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ACT Math : How to find an angle with cosine

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Example Questions

Example Question #1 : Cosine

Soh_cah_toa

In the above triangle, a = 15 and h = 25 . Find $\theta$ .

Possible Answers:

$59.0^{\circ}$

$0.6^{\circ}$

$36.9^{\circ}$

$53.1^{\circ}$

$31.0^{\circ}$

Correct answer:

$53.1^{\circ}$

Explanation:

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. With this information, we can use the cosine function to find the angle.

$\cos = \frac{adjacent}{hypotenuse}$

$\cos\left ( \theta\right ) = \frac{15}{25} = 0.6$

$\arccos\left ( 0.6\right ) = \theta$

$53.1^{\circ}= \theta$

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

Soh_cah_toa

For the above triangle, a = 12 and h = 30 . Find $\theta$ .

Possible Answers:

$0.4^{\circ}$

$66.4^{\circ}$

$25.6^{\circ}$

$68.2^{\circ}$

$21.8^{\circ}$

Correct answer:

$66.4^{\circ}$

Explanation:

$\cos = \frac{adjacent}{hypotenuse}$

$\cos\left ( \theta\right ) = \frac{12}{30} = 0.4$

$\arccos\left ( 0.4\right ) = \theta$

$66.4^{\circ}= \theta$

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

Soh_cah_toa

For the above triangle, a = 17 and h = 13 . Find $\theta$ .

Possible Answers:

$37.4^{\circ}$

$49.9^{\circ}$

$40.1^{\circ}$

This triangle cannot exist.

$52.6^{\circ}$

Correct answer:

This triangle cannot exist.

Explanation:

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. However, if we plug the given values into the formula for cosine, we get:

$\cos = \frac{adjacent}{hypotenuse}$

$\cos\left ( \theta\right ) = \frac{17}{13} = 1.3$

$\arccos\left ( 1.3\right ) = \theta = \textup{no solution}$

This problem does not have a solution. The sides of a right triangle must be shorter than the hypotenuse. A triangle with a side longer than the hypotenuse cannot exist. Similarly, the domain of the arccos function is $\left [ -1, 1\right ]$ . It is not defined at 1.3.

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

A $50\textup{-foot}$ rope is thrown down from a building to the ground and tied up at a distance of $27\textup{ feet}$ from the base of the building. What is the angle measure between the rope and the ground? Round to the nearest hundredth of a degree.

Possible Answers:

$32.68$^{\circ}$$

$28.37$^{\circ}$$

$22.44$^{\circ}$$

$57.32$^{\circ}$$

$31.34$^{\circ}$$

Correct answer:

$57.32$^{\circ}$$

Explanation:

You can draw your scenario using the following right triangle:

Theta3

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

$\small \small \Theta = cos^-^1(\frac{27}{50})=57.31636115374206$ or $\small 57.32$ degrees.

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Example Question #1 : Cosine

Theta6

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

Possible Answers:

$70.38$^{\circ}$$

$18.56$^{\circ}$$

$68.49$^{\circ}$$

$71.44$^{\circ}$$

$19.62$^{\circ}$$

Correct answer:

$70.38$^{\circ}$$

Explanation:

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

$\small \Theta = cos^-^1(\frac{47}{140})=70.38402086459453$ or $70.38$^{\circ}$$ .

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

A support beam (buttress) lies against a building under construction. If the beam is feet long and strikes the building at a point feet up the wall, what angle does the beam strike the building at? Round to the nearest degree.

Possible Answers:

$16^{\circ}$

$23^{\circ}$

$18^{\circ}$

$22^{\circ}$

$7^{\circ}$

Correct answer:

$16^{\circ}$

Explanation:

Our answer lies in inverse functions. If the buttress is feet long and is feet up the ladder at the desired angle, then:

$\textup{cos }x^{\circ} = \frac{24}{25}$

Thus, using inverse functions we can say that $x = \textup{cos}^{-1}\frac{24}{25} \approx 16.26$

Thus, our buttress strikes the buliding at approximately a $16^{\circ}$ angle.

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

A stone monument stands as a tourist attraction. A tourist wants to catch the sun at just the right angle to "sit" on top of the pillar. The tourist lies down on the ground meters away from the monument, points the camera at the top of the monument, and the camera's display reads "DISTANCE -- METERS". To the nearest degree, what angle is the sun at relative to the horizon?

Possible Answers:

$18.13^{\circ}$

$48.19^{\circ}$

$78.44^{\circ}$

$23.20^{\circ}$

$62.82^{\circ}$

Correct answer:

$48.19^{\circ}$

Explanation:

Our answer lies in inverse functions. If the monument is meters away and the camera is meters from the monument's top at the desired angle, then:

$\textup{cos} x^{\circ} = \frac{8}{12}$

Thus, using inverse functions we can say that $x = \textup{cos}^{-1}\frac{8}{12} \approx 48.189$

Thus, our buttress strikes the buliding at approximately a $48.19^{\circ}$ angle.

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ACT Math : How to find an angle with cosine

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Cosine

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

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

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

Example Question #1 : Cosine

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

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

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