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Precalculus : Graphing the Sine and Cosine Functions

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Example Questions

Example Question #53 : The Unit Circle And Radians

What is the reference angle for $\frac{16\pi}{3}$ ?

Possible Answers:

$\frac{4\pi}{3}$

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$\frac{3\pi}{4}$

Correct answer:

$\frac{4\pi}{3}$

Explanation:

To find the reference angle, subtract $2\pi$ (1 trip around the unit circle) from the given angle until you reach an angle which is less than $2\pi$ .

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Example Question #31 : Trigonometry

What is $\sin(\pi)$ ?

Possible Answers:

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

$\frac{1}{2}$

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

Correct answer:

Explanation:

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

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Example Question #21 : Graphs And Inverses Of Trigonometric Functions

What is $\cos(\pi)$ ?

Possible Answers:

$\frac{\sqrt2}{2}$

$\frac{\sqrt3}{2}$

Correct answer:

Explanation:

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

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Example Question #21 : Trigonometry

Which one of these is positive in quadrant III?

Possible Answers:

Cosine

Sine

All trig functions

Tangent

No trig functions

Correct answer:

Tangent

Explanation:

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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Example Question #1 : Understanding Coterminal Angles

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

Possible Answers:

$300^{\circ}$

$360^{\circ}$

$-15^{\circ}$

$85^{\circ}$

$305^{\circ}$

Correct answer:

$305^{\circ}$

Explanation:

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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Example Question #2 : Understanding Coterminal Angles

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

Possible Answers:

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

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

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

Each angle given in the other choices is coterminal with $\frac{5\pi}{9}$ .

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

Correct answer:

Each angle given in the other choices is coterminal with $\frac{5\pi}{9}$ .

Explanation:

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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Example Question #5 : Trigonometry

$\sin(x^\circ)=\frac{1}{2}$

What is ?

Possible Answers:

$60^\circ$

$1^\circ$

$30^\circ$

$0.009^\circ$

$26.57^\circ$

Correct answer:

$30^\circ$

Explanation:

To get rid of $\sin(x^\circ)$ , we take the $\arcsin$ or $\sin^{-1}$ of both sides.

$\sin(x^\circ)=\frac{1}{2}$

$\sin^{-1}(\sin(x^\circ))=\sin^{-1}(\frac{1}{2})$

$x^\circ=\sin^{-1}(\frac{1}{2})$

$x^\circ=30^\circ$

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Example Question #3 : Trigonometric Operations

Trig_id

$o=7.6, h=15.7, \Theta = ?$

Possible Answers:

$0.50^{\circ}$

$85.92^{\circ}$

$59.28^{\circ}$

$50.53^{\circ}$

$28.95^{\circ}$

Correct answer:

$28.95^{\circ}$

Explanation:

In order to find $\theta$ we need to utilize the given information in the problem. We are given the opposite and hypotenuse sides. We can then, by definition, find the $\sin$ of $\theta$ and its measure in degrees by utilizing the $\arcsin$ function.

$\sin=\frac{opposite}{hypotenuse}$

$\sin=\frac{7.6}{15.7}$

$\sin=0.4841$

Now to find the measure of the angle using the $\arcsin$ function.

$\theta=\arcsin0.4841$

$\rightarrow 28.95^{\circ}$

If you calculated the angle's measure to be $\dpi{100} 0.50^{\circ}}$ then your calculator was set to radians and needs to be set on degrees.

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Example Question #21 : Graphing The Sine And Cosine Functions

What is the amplitude of the following equation?

$f(x)=4\sin{3x}+2$

Possible Answers:

Correct answer:

Explanation:

Based on the generic form $a\sin{bx}+c$ , a is the amplitude. Thus, the amplitude is 4.

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Example Question #82 : Pre Calculus

What is one possible length of side if right triangle ABC has $\angle A = 30^{\circ}$ and side AC = 6 ?

(Hint: There are two possible answers, but only one of them is listed.)

Possible Answers:

$3\sqrt{2}$

$3\sqrt{3}$

Correct answer:

$3\sqrt{3}$

Explanation:

First we must set up our equation given the information.

$\cos 30 = \frac{AB}{6}$

$AB = 6\cos30$

$AB=6*\frac{\sqrt3}{2}$

$AB = 3\sqrt{3}$

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Precalculus : Graphing the Sine and Cosine Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #53 : The Unit Circle And Radians

Example Question #31 : Trigonometry

Example Question #21 : Graphs And Inverses Of Trigonometric Functions

Example Question #21 : Trigonometry

Example Question #1 : Understanding Coterminal Angles

Example Question #2 : Understanding Coterminal Angles

Example Question #5 : Trigonometry

Example Question #3 : Trigonometric Operations

Example Question #21 : Graphing The Sine And Cosine Functions

Example Question #82 : Pre Calculus

All Precalculus Resources

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