6 ## SOLVING RIGHT TRIANGLESThis is a topic in traditional trigonometry. It does not come up in calculus. To SOLVE A TRIANGLE means to know all three sides and all three angles. When we know the ratios of the sides, we use the method of similar figures. That is the method to use when solving an isosceles right triangle or a 30°-60°-90° triangle. When we do not know the ratio numbers, then we must use the Table of ratios. The following example illustrates how. The general method
Example 1. Given an acute angle and one side. Solve the right triangle ABC if angle A is 36°, and side
To find an unknown side, say
Problem 1. Solve the triangle for side To see the answer, pass your mouse over the colored area.
Problem 2. To measure the width of a river. Two trees stand opposite one another, at points A and B, on opposite banks of a river. Distance AC along one bank is perpendicular to BA, and is measured to be 100 feet. Angle ACB is measured to be 79°. How far apart are the trees; that is, what is the width
from the Table. (To measure the height of a flagpole, and for the meaning of the angle of elevation, see the Example in Topic 3.) Example 2. Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°.
Now, the triangle formed by the lighthouse and the distance If
Therefore,
Example 3. Given two sides of a right triangle. Solve the right triangle ABC given that side
Next, to find angle A, we have
(See Skill in Arithmetic: Fractions into decimals.) We must now inspect the Table to find the angle whose cosine is closest to We find cos 16° = Therefore, Angle A 16°. Finally, Angle B = 90° − 16° = 74°. We have solved the triangle.
Problem 3. Solve the right triangle ABC given that
To find the remaining side
To find angle A, we have
Now inspect the Table to find the angle whose cosine is closest to Find cos 37° = Therefore, Angle A37°. Angle B = 90° − 37° = 53°. Next Topic: The Law of Cosines Copyright © 2017 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |