot(A) = \frac{1}{\frac{\sqrt{5}}{15}}$

STEP 23

implify $\cot(A)$ by multiplying the numerator and denominator by $15$ .
$\cot(A) = \frac{15}{\sqrt{5}}$

STEP 24

Rationalize the denominator to simplify $\cot(A)$ .
$\cot(A) = \frac{15\sqrt{}}{20}$

STEP 25

implify $\cot(A)$ by dividing the numerator and denominator by $5$ .
$\cot(A) = \frac{3\sqrt{5}}{4}$ The exact values of the trigonometric functions of angle $A$ are $\sin (A)=\frac{}{7}$ $\cos (A)=\frac{3\sqrt{5}}{7}$ $\tan (A)=\frac{\sqrt{5}}{15}$ $\sec (A)=\frac{7\sqrt{5}}{15}$ $\csc (A)=\frac{7}{}$ $\cot (A)=\frac{3\sqrt{5}}{4}$

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