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ACT Math : Sine

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Example Questions

Example Question #81 : Trigonometry

In a given right triangle $\Delta ABC$ , hypotenuse AC = 17 and $\angle A = 45^{\circ}$ . Using the definition of $\sin$ , find the length of leg . Round all calculations to the nearest hundredth.

Possible Answers:

12.02

14.40

12.04

18.52

23.03

Correct answer:

12.02

Explanation:

In right triangles, SOHCAHTOA tells us that $\sin A = \frac{\textup{opposite}}{\textup{hypotenuse}}$ , and we know that $\angle A = 45^{\circ}$ and hypotenuse AC = 17 . Therefore, a simple substitution and some algebra gives us our answer.

$\sin 45^{\circ} = \frac{CB}{17}$

$17\cdot \sin 45^{\circ}= CB$ Isolate the variable term.

Thus, 12.02= CB .

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Example Question #2991 : Act Math

Varisty_tutor_images_1

What is the sine of $\angle ACB$ ?

Possible Answers:

$60^{\circ}$

$\frac{12}{5}$

$\frac{5}{12}$

$\frac{12}{13}$

$\frac{5}{13}$

Correct answer:

$\frac{5}{13}$

Explanation:

Sine can be found using the SOH CAH TOA method. For sine we do $\frac{opposite}{hypotenuse}$ .

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

Triangle

See right triangle ABC. If the length AB is 8 and the length of BC is 6, what is the sine of angle A?

Possible Answers:

0.8

0.6

Correct answer:

0.6

Explanation:

Sine A = Opposite / Hypotenuse = BC / AC

To find AC, use Pythagorean Theorum

AB² + BC² = AC²

8² + 6² = AC²

64 + 36 = AC²

100 = AC²

AC = 10

Sine A = BC / AC = 6 / 10 = 0.6

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

Solve for $\sin^2{Q}+3\sin{Q}=-2$ over the interval $0\leq Q \leq 2\pi$

Possible Answers:

Q = π or 2π

Q = π or does not exist 2

Q = π or 3π 2 2

Q = 3π or does not exist 2

Correct answer:

Q = 3π or does not exist 2

Explanation:

Substitute x = sinQ and solve the new equation x² + 3x = –2 by factoring. Be sure to change variables back to Q. As a result, sinQ = –1 or sinQ = –2. This function is bounded between –1 and 1 so sinQ can never be –2 and sinQ is –1 only at 3π/2 or 270 °.

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Example Question #1 : 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:

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

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

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

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

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

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 #2 : 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:

$\frac{12}{13}$

$\frac{5}{13}$

$\frac{13}{5}$

$\frac{5}{12}$

$\frac{12}{5}$

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 #1 : 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 #5 : 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(x)

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

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

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

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

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

Sinx

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

Possible Answers:

0.77

18.28

9.41

15.84

10.72

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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ACT Math : Sine

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #81 : Trigonometry

Example Question #2991 : Act Math

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

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

Example Question #1 : How To Find The Sine Of An Angle

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

Example Question #1 : How To Find The Sine Of An Angle

Example Question #5 : How To Find The Sine Of An Angle

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

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

All ACT Math Resources

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