Chapter 4.4, Problem 106E

a.

To determine

Find $\sin t$ and $\cos t$ using the unit circle definitions of sine and cosine

Answer to Problem 106E

  $\boxed{\begin{array}{l}\sin t=y\\ \cos t=x\end{array}}$

Explanation of Solution

Given information:

The figure shows point $P\left(x,y\right)$ on a unit circle and right triangle $OAP$ .

  

Find $\sin t$ and $\cos t$ using the unit circle definitions of sine and cosine.

Calculation:

Let us consider the following figure Also let this is a unit circle with radius $r$ and a point $P$ on it.

  $P=\left(x,y\right)$

Hence, from the unit circle definitions of sine and cosine,

  $\boxed{\begin{array}{l}\sin t=y\\ \cos t=x\end{array}}$

b.

To determine

What is the value of $r$ ?

Answer to Problem 106E

  $\boxed{r=1}$

Explanation of Solution

Given information:

The figure shows point $P\left(x,y\right)$ on a unit circle and right triangle $OAP$ .

  

What is the value of $r$ ? Explain.

Calculation:

Let us consider the following figure Also let this is a unit circle with radius $r$ and a point $P$ on it.

  $P=\left(x,y\right)$

The value of $r$ is equal to $1$ unit as the circle is a unit circle with radius $1$ unit given by the equation,

  $x^2+y^2=1$

Hence, we get

  $\boxed{r=1}$

c.

To determine

Find $\text{sin}\theta$ and $\cos \theta$ . Write your answers in terms of $x$ and $y$ .

Answer to Problem 106E

  $\boxed{\begin{array}{l}\sin \theta =y\\ \cos \theta =x\end{array}}$

Explanation of Solution

Given information:

The figure shows point $P\left(x,y\right)$ on a unit circle and right triangle $OAP$ .

  

Use the definitions of sine and cosine given in this section to find $\text{sin}\theta$ and $\cos \theta$ . Write your answers in terms of $x$ and $y$ .

Calculation:

Let us consider the following figure Also let this is a unit circle with radius $r$ and a point $P$ on it.

  $P=\left(x,y\right)$

Using the definition of sine and cosine,

  $\sin \theta =\frac{y}{r}$

  $\cos \theta =\frac{x}{r}$

As, $r=1$ , thus we have,

  $\boxed{\begin{array}{l}\sin \theta =y\\ \cos \theta =x\end{array}}$

d.

To determine

what can you conclude from the part (a) and (c).

Answer to Problem 106E

  $\boxed{Equal}$

Explanation of Solution

Given information:

The figure shows point $P\left(x,y\right)$ on a unit circle and right triangle $OAP$ .

  

Based on your answers to parts (a) and (c), what can you conclude?

Calculation:

Let us consider the following figure Also let this is a unit circle with radius $r$ and a point $P$ on it.

  $P=\left(x,y\right)$

From the answers in part (a) and (b), we can conclude that the value of trigonometric functions from unit circle definitions is equal to the value of trigonometric functions from rectangular co-ordinate definitions.

Hence, the trigonometric functions from unit circle is equal to the trigonometric functions from rectangular co-ordinate.