Problem 50 - Precalculus: Functions and Graphs

Question

If \(\alpha\) and \(\beta\) are complementary angles, show that $$ \sin ^{2} \alpha+\sin ^{2} \beta=1 $$

Step-by-Step Solution

Answer

The angles are complementary, so \(\sin^2 \alpha + \sin^2 \beta = 1\) using trigonometric identities.

Step 1: Understanding Complementary Angles

Two angles are complementary if their sum is 90°. This means if the measure of one angle is \(\alpha\), then the other angle \(\beta\) is given by \(\beta = 90° - \alpha\).

Step 2: Using the Complementary Angle Identity

The identity \(\sin(90° - x) = \cos(x)\) can be applied to complementary angles. For angle \(\beta\), use: \(\sin(\beta) = \sin(90° - \alpha) = \cos(\alpha)\).

Step 3: Square and Sum the Sine Functions

We are tasked to prove \(\sin^2 \alpha + \sin^2 \beta = 1\). Using the identity from the previous step, \(\sin^2 \beta = \cos^2 \alpha\). Thus, the expression becomes \(\sin^2 \alpha + \cos^2 \alpha\).

Step 4: Apply the Pythagorean Identity

According to the Pythagorean Identity, \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since we have arrived at this identity, \(\sin^2 \alpha + \sin^2 \beta\) indeed equals 1.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving the trigonometric functions that are true for every value of the variables involved. These identities help us solve complex trigonometric problems more easily.
One of the key identities used in solving problems with complementary angles is the Complementary Angle Identity. This states that the sine of one angle is equal to the cosine of its complement. Expressed mathematically, it is written as follows:

• \( \sin(90° - x) = \cos(x) \)

This simple yet powerful relationship is useful when handling angles that sum to 90°. When tackling trigonometric problems, always look for opportunities to apply these identities, as they simplify expressions and lead to quicker solutions.

Sine and Cosine Functions

The sine and cosine functions are two of the most essential functions in trigonometry. They describe the relationships of the angles of a triangle to the lengths of its sides in the context of a right triangle. Understanding these

• The sine function (\( \sin(x) \)) relates a given angle to the ratio of the opposite side to the hypotenuse.
• The cosine function (\( \cos(x) \)) is the ratio of the adjacent side to the hypotenuse.

When working with complementary angles, the relationship between sine and cosine becomes especially important.
For complementary angles \( \alpha \) and \( \beta \), the important identity is that \( \sin(\beta) \) equals \( \cos(\alpha) \). This fundamentally connects these functions when dealing with right triangles and their angles.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstones of trigonometry. Derived from the Pythagorean theorem, it connects the sine and cosine functions together in a powerful relationship:

• \( \sin^2(x) + \cos^2(x) = 1 \)

This identity holds true for any angle \( x \) and arises from considering a right triangle where the sides squared add up to the hypotenuse squared. It illustrates how the basic geometric properties of triangles are reflected in trigonometry.
In the context of complementary angles like \( \alpha \) and \( \beta \), we see its strength. By expressing \( \sin^2(\beta) \) as \( \cos^2(\alpha) \), we ultimately demonstrate that \( \sin^2(\alpha) + \sin^2(\beta) = 1 \) through this very identity. Always count on the Pythagorean identity when you need to transition between squares of these functions.