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High School Math : The Unit Circle and Radians

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Example Questions

Example Question #1 : The Unit Circle And Radians

What is $\sin(\frac{\pi}{2})$ $\displaystyle \sin(\frac{\pi}{2})$ ?

Possible Answers:

$\displaystyle -1$

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

$2\pi$ $\displaystyle 2\pi$

$\displaystyle 0$

$\displaystyle 1$

Correct answer:

$\displaystyle 1$

Explanation:

If you examine the unit circle, you'll see that the the $\sin(\frac{\pi}{2})=1$ $\displaystyle \sin(\frac{\pi}{2})=1$ . If you were to graph a sine function, you would also see that it crosses through the point $(\frac{\pi}{2},1)$ $\displaystyle (\frac{\pi}{2},1)$ .

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Example Question #2 : The Unit Circle And Radians

What is $\cos(\frac{2\pi}{3})$ $\displaystyle \cos(\frac{2\pi}{3})$ ?

Possible Answers:

$\displaystyle 0$

$-\frac{1}{2}$ $\displaystyle -\frac{1}{2}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 1$

$\displaystyle -1$

Correct answer:

$-\frac{1}{2}$ $\displaystyle -\frac{1}{2}$

Explanation:

If you look at the unit circle, you'll see that $\cos(\frac{2\pi}{3})=\frac{1}{2}$ $\displaystyle \cos(\frac{2\pi}{3})=\frac{1}{2}$ . You can also think of this as the cosine of $60^\circ$ $\displaystyle 60^\circ$ , which is also $\frac{1}{2}$ $\displaystyle \frac{1}{2}$ .

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Example Question #1 : Using The Unit Circle

What is $\sin(\frac{2\pi}{3})$ $\displaystyle \sin(\frac{2\pi}{3})$ ?

Possible Answers:

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle -1$

$\displaystyle 0$

$-\frac{\sqrt3}{2}$ $\displaystyle -\frac{\sqrt3}{2}$

$\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$

Correct answer:

$\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$

Explanation:

If you look at the unit circle, you'll see that $\sin(\frac{2\pi}{3})=\frac{\sqrt3}{2}$ $\displaystyle \sin(\frac{2\pi}{3})=\frac{\sqrt3}{2}$ . You can also think of this as the sine of $60^\circ$ $\displaystyle 60^\circ$ , which is also $\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$ .

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Example Question #3 : The Unit Circle And Radians

What is $\cos(\frac{\pi}{4})$ $\displaystyle \cos(\frac{\pi}{4})$ ?

Possible Answers:

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

$\frac{2\sqrt2}{2}$ $\displaystyle \frac{2\sqrt2}{2}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 1$

$\displaystyle 0$

Correct answer:

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

Explanation:

Using the unit circle, $\cos(\frac{\pi}{4})=\frac{\sqrt2}{2 }$ $\displaystyle \cos(\frac{\pi}{4})=\frac{\sqrt2}{2 }$ . You can also think of this as the cosine of $45^\circ$ $\displaystyle 45^\circ$ , which would also be $\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$ .

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Example Question #3 : The Unit Circle And Radians

What is $\cos(\frac{\pi}{2})$ $\displaystyle \cos(\frac{\pi}{2})$ ?

Possible Answers:

$\frac{1}{3}$ $\displaystyle \frac{1}{3}$

$\frac{\sqrt2}{}2$ $\displaystyle \frac{\sqrt2}{}2$

$\displaystyle 1$

$\displaystyle -1$

$\displaystyle 0$

Correct answer:

$\displaystyle 0$

Explanation:

Using the unit circle, you can see that $\cos(\frac{\pi}{2})=0$ $\displaystyle \cos(\frac{\pi}{2})=0$ . If you were to graph a cosine function, you would also see that it crosses through the point $(\frac{\pi}{2},0)$ $\displaystyle (\frac{\pi}{2},0)$ .

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Example Question #4 : The Unit Circle And Radians

What is $\sin(\frac{3\pi}{4})$ $\displaystyle \sin(\frac{3\pi}{4})$ ?

