Angles - Math

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Question

What angle is supplementary to $131^\circ$ ?

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Answer

Supplementary angles add up to $180^\circ$ . That means:

$x+131^\circ=180^\circ$

$x=180^\circ-131^\circ$

$x=49^\circ$

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Question

Solve for and .

(Figure not drawn to scale).

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Answer

The angles containing the variable all reside along one line, therefore, their sum must be .

$\small 5x+4x+3x=180^o$

$\small 12x=180^o$

$\small x=15^o$

Because and are opposite angles, they must be equal.

$\small 2y=5x$

$\small x=15^o$

$\small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$

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Question

Which one of these is positive in quadrant III?

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Answer

The pattern for positive functions is All Student Take Calculus. In quandrant I, all trigonometric functions are positive. In quadrant II, sine is positive. In qudrant III, tangent is positive. In quadrant IV, cosine is positive.

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Question

Solve for .

(Figure not drawn to scale).

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Answer

The angles are supplementary, therefore, the sum of the angles must equal .

$\small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$

$\small 12n=168^o$

$\small n=14^o$

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Question

Are $129^{\circ}$ and $51^{\circ}$ supplementary angles?

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Answer

Since supplementary angles must add up to $180^{\circ}$ , the given angles are indeed supplementary.

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Question

Are $73^{\circ}$ and $17^{\circ}$ complementary angles?

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Answer

Complementary angles add up to $90^{\circ}$ . Therefore, these angles are complementary.

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Question

Which of the following angles is supplementary to $15^\circ$ ?

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Answer

When two angles are supplementary, they add up to $180^\circ$ .

For this problem, we can set up an equation and solve for the supplementary angle:

$x^\circ+15^\circ=180^\circ$

$x^\circ=180^\circ-15^\circ$

$x^\circ=165^\circ$

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Question

What angle is complementary to $34^\circ$ ?

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Answer

Two complementary angles add up to $90^\circ$ .

Therefore, $34^\circ+x^\circ=90^\circ$ .

$x^\circ=90^\circ-34^\circ$

$x^\circ=56^\circ$

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Question

Which of the following angles is complementary to $32^\circ$ ?

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Answer

Two complementary angles add up to $90^\circ$ .

$90^\circ=32^\circ+x^\circ$

$90^\circ-32^\circ=x^\circ$

$58^\circ=x^\circ$

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Question

What angle is supplementary to $142^\circ$ ?

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Answer

When two angles are supplementary, they add up to $180^\circ$ .

$142^\circ+x=180^\circ$

Solve for :

$142^\circ+x=180^\circ$

$x=180^\circ-142^\circ$

$x=38^\circ$

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Question

Find a coterminal angle for $-55^{\circ}$ .

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Answer

Coterminal angles are angles that, when drawn in the standard position, share a terminal side. You can find these angles by adding or subtracting 360 to the given angle. Thus, the only angle measurement that works from the answers given is $305^{\circ}$ .

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Question

Which of the following angles is coterminal with $\frac{2\pi}{7}$ ?

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Answer

For an angle to be coterminal with $\theta$ , that angle must be of the form $\theta + 2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all five choices.

$\frac{9\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{9\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{7\pi}{7} \right ) =\left ( \frac{1}{2\pi } \right ) \pi = \frac{1}{2 }$

$\frac{12\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{12\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{10\pi}{7} \right ) =- \frac{5}{7}$

$-\frac{2\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{2\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{2\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{4\pi}{7} \right ) = \frac{2}{7}$

$-\frac{9\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{9\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{9\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{11\pi}{7} \right ) = \frac{11}{14 }$

$-\frac{12\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{12\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{12\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{14\pi}{7} \right ) = \left ( \frac{1}{2\pi } \right ) \cdot2\pi = 1$

$-\frac{12\pi}{7}$ is the correct choice, since only that choice passes our test.

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Question

Which of the following angles is coterminal with $\frac{5\pi}{9}$ ?

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Answer

For an angle to be coterminal with $\theta$ , that angle must be of the form $\theta + 2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all four choices.

$\frac{41\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (\frac{41\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{36\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )4 \pi =2$

$\frac{23\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (\frac{23\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{18\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )2 \pi =1$

$-\frac{13\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (-\frac{13\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{-18\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )\left (- 2 \pi \right )=-1$

$-\frac{31\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (-\frac{31\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{-36\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )\left (- 4 \pi \right )=-2$

All four choices pass the test, so all four angles are coterminal with $\frac{5\pi}{9}$ .

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Question

Which of the following choices represents a pair of coterminal angles?

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Answer

For two angles to be coterminal, they must differ by $2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all five choices.

$\frac{17\pi }{2} \textrm{ and } \frac{23\pi }{2}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{2} - \frac{17\pi }{2} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{2} = \frac{3 }{2}$

$\frac{17\pi }{3} \textrm{ and } \frac{23\pi }{3}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{3} - \frac{17\pi }{3} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{3} = 1$

$\frac{17\pi }{4} \textrm{ and } \frac{23\pi }{4}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{4} - \frac{17\pi }{4} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{4} = \frac{3 }{4}$

$\frac{17\pi }{5} \textrm{ and } \frac{23\pi }{5}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{5} - \frac{17\pi }{5} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{5} = \frac{3 }{5}$

$\frac{17\pi }{6} \textrm{ and } \frac{23\pi }{6}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{6} - \frac{17\pi }{6} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{6} = \frac{1 }{2}$

The only angles that pass the test - and are therefore coterminal - are $\frac{17\pi }{3} \textrm{ and } \frac{23\pi }{3}$ .

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Question

$\textup{Which of the following angle measures, given in radians, is coterminal with an}$

$\textup{an angle of 90 degrees?}$

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Answer

$90\textup{ degrees }=\frac{\pi }{2}:: \textup{ Coterminal angles share the same position on the unit circle.}$

$\textup{Therefore, coterminal angles can be found by adding or subtracting multiples of }2\pi$ .

$\frac{\pi }{2}+2\pi=\frac{5\pi }{2}$

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Question

What angle is supplementary to $131^\circ$ ?

Tap to reveal answer

Answer

Supplementary angles add up to $180^\circ$ . That means:

$x+131^\circ=180^\circ$

$x=180^\circ-131^\circ$

$x=49^\circ$

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Question

Solve for and .

(Figure not drawn to scale).

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Answer

The angles containing the variable all reside along one line, therefore, their sum must be .

$\small 5x+4x+3x=180^o$

$\small 12x=180^o$

$\small x=15^o$

Because and are opposite angles, they must be equal.

$\small 2y=5x$

$\small x=15^o$

$\small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$

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Question

Which one of these is positive in quadrant III?

Tap to reveal answer

Answer

The pattern for positive functions is All Student Take Calculus. In quandrant I, all trigonometric functions are positive. In quadrant II, sine is positive. In qudrant III, tangent is positive. In quadrant IV, cosine is positive.

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Question

Solve for .

(Figure not drawn to scale).

Tap to reveal answer

Answer

The angles are supplementary, therefore, the sum of the angles must equal .

$\small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$

$\small 12n=168^o$

$\small n=14^o$

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Question

Are $129^{\circ}$ and $51^{\circ}$ supplementary angles?

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Answer

Since supplementary angles must add up to $180^{\circ}$ , the given angles are indeed supplementary.

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