Chapter 4.3, Problem 76E

Solution

a.

To identify: The why $\sin \left(\pi -t\right)=\sin t$

  

The $\sin \left(\pi -t\right)=\sin t$

Given information:

The given is t is an angle such that $\frac{\pi }{2}<t<\pi$

And P is a point on the terminal side of angle t

  

Explanation:

Consider the given information.

It is needed to show

  $\sin \left(\pi -t\right)=\sin t$

First evaluate $\sin t$

See the figure above point on the terminal side of angle t is $P\left(a,b\right)$

Since it is a unit circle so distance from P to origin is 1

So, reference triangle is in 1^ST quadrant with sides length 1 a and b

So $x=a,y=b,r=1$

See that

  $\begin{array}{l}\sin t=\frac{y}{r}\\ \sin t=\frac{b}{1}\mathrm{...}\left(1\right)\\ \sin t=b\end{array}$

Now evaluate $\sin \left(\pi -t\right)$

For this determine reference triangle

See the figure above point on the terminal side of angle t is $Q\left(-b,a\right)$

Since it is a unit circle so distance from Q to origin is 1

So reference triangle is in 2 quadrant with sides length 1 a and b

  $x=-b,y=a,r=1$

See that

  $\begin{array}{l}\sin \left(\pi -t\right)=\frac{y}{r}\\ \sin \left(\pi -t\right)=\frac{b}{1}\\ \sin \left(\pi -t\right)=b\end{array}$

Hence equation (1)

  $\sin \left(\pi -t\right)=\sin t$

.

b.

To identify: The why $\cos \left(\pi -t\right)=-\cos t$

  

The $\cos \left(\pi -t\right)=-\cos t$

Given information:

The given is t is an angle such that $\frac{\pi }{2}<t<\pi$

And P is a point on the terminal side of angle t

  

Explanation:

Consider the given information.

It is needed to show

  $\cos \left(\pi -t\right)=-\cos t$

First evaluate $\cos t$

See the figure above point on the terminal side of angle t is $P\left(a,b\right)$

Since it is a unit circle so distance from P to origin is 1

So, reference triangle is in 1^ST quadrant with sides length 1 a and b

So $x=a,y=b,r=1$

See that

  $\begin{array}{l}\cos t=\frac{x}{r}\\ \cos t=\frac{a}{1}\mathrm{...}\left(2\right)\\ \cos t=a\end{array}$

Now evaluate $\cos \left(\pi -t\right)$

For this determine reference triangle

See the figure above point on the terminal side of angle t is $Q\left(-b,a\right)$

Since it is a unit circle so distance from Q to origin is 1

So reference triangle is in 2 quadrant with sides length 1 a and b

  $x=-a,y=b,r=1$

See that

  $\begin{array}{l}\cos \left(\pi -t\right)=\frac{x}{r}\\ \cos \left(\pi -t\right)=\frac{-a}{1}\\ \cos \left(\pi -t\right)=-a\end{array}$

Hence equation (a)

  $\cos \left(\pi -t\right)=-\cos t$