Possible Answers:

$\displaystyle 1$

$\frac{\sqrt{2}}{2}$ $\displaystyle \frac{\sqrt{2}}{2}$

$\displaystyle -1$

$\frac{-\sqrt2}{2}$ $\displaystyle \frac{-\sqrt2}{2}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

Correct answer:

$\frac{\sqrt{2}}{2}$ $\displaystyle \frac{\sqrt{2}}{2}$

Explanation:

Using the unit circle, $\sin(\frac{3\pi}{4})=\frac{\sqrt2}{2 }$ $\displaystyle \sin(\frac{3\pi}{4})=\frac{\sqrt2}{2 }$ . You can also think of this as the sine of $135^\circ$ $\displaystyle 135^\circ$ , which would also be $\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$ .

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Example Question #4 : The Unit Circle And Radians

What is $\cos (\frac{3\pi}{4})$ $\displaystyle \cos (\frac{3\pi}{4})$ ?

Possible Answers:

$\frac{2\sqrt2}{2}$ $\displaystyle \frac{2\sqrt2}{2}$

$-\frac{\sqrt2}{2}$ $\displaystyle -\frac{\sqrt2}{2}$

$\displaystyle 1$

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

$\displaystyle -1$

Correct answer:

$-\frac{\sqrt2}{2}$ $\displaystyle -\frac{\sqrt2}{2}$

Explanation:

Using the unit circle, you can see that the $\cos (\frac{3\pi}{4})=-\frac{\sqrt2}{2}$ $\displaystyle \cos (\frac{3\pi}{4})=-\frac{\sqrt2}{2}$ . Since the angle is in Qudrant II, sine is positive and cosine is negative.

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Example Question #2 : The Unit Circle And Radians

What is $\sin(\frac{\pi}{4})$ $\displaystyle \sin(\frac{\pi}{4})$ ?

Possible Answers:

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 1$

$-\frac{\sqrt2}{2}$ $\displaystyle -\frac{\sqrt2}{2}$

$\displaystyle -1$

Correct answer:

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

Explanation:

Using the unit circle, $\sin(\frac{\pi}{4})=\frac{\sqrt2}{2 }$ $\displaystyle \sin(\frac{\pi}{4})=\frac{\sqrt2}{2 }$ . You can also think of this as the sine of $45^\circ$ $\displaystyle 45^\circ$ , which would also be $\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$ .

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Example Question #21 : Graphing The Sine And Cosine Functions

What is $\sin(\pi)$ $\displaystyle \sin(\pi)$ ?

Possible Answers:

$\frac{\sqrt{2}}{2}$ $\displaystyle \frac{\sqrt{2}}{2}$

$\frac{\sqrt{3}}{2}$ $\displaystyle \frac{\sqrt{3}}{2}$

$\displaystyle 0$

$\displaystyle 1$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

Correct answer:

$\displaystyle 0$

Explanation:

If you examine the unit circle, you'll see that that $\sin(\pi)=0$ $\displaystyle \sin(\pi)=0$ .

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Example Question #22 : Graphing The Sine And Cosine Functions

What is $\cos(\pi)$ $\displaystyle \cos(\pi)$ ?

Possible Answers:

$\displaystyle 1$

$\displaystyle 0$

$\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$

$\displaystyle -1$

$\frac{\sqrt2}{2}$ $\displaystyle \frac{\sqrt2}{2}$

Correct answer:

$\displaystyle -1$

Explanation:

If you examine the unit circle, you'll see that the value of $\cos(\pi)=-1$ $\displaystyle \cos(\pi)=-1$ . You can also get this by examining a cosine graph and you'll see it crosses the point $(\pi,-1)$ $\displaystyle (\pi,-1)$ .

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High School Math : The Unit Circle and Radians

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : The Unit Circle And Radians

Example Question #2 : The Unit Circle And Radians

Example Question #1 : Using The Unit Circle

Example Question #3 : The Unit Circle And Radians

Example Question #3 : The Unit Circle And Radians

Example Question #4 : The Unit Circle And Radians

Example Question #4 : The Unit Circle And Radians

Example Question #2 : The Unit Circle And Radians

Example Question #21 : Graphing The Sine And Cosine Functions

Example Question #22 : Graphing The Sine And Cosine Functions

All High School Math Resources

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