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Precalculus : Pre-Calculus

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Example Questions

Example Question #2 : Graphs And Inverses Of Trigonometric Functions

Trig_id

What is $\theta$ if o=10 and a=8 ?

Possible Answers:

$89.60^{\circ}$

$51.34^{\circ}$

$0.90^{\circ}$

$53.14^{\circ}$

$55.14^{\circ}$

Correct answer:

$51.34^{\circ}$

Explanation:

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

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

$\tan=\frac{10}{8}$

$\tan=1.25$

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

$\theta=\arctan1.25$

$\rightarrow 51.34^{\circ}$

If you calculated the angle's measure to be $0.90^{\circ}}$ then your calculator was set to radians and needs to be set on degrees.

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Example Question #3 : Graphs And Inverses Of Trigonometric Functions

Trig_id

If $\theta$ equals $55^{\circ}$ and is 15ft , how long is ?

Possible Answers:

Not enough information to solve

13.2ft

8.6ft

11.3ft

12.3ft

Correct answer:

12.3ft

Explanation:

This problem can be easily solved using trig identities. We are given the hypotenuse (15ft) and $\theta (55^{\circ})$ . We can then calculate side using the $\sin$ .

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

$\sin55^{\circ}=\frac{o}{15ft}$

Rearrange to solve for .

$o=\sin55^{\circ}*15ft$

$\dpi{100} \rightarrow 12.3ft$

If you calculated the side to equal 8.6ft then you utilized the $\cos$ function rather than the $\sin$ .

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Example Question #2 : Finding Sides

Triangle

What is the length of CB?

Possible Answers:

$\frac{AB}{\tan C}$

$\frac{AB}{\sin A}$

$\frac{AC}{\sin A}$

$\frac{\tan C}{AB}$

$\frac{\sin B}{AC}$

Correct answer:

$\frac{AB}{\tan C}$

Explanation:

$\tan X= \frac{OPPOSITE}{ADJACENT}$

$\tan C=\frac{AB}{CB}$

$CB= \frac{AB}{\tan C}$

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Example Question #1 : Law Of Sines

Rt_triangle_letters

In this figure, angle $a=30^\circ$ . If side Z =12 and Y=20 , what is the value of angle ?

Possible Answers:

Undefined

$90^\circ$

$24^\circ$

$0.83^\circ$

$56.44^\circ$

Correct answer:

$56.44^\circ$

Explanation:

For this problem, use the law of sines:

$\frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}=\frac{\text{side opposite }c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$\frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}$

$\frac{Z}{\sin(a)}=\frac{Y}{\sin{(b)}}$

$\frac{12}{\sin(30^\circ)}=\frac{20}{\sin{(b)}}$

$24=\frac{20}{\sin(b)}$

$\sin(b)=\frac{20}{24}$

$b=\sin^{-1}(\frac{20}{24})$

$b=56.44^\circ$

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Example Question #2 : Graphing The Sine And Cosine Functions

Rt_triangle_letters In this figure, if angle $a=18.5^\circ$ , side Z =30.2 , and side Y=17.2 , what is the value of angle ?

(NOTE: Figure not necessarily drawn to scale.)

Possible Answers:

$71.5^\circ$

$10.41^\circ$

$33.6^\circ$

$61.22^\circ$

Undefined

Correct answer:

$10.41^\circ$

Explanation:

First, observe that this figure is clearly not drawn to scale. Now, we can solve using the law of sines:

$\frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}=\frac{\text{side opposite }c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$\frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}$

$\frac{Z}{\sin(a)}=\frac{Y}{\sin{(b)}}$

$\frac{30.2}{\sin(18.5^\circ)}=\frac{17.2}{\sin{(b)}}$

$95.18=\frac{17.2}{\sin(b)}$

$\sin(b)=\frac{17.2}{95.18}$

$b=\sin^{-1}(\frac{17.2}{95.18})$

$b=10.41^\circ$

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Example Question #4 : Graphs And Inverses Of Trigonometric Functions

Rt_triangle_letters

In this figure, if angle $c=90^\circ$ , side X=12 , and side Z=8 , what is the measure of angle ?

Possible Answers:

Undefined

$41.8^\circ$

$90^\circ$

$\frac{2}{3}^\circ$

$48.2^\circ$

Correct answer:

$41.8^\circ$

Explanation:

Since $c=90^\circ$ , we know we are working with a right triangle.

That means that $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$ .

In this problem, that would be:

$\sin(a)=\frac{\text{Z}}{\text{X}}$

Plug in our given values:

$\sin(a)=\frac{8}{12}$

$\sin(a)=\frac{2}{3}$

$a=\sin^{-1}(\frac{2}{3})$

$a=41.8^\circ$

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Example Question #2 : Graphing The Sine And Cosine Functions

Rt_triangle_letters

In this figure, X=12 , $Y=6\sqrt{3}$ , and Z=6 . What is the value of angle ?

Possible Answers:

$60^\circ$

$30^\circ$

$90^\circ$

Undefined

$45^\circ$

Correct answer:

$30^\circ$

Explanation:

Notice that these sides fit the pattern of a 30:60:90 right triangle: $x:x\sqrt{3}:2x$ .

In this case, x=6 .

Since angle is opposite , it must be $30^\circ$ .

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Example Question #3 : Graphing The Sine And Cosine Functions

A triangle has angles of $30^\circ:60^\circ:90^\circ$ . If the side opposite the $30^\circ$ angle is , what is the length of the side opposite $60^\circ$ ?

Possible Answers:

$7\sqrt{2}$

$14\sqrt{3}$

$\frac{7\sqrt{2}}{2}$

$7\sqrt{3}$

Correct answer:

$7\sqrt{3}$

Explanation:

The pattern for $30^\circ:60^\circ:90^\circ$ is that the sides will be $x:x\sqrt{3}:2x$ .

If the side opposite $30^\circ$ is , then the side opposite $60^\circ$ will be $7\sqrt{3}$ .

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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 #11 : Graphs And Inverses Of Trigonometric Functions

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

Possible Answers:

112.4

47.9

26.8

41.1

A triangle with these characteristics cannot exist.

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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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #2 : Graphs And Inverses Of Trigonometric Functions

Example Question #3 : Graphs And Inverses Of Trigonometric Functions

Example Question #2 : Finding Sides

Example Question #1 : Law Of Sines

Example Question #2 : Graphing The Sine And Cosine Functions

Example Question #4 : Graphs And Inverses Of Trigonometric Functions

Example Question #2 : Graphing The Sine And Cosine Functions

Example Question #3 : Graphing The Sine And Cosine Functions

Example Question #2 : Applying The Law Of Cosines

Example Question #11 : Graphs And Inverses Of Trigonometric Functions

All Precalculus Resources

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