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Example Questions

Example Question #2 : How To Find The Sine Of An Angle

If $\small a=3$ , $\small b=4$ , and $\small c=5$ , what is the sine of $\small \angle A$ ?

Possible Answers:

$\small \frac{5}{3}$

$\small \frac{3}{5}$

$\small \frac{4}{5}$

$\small \frac{3}{4}$

$\small \frac{4}{3}$

Correct answer:

$\small \frac{3}{5}$

Explanation:

Recall that sin = opposite / hypotenuse. Based on the figure shown, we see that is the opposite side needed and is the hypotenuse. Plug these values in to solve.

$\small \sin \angle A = \frac{a}{c} = \frac{3}{5}$

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Example Question #3 : How To Find The Sine Of An Angle

Triangle

Triangle ABC shown is a right triangle. If line AB=12 and line BC=5 , what is the sine of the angle at ?

Possible Answers:

$\frac{5}{12}$

$\frac{13}{5}$

$\frac{12}{13}$

$\frac{12}{5}$

$\frac{5}{13}$

Correct answer:

$\frac{5}{13}$

Explanation:

$Sine(A)=\frac{Opp}{Hyp}=\frac{BC}{AC}=\frac{5}{AC}$

Now solve for using Pythagorean Theorem:

$AB^{2}+BC^{2}=AC^{2}$

$12^{2}+5^{2}=AC^{2}$

$144+25=AC^{2}$

$169=AC^{2}$

AC=13

$Sine(A)=\frac{5}{13}$

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Example Question #3 : How To Find The Sine Of An Angle

If $tan \theta = -5/12$ , and if $\theta$ is an angle between and 180 degrees, which of the following equals $sin \theta$ ?

Possible Answers:

12/13

5/13

-5/13

5/12

Correct answer:

5/13

Explanation:

An angle between and 180 degrees means that the angle is located in the second quadrant.

The tangent function is derived from taking the side opposite to the angle and dividing by the side adjacent to the angle ( y/x , as shown in the image).

Amsler grid

Hence, the side is units long and side is units high. Therefore, according to Pythagorean Theorem rules, the side must be units long (since 5^2 + 12^2 = 13^2 ).

The sine function is positive in the second quadrant. It is also equivalent to the side opposite the angle () divided by the hypotenuse ().

This makes $sin\theta = 5/13$ .

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Example Question #4 : How To Find The Sine Of An Angle

A sine function has a period of , a -intercept of , an amplitude of and no phase shift. These describe which of these equations?

Possible Answers:

$f(x) = 2sin(2\pi x)$

$f(x) = 2sin(\pi x) + 1$

$f(x) = sin(\pi x)$

$f(x) = 2sin(\pi x)$

f(x) = 2sin(x)

Correct answer:

$f(x) = 2sin(\pi x)$

Explanation:

Looking at this form of a sine function:

f(x) = asin(bx + c) + d

We can draw the following conclusions:

because the amplitude is specified as .
$b = \pi$ because of the specified period of since $\frac{2\pi}{\pi} = 2$ .
because the problem specifies there is no phase shift.
because the -intercept of a sine function with no phase shift is .

Bearing these in mind, $f(x) = 2sin(\pi x)$ is the only function that fits all four of those.

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Example Question #1 : How To Find The Sine Of A Missing Side

In the triangle below, AB = 2 units and AC = 6 units. What is $\sin C$ ?

Triangle

Possible Answers:

$3\sqrt10$

$\sqrt10$

$\frac{2}{\sqrt 10}$

$\frac{1}{\sqrt 10}$

$\frac{\sqrt 10}{2}$

Correct answer:

$\frac{1}{\sqrt 10}$

Explanation:

Because $\sin C =\frac{AB}{BC}$ , we need to first find the length of BC.

Using the Pythagorean theorem,

AB^2+AC^2=BC^2

$BC=\sqrt {AB^2+AC^2}$

$BC=\sqrt {6^2+2^2}$

$BC=\sqrt{36+4} = \sqrt {40} =\sqrt{4\cdot10}$

$BC=2\sqrt10$ .

