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ACT Math : Sine

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Example Questions

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 + cos2Θ

1 + sin2Θ

None of the answers are correct

2sinΘcosΘ -1

cos2Θ -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:

$58^{\circ}$

$38^{\circ}$

None of the other answers

$32^{\circ}$

$52^{\circ}$

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

Theta2

What is the value of $\small \Theta$ in the right triangle above? Round to the nearest hundredth of a degree.

Possible Answers:

$21.80$^{\circ}$$

$23.58$^{\circ}$$

$20.11$^{\circ}$$

$66.42$^{\circ}$$

$27.84$^{\circ}$$

Correct answer:

$23.58$^{\circ}$$

Explanation:

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:

$\Theta = sin^-^1(\frac{30}{75})=23.57817847820183$ or $23.58$^{\circ}$$ .

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ACT Math : Sine

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

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