Chapter 7.1, Problem 5AYU

a.

To determine

To calculate: The exact value of $\tan \frac{\pi }{4}$ .

Answer to Problem 5AYU

The exact value of $\tan \frac{\pi }{4}$ is $1$ .

Explanation of Solution

Given information:

The trigonometric expression $\tan \frac{\pi }{4}$ .

Formula used:

The trigonometric expression for tangentis

  $\begin{array}{c}\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}\\ =\frac{y}{x}\end{array}$

Calculation:

Consider a point P on the unit circle $x^2+y^2=1$ that corresponds to $\theta =\frac{\pi }{4}$ is $P=\left(1,0\right)$ .

P must lie on the line that bisects quadrant I.

It follows that,

  $y=x$

  

And

  $\begin{array}{c}{x}^2+y^2=1\\ x^2+x^2=1\\ 2x^2=1\\ x^2=\frac{1}{2}\\ x=\frac{1}{\sqrt{2}}\\ x=\frac{\sqrt{2}}{2}\end{array}$

Then,

  $y=\frac{\sqrt{2}}{2}$

Therefore,

  $\begin{array}{c}\tan \theta =\frac{y}{x}\\ \tan \frac{\pi }{4}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\\ \tan \frac{\pi }{4}=1\end{array}$

Thus, the exact value of $\tan \frac{\pi }{4}$ is $1$ .

b.

To determine

To calculate: The exact value of $\sin \frac{\pi }{3}$ .

Answer to Problem 5AYU

The exact value of $\sin \frac{\pi }{3}$ is $\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given information:

The trigonometric expression $\sin \frac{\pi }{3}$ .

Formula used:

The trigonometric expression for sine is

  $\begin{array}{c}\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}\\ =\frac{y}{1}\end{array}$

Calculation:

Consider a point P on the unit circle $x^2+y^2=1$ that corresponds to $\theta =\frac{\pi }{3}$ is $P=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ .

With hypotenuse $=1$ .

It follows that,

  $y=\frac{\sqrt{3}}{2}$

  

And

  $\begin{array}{l}\sin \theta =\frac{y}{1}\\ \sin \frac{\pi }{3}=\frac{\frac{\sqrt{3}}{2}}{1}\\ \sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}\end{array}$

Thus, the exact value of $\sin \frac{\pi }{3}$ is $\frac{\sqrt{3}}{2}$ .