# To prove : sin ( − θ ) = − sin ( θ ) ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter C, Problem 19E

(a)

To determine

## To prove : sin(−θ)=−sin(θ)

Expert Solution

### Explanation of Solution

Given information :

sin(θ)=sin(θ)

Concept used :

There are 4 quadrants in all and we know that in 1st quadrant , all 6 trigonometric functions i.esinθ,cosθ,secθ,cscθ,tanθ and cotθ take positive values.

In 2nd quadrant , only sinθ and csc functions take positive values and other left 4 functions take negative values .

Similarly, in 3rd quadrant , tanθ and cotθ functions take positive values and left 4 functions take negative values.

And in 4th quadrant , cosθ and secθ functions will take positive values and left ones will take negative values .

Proof:

sin(θ) can be written as sin(0θ)

Now, referring to 4 quadrants we have in geometry, we will see that (0θ) lies in 4th quadrant

And since only cos and sec functions take positive values in 4th quadrant

sin(0θ) or sin(θ) comes out as sin(θ) because sin function takes negative value in 4th quadrant.

Hence, proved

(b)

To determine

Expert Solution

### Explanation of Solution

Given information :

cos(θ)=cos(θ)

Concept used :

There are 4 quadrants in all and we know that in 1st quadrant , all 6 trigonometric functions i.esinθ,cosθ,secθ,cscθ,tanθ and cotθ take positive values.

In 2nd quadrant , only sinθ and csc functions take positive values and other left 4 functions take negative values .

Similarly, in 3rd quadrant , tanθ and cotθ functions take positive values and left 4 functions take negative values.

And in 4th quadrant , cosθ and secθ functions will take positive values and left ones will take negative values .

Proof:

cos(θ) can be written as cos(0θ)

Now, referring to 4 quadrants we have in geometry, we will see that (0θ) lies in 4th quadrant

And since only cos and sec functions take positive values in 4th quadrant

cos(0θ) or cos(θ) comes out as cos(θ) because cos function takes positive value in 4th quadrant.

Hence, proved

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