BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter C, Problem 12E
To determine

To calculate:

The value of the exact trigonometric ratio of the given radian angle:

  4π3

Expert Solution

Answer to Problem 12E

The exact trigonometric ratio of the 4π3

  sin4π3=32cosec4π3=23cos4π3=12sec4π3=2tan4π3=cot4π3=3

Explanation of Solution

Given information:

  4π3

Formula Used:

Trigonometric ratios are given by the following formula:

  sinθ=yrcosecθ=rycosθ=xrsecθ=rxtanθ=yxcotθ=xy

Calculation:

The given value of radian is 4π3 . It is positive angle value that means the rotation is measured from the x axis as counter-clockwise direction.

Know that:

In one full rotation the angle is 2π . And the relation between the degree and radian is shown by below expression:

  πrad=180°1rad=180°π

Now, rewrite the given angle in the standard position:

  4π3=3π3+π34π3=π+π3

Thus, the given angle 4π3 measures angle half revolution plus π3 from the positive x axis.

This angle is drawn as below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter C, Problem 12E

Here, the triangle is fixed at terminal (x,y)=(1,3) and r=(1)2+(3)2=4=2

Therefore, trigonometric ratios are given as:

  sin4π3=yr=32cosec4π3=ry=23cos4π3=xr=12sec4π3=rx=21=2tan4π3=yx=31=3cot4π3=xy=13=3

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