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Precalculus : Graphs and Inverses of Trigonometric Functions

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Example Questions

Example Question #2 : Applying The Law Of Cosines

A triangle has sides of length 12, 17, and 22. Of the measures of the three interior angles, which is the greatest of the three?

Possible Answers:

$86.4^{\circ }$

$130.1^{\circ }$

$97.2^{\circ }$

$119.2^{\circ }$

$93.6^{\circ }$

Correct answer:

$97.2^{\circ }$

Explanation:

We can apply the Law of Cosines to find the measure of this angle, which we will call :

$c^{2} = a^{2} + b^{2} -2ab \cos C$

The widest angle will be opposite the side of length 22, so we will set:

c=22 , a=12 , b=17

$22^{2} = 12^{2} + 17^{2} -2\cdot 12\cdot 17 \cos C$

$484 = 144 + 289 -408 \cos C$

$51= -408 \cos C$

$\cos C = -\frac{51}{408} = -0.125$

$C = \cos^{-1} 0.125 = 97.2^{\circ }$

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Example Question #1 : Triangles

In $\Delta ABC$ , $m \angle A = 66^{\circ }$ , AB = 26 , and AC = 23 . To the nearest tenth, what is ?

Possible Answers:

26.8

41.1

112.4

A triangle with these characteristics cannot exist.

47.9

Correct answer:

26.8

Explanation:

By the Law of Cosines:

$\left ( BC\right )^{2} = \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A$

or, equivalently,

$BC =\sqrt{ \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A}$

Substitute:

$BC =\sqrt{ 26^{2} +23^{2} - 2 \cdot 26 \cdot 23 \cdot \cos 66 ^{\circ }}$

$\approx \sqrt{ 676 +529 - 1,196 \cdot 0.4067}$

$\approx \sqrt{ 718.6} \approx 26.8$

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Example Question #12 : Graphs And Inverses Of Trigonometric Functions

What is the period of

$\tan \left ( \frac{x}{\pi } \right )$ ?

Possible Answers:

$\frac{1}{\pi }$

$2\pi$

$\pi^{2}$

$\pi$

Correct answer:

$\pi^{2}$

Explanation:

The period for $\tan(x)$ is $\pi$ . However, if a number is multiplied by , you divide the period $\pi$ by what is being multiplied by . Here, $(1/\pi )$ is being multiplied by . $\pi/(1/\pi )$ equals $\pi^{2}$ .

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Example Question #1 : Understanding Trigonometric Functions

Which of the following is not in the range of the function $y = \sin{x}$ ?

Possible Answers:

$\frac{2}{3}$

$-\frac{1}{2}$

$\frac{3}{4}$

$\frac{3}{2}$

Correct answer:

$\frac{3}{2}$

Explanation:

The range of the function $y= \sin x$ is all numbers between and (the sine wave never goes above or below this).

Of the choices given, $\frac{3}{2}$ is greater than and thus not in this range.

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Example Question #2 : Trigonometric Identities

Trig_id

What is the $\sin$ of $\theta$ ?

Possible Answers:

$\frac{o}{a}$

$\frac{h}{a}$

$\frac{h}{o}$

$\frac{o}{h}$

$\frac{a}{h}$

Correct answer:

$\frac{o}{h}$

Explanation:

When working with basic trigonometric identities, it's easiest to remember the mnemonic: SOHCAHTOA .

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

$\cos=\frac{adjacent}{hypotenuse}$ ,

$\tan=\frac{opposite}{adjacent}$

When one names the right triangle, the opposite side is opposite to the angle, the adjacent side is next to the angle, and the hypotenuse spans the two legs of the right angle.

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Example Question #3 : Trigonometric Identities

Trig_id

What is the $\cos$ of $\theta$ ?

Possible Answers:

$\frac{a}{h}$

$\frac{h}{a}$

$\frac{o}{a}$

$\frac{o}{h}$

$\frac{h}{o}$

Correct answer:

$\frac{a}{h}$

Explanation:

When working with basic trigonometric identities, it's easiest to remember the mnemonic: SOHCAHTOA .

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

$\cos=\frac{adjacent}{hypotenuse}$ ,

$\tan=\frac{opposite}{adjacent}$

When one names the right triangle, the opposite side is opposite to the angle, the adjacent side is next to the angle, and the hypotenuse spans the two legs of the right angle.

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Example Question #4 : Solving Trigonometric Equations

In a right triangle, if

$cos(x)=\frac{3}{5}$

and

$cos(y)=\frac{4}{5}$

then what does $\tan(x)$ equal?

Possible Answers:

$\frac{5}{4}$

$\frac{3}{4}$

$\frac{3}{5}$

$\frac{4}{3}$

Correct answer:

$\frac{4}{3}$

Explanation:

One can draw a right triangle with acute angles $\angle x$ and $\angle y$ . The side adjacent to $\angle y$ is 4, and the side adjacent to $\angle x$ is 3.

$tan(x)=\frac{opposite}{adjacent}$

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Example Question #41 : Understanding Radians And Conversions

Bob manages a pizza store. He bought a new machine that tracks how big his employees are cutting the pizza slices. The machine measures the average angle size of each slice of each pizza. Unfortunately, the angle is given as 0.7854 radians which Bob does not understand. Help Bob by converting the radian angle into degrees. In degrees, what is the size of the angle for an average pizza slice.

Possible Answers:

$45^{\circ}$

$60^{\circ}$

$30^{\circ}$

$75^{\circ}$

$25^{\circ}$

Correct answer:

$45^{\circ}$

Explanation:

To convert we use a common conversion amount. It may be easiest to remember the full circle example. In degrees, a full circle is $360^{\circ}$ around. In terms of radians, a full circle is $2\pi$ . So to get our answer

$0.7854\ \text{Radians} *\frac{360^{\circ}}{2\pi\ \text{Radians}}= 45^{\circ}$

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Example Question #1801 : High School Math

Convert $120^{\circ}$ into radians.

Possible Answers:

$\pi$

$2\pi$

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

$\frac{2\pi }{3}$

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

Correct answer:

$\frac{2\pi }{3}$

Explanation:

To convert from degrees to radians, one multiplies by $\frac{\pi }{180}$ .

$120\cdot \frac{\pi }{180}=\frac{2\pi}{3}$

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Example Question #51 : The Unit Circle And Radians

In the unit circle, what is the angle in radians that corresponds to the point (0, -1)?

Possible Answers:

$\pi$

$\frac{3\pi }{2}$

$\frac{\pi }{2}$

$2\pi$

Correct answer:

$\frac{3\pi }{2}$

Explanation:

On the unit circle, (0,-1) is the point that falls between the third and fourth quadrant. This corresponds to $\frac{3\pi }{2}$ .

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Precalculus : Graphs and Inverses of Trigonometric Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #2 : Applying The Law Of Cosines

Example Question #1 : Triangles

Example Question #12 : Graphs And Inverses Of Trigonometric Functions

Example Question #1 : Understanding Trigonometric Functions

Example Question #2 : Trigonometric Identities

Example Question #3 : Trigonometric Identities

Example Question #4 : Solving Trigonometric Equations

Example Question #41 : Understanding Radians And Conversions

Example Question #1801 : High School Math

Example Question #51 : The Unit Circle And Radians

All Precalculus Resources

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