$\sin C =\frac{AB}{BC}=\frac{2}{2\sqrt10}=\frac{1}{\sqrt10}$

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Example Question #3001 : Act Math

Sinx

What is the value of in the triangle above? Round to the nearest hundredth.

Possible Answers:

15.84

0.77

9.41

10.72

18.28

Correct answer:

10.72

Explanation:

We know that the sine of an angle is:

$\frac{opposite}{hypotenuse}$

Therefore, we can write for this question:

$sin(50)=\frac{x}{14}$

This allows us to solve for easily:

x=14sin(50)=10.72462220366569

Rounding, this gives us 10.72 .

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Example Question #1 : How To Find The Sine Of A Missing Side

A right triangle has leg lengths and . What is the sine of the angle opposite from the side of length ?

Possible Answers:

$\frac{4}{\sqrt{41}}{}$

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

$\frac{4}{5}$

$\frac{5}{\sqrt{41}}{}$

Correct answer:

$\frac{5}{\sqrt{41}}{}$

Explanation:

Using SOHCAHTOA, the sine of an angle is simply the length of the side opposite to it over the hypotenuse; however, we do not have the length of the hypotenuse yet.

Using the Pythagorean Theorem, we can solve for it:

$8^2+10^2=h^2{}$

$h=\sqrt{164}=2\sqrt{41}{}$

So, the sine of this angle is:

$\frac{10}{2\sqrt{41}} = \frac{5}{\sqrt{41}}{}$

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Example Question #1 : How To Find An Angle With Sine

Simplify: (sinΘ + cosΘ)²

Possible Answers:

1 + sin2Θ

None of the answers are correct

cos2Θ -1

1 + cos2Θ

2sinΘcosΘ -1

Correct answer:

1 + sin2Θ

Explanation:

Using the foil method, multiply. Simplify using the Pythagorean identity sin²Θ + cos²Θ = 1 and the double angle identity sin2Θ = 2sinΘcosΘ.

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Example Question #1 : How To Find An Angle With Sine

For the triangle Using_sin_to_find_angle , find $\angle A$ in degrees to the nearest integer

Note: The triangle is not necessarily to scale

Possible Answers:

None of the other answers

$38^{\circ}$

$58^{\circ}$

$52^{\circ}$

$32^{\circ}$

Correct answer:

$38^{\circ}$

Explanation:

To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. Looking at the problem statement, we are given the side opposite of the angle we are trying to find as well as the hypotenuse. Therefore, we will be using the SOH part of our mnemonic. Inserting our values, this becomes $SIN(\angle A) = \frac{8}{13}$ . Then, we can write $\angle A = SIN^{-1}\left ( \frac{8}{13} \right )$ . Solving this, we get $\angle A = 38^{\circ}$

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Example Question #2 : How To Find An Angle With Sine

A fifteen foot ladder is leaned up against a twelve foot building, reaching the top of the building. What is the angle made between the ladder and the ground? Round to the nearest hundredth of a degree.

Possible Answers:

$37.14$^{\circ}$$

$51.34$^{\circ}$$

$41.99$^{\circ}$$

$36.87$^{\circ}$$

$53.13$^{\circ}$$

Correct answer:

$53.13$^{\circ}$$

Explanation:

You can draw your scenario using the following right triangle:

Theta1

Recall that the sine of an angle is equal to the ratio of the opposite side to the hypotenuse of the triangle. You can solve for the angle by using an inverse sine function:

$\small \small \Theta = sin^-^1(\frac{12}{15})=53.13010235415598$ or $53.13$^{\circ}$$ .

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ACT Math : ACT Math

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : How To Find The Sine Of An Angle

Example Question #3 : How To Find The Sine Of An Angle

Example Question #3 : How To Find The Sine Of An Angle

Example Question #4 : How To Find The Sine Of An Angle

Example Question #1 : How To Find The Sine Of A Missing Side

Example Question #3001 : Act Math

Example Question #1 : How To Find The Sine Of A Missing Side

Example Question #1 : How To Find An Angle With Sine

Example Question #1 : How To Find An Angle With Sine

Example Question #2 : How To Find An Angle With Sine

All ACT Math Resources

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