Problem 63 - Precalculus Essentials

Question

What does it mean to solve a right triangle?

Step-by-Step Solution

Answer

To solve a right triangle means to find the lengths of all of its sides and the measures of all its angles using the Pythagorean theorem and trigonometric functions.

Step 1: Understanding the triangle properties

A right triangle has one angle exactly equal to 90 degrees. The side opposite this angle is known as the hypotenuse. The other two sides are known as the 'base' and the 'perpendicular', based on which angle we are concerned with. For any angle in the triangle other than the right angle, the 'base' is the side adjacent to it and the 'perpendicular' is the side opposite to it.

Step 2: Applying the Pythagorean theorem

If two sides of the triangle are known, the remaining side can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse (h) is equal to the sum of the squares of the lengths of the other two sides (b and p). Mathematically, this can be written as: \[ h^2 = b^2 + p^2 \].

Step 3: Using trigonometric functions

If an angle and a side are given, the remaining sides can be found using sine, cosine or tangent functions. For any angle A in the triangle, sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, cosine is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and tangent is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, these can be written as: \[ \sin(A) = p/h, \cos(A) = b/h, \tan(A) = p/b \]. To find the unknown sides, rearrange these formulae.

Step 4: Finding the Unknown Angles

If two sides are known, the unknown angle can be found using inverse trigonometric functions (also known as arcsin, arccos, and arctan). For example, if p and h are known, then angle A can be found as: \[ A = \arcsin(p/h) \]. Similarly, if b and h are known, then angle A can be found as: \[ A = \arccos(b/h) \]. If p and b are known, then angle A can be found as: \[ A = \arctan(p/b) \].