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Chapter 10: Problem 52

If the measure of one of the acute angles and the hypotenuse of a right triangle are known, describe how to find the measure of the remaining parts of the triangle.

Short Answer

Expert verified
To find the measure of the remaining parts of the triangle, use the sine of the known angle to find the length of the opposite side, the cosine to find the length of the adjacent side, and subtract the given angle from 90掳 to find the other angle.

Step by step solution

01

Identify Given Information

Identify the given information from the problem. Here, it is given that we know the measure of one acute angle and the length of the hypotenuse.
02

Use Sine Ratio

Utilize the properties of a right triangle and use the sine of the known angle to find the length of the opposite side. The sine of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. Rearrange the formula to solve for the length of the opposite side: \( \text{opposite side} = \text{hypotenuse} \times \sin(\text{angle}) \).
03

Use Cosine Ratio

Next, use the cosine of the angle to find the length of the adjacent side. The cosine of an angle in a right triangle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. Rearrange to solve for the length of the adjacent side: \( \text{adjacent side} = \text{hypotenuse} \times \cos(\text{angle}) \).
04

Find the measure of the remaining angle

Find the measure of the other acute angle of the triangle by subtracting the known angle from 90 degrees, because in a right triangle the two acute angles always add to 90 degrees: \( \text{remaining angle} = 90掳 - \text{known angle} \).

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Most popular questions from this chapter

The Great Pyramid outside Cairo, Egypt, has a square base measuring 756 feet on a side and a height of 480 feet. a. What is the volume of the Great Pyramid, in cubic yards? b. The stones used to build the Great Pyramid were limestone blocks with an average volume of \(1.5\) cubic yards. How many of these blocks were needed to construct the Great Pyramid?

Describe each type of angle: acute, right, obtuse, and straight.

In Exercises 15-20, find the measure of the complement and the supplement of each angle. \(52^{\circ}\)

Explain why the sine or cosine of an acute angle cannot be greater than or equal to 1 .

What general assumption did Euclid make about a point and a line in order to prove that the sum of the measures of the angles of a triangle is \(180^{\circ}\) ?
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