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Precalculus : Graphing the Sine and Cosine Functions

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Example Questions

Example Question #1 : Graphs And Inverses Of Trigonometric Functions

Triangle

What is the $\sin C$ ?

Possible Answers:

$\frac{AB}{BC}$

$\frac{AC}{AB}$

$\frac{BC}{AB}$

$\frac{AB}{AC}$

$\frac{BC}{AC}$

Correct answer:

$\frac{AB}{AC}$

Explanation:

$\sin X = \frac{OPPOSITE}{HYPOTENUSE}$

$\sin C = \frac{AB}{AC}$

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Example Question #2 : Graphs And Inverses Of Trigonometric Functions

Triangle

In the right triangle above, which of the following expressions gives the length of y?

Possible Answers:

$x\sin\theta$

$x\cos\theta$

$\frac{\sin(x)}{\theta}$

$x\tan\theta$

$\cos(x\times \theta)$

Correct answer:

$x\cos\theta$

Explanation:

$\cos\theta$ is defined as the ratio of the adjacent side to the hypotenuse, or in this case $\frac{y}{x}$ . Solving for y gives the correct expression.

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Example Question #1 : Arcsin, Arccos, Arctan

Trig_id

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

Possible Answers:

$53.14^{\circ}$

$89.60^{\circ}$

$0.90^{\circ}$

$55.14^{\circ}$

$51.34^{\circ}$

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 #1 : Graphs And Inverses Of Trigonometric Functions

Trig_id

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

Possible Answers:

12.3ft

13.2ft

8.6ft

Not enough information to solve

11.3ft

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 : Graphs And Inverses Of Trigonometric Functions

Triangle

What is the length of CB?

Possible Answers:

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

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

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

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

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

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 #3 : Graphs And Inverses Of Trigonometric Functions

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:

$0.83^\circ$

$24^\circ$

Undefined

$90^\circ$

$56.44^\circ$

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 #1 : 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:

$61.22^\circ$

$10.41^\circ$

$33.6^\circ$

$71.5^\circ$

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 #2 : Graphing The Sine And Cosine 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

$90^\circ$

$48.2^\circ$

$41.8^\circ$

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

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 #1 : Understanding 30 60 90 Triangles

Rt_triangle_letters

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

Possible Answers:

Undefined

$45^\circ$

$90^\circ$

$30^\circ$

$60^\circ$

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 #2 : Understanding 30 60 90 Triangles

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:

$7\sqrt{2}$

$7\sqrt{3}$

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

$14\sqrt{3}$

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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Precalculus : Graphing the Sine and Cosine Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Graphs And Inverses Of Trigonometric Functions

Example Question #2 : Graphs And Inverses Of Trigonometric Functions

Example Question #1 : Arcsin, Arccos, Arctan

Example Question #1 : Graphs And Inverses Of Trigonometric Functions

Example Question #2 : Graphs And Inverses Of Trigonometric Functions

Example Question #3 : Graphs And Inverses Of Trigonometric Functions

Example Question #1 : Graphing The Sine And Cosine Functions

Example Question #2 : Graphing The Sine And Cosine Functions

Example Question #1 : Understanding 30 60 90 Triangles

Example Question #2 : Understanding 30 60 90 Triangles

All Precalculus Resources

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