Chapter 8, Problem 31RE

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

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

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

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Evaluating trigonometric Functions In Exercises 25–32, evaluate the six trigonometric functions of the angle without using a calculator. − 5 π 6

To determine

To calculate: The value of six trigonometric functions sinθ, cosθ, tanθ, cotθ, secθ and cscθ at θ=5π6 without using a calculator.

Explanation

Given Information:

The provided angle is 5π6.

Formula used:

The reciprocal trigonometric identities:

cscθ=1sinθ,

secθ=1cosθ,

And,

cotθ=1tanθ

The trigonometric value for some common angles:

 θ(degrees) 0° 30° 45° 60° 90° 180° 270° θ(radians) 0 π6 π4 π3 π2 π 3π2 sinθ 0 12 22 32 1 0 −1 cosθ 1 32 22 12 0 −1 0 tanθ 0 33 1 3 Undefined 0 Undefined

Calculation:

Consider the provided angle.

θ=5π6

Here, negative sign shows that measured angle is in clockwise direction. Therefore, measured angle in counter clockwise direction can be calculated as:

θ=2π+θ=2π5π6=12π5π6=7π6

Draw the diagram for reference angle.

First, find the reference angle for θ=7π6.

Reference angle of 7π6=π7π6=6π7π6=π6

Since, angle θ=7π6 is in III quadrant. Therefore, tangent and Cotangent are positive, and remaining trigonometric functions are negative. So, write the six trigonometric functions as:

sin(5π6)=sinπ6,cos(5π6)=cosπ6,tan(5π6)=tanπ6,csc(5π6)=cscπ6

Remaining trigonometric functions as:

sec(5π6)=secπ6,cot(5π6)=cotπ6

So, evaluate the value of six trigonometric functions for θ=5π6 value of the trigonometric identities for common angles

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