   Chapter 8.2, Problem 7E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Evaluating Trigonometric Functions In Exercises 7–12, sketch a right triangle corresponding to the trigonometric function of the acute angle θ and find the values of the other five trigonometric functions of θ. sin   θ = 1 3

To determine

To calculate: The values of other five trigonometric functions of θ if sinθ=13.

Explanation

Given Information:

The value of trigonometric ratio is sinθ=13.

Formula used:

Circular function definition:

sinθ=yr         cscθ=rycosθ=xr         secθ=rxtanθ=yx         cotθ=xy

Where, x is the length of the side adjacent to θ, y is the length of the side opposite to θ, and r=x2+y2 is the length of hypotenuse which is not equal to 0.

Calculation:

From the provided trigonometric function, the value of y is 1 and r is 3. So, the value of x will be:

r=x2+y23=x2+123=x2+1

Take square both sides.

32=(x2+1)29=x2+1

Subtract one from both sides of the above equation.

91=x2+118=x2x2=8

Now, take square root both sides.

x2=8x=22

Thus, x=22.

Use the obtained value of x and the provided value of y and r to create a right triangle corresponding to θ:

Now, substitute the value of x, y, and r to evaluate the exact value of the remaining trigonometric functions using circular function definition.

Now, for the value of cscθ.

cscθ=ry=31=3

Now, for the value of cosθ

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