Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

\( \sin^2 \alpha + \sin^2 \beta = 1 \) if \( \alpha + \beta = 90^\circ \).

Step 1: Understanding Complementary Angles

Two angles are complementary if their sum is 90 degrees. Therefore, we have \( \alpha + \beta = 90^\circ \). This is the key relationship we will use in solving the problem.

Step 2: Express \(\beta\) in Terms of \(\alpha\)

Since \( \beta = 90^\circ - \alpha \), we can substitute this expression for \( \beta \) in terms of \( \alpha \) into trigonometric identities to simplify our calculations.

Step 3: Applying Trigonometric Identity for Sine

We know from trigonometric identities that \( \sin(90^\circ - \alpha) = \cos \alpha \). Using this identity, we can express \( \sin \beta = \cos \alpha \).

Step 4: Calculate \( \sin^2 \alpha + \sin^2 \beta \)

Substitute \( \sin \beta = \cos \alpha \) into the expression \( \sin^2 \alpha + \sin^2 \beta \):\[ \sin^2 \alpha + \sin^2 \beta = \sin^2 \alpha + \cos^2 \alpha. \]

Step 5: Apply Pythagorean Identity

The Pythagorean Identity states that \( \sin^2 \alpha + \cos^2 \alpha = 1 \) for any angle \( \alpha \). Thus, substituting this in gives us:\[ \sin^2 \alpha + \sin^2 \beta = 1. \]

Step 6: Conclusion

Thus, we have shown that if \( \alpha \) and \( \beta \) are complementary angles, \( \sin^2 \alpha + \sin^2 \beta = 1 \).

Key Concepts

Complementary Angles

Complementary angles are an essential part of trigonometry and geometry. Two angles are complementary when their sum equals 90 degrees. This concept is critical because it allows us to find relationships between different trigonometric functions. In this exercise, we have \( \alpha \) and \( \beta \) as complementary angles, which means \( \alpha + \beta = 90^\circ \). By understanding this relationship, one can deduce other angle properties or find missing angle measurements. This "complementarity" aids in various mathematical solutions, ensuring that angles used together cover half of a right angle. This property becomes a powerful tool, especially when utilizing trigonometric identities, as it is the foundational concept for deriving other identities related to sine and cosine.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that expresses the relationship between the sine and cosine of an angle. It states that for any angle \( \alpha \): \[ \sin^2 \alpha + \cos^2 \alpha = 1. \]
This identity stems from the Pythagorean theorem, applied within the context of a unit circle, where the radius (or hypotenuse) is one unit. It has powerful implications in simplifying complex trigonometric expressions. In this case, it directly justifies why \( \sin^2 \alpha + \sin^2 \beta = 1 \). By recognizing that \( \sin \beta = \cos \alpha \), we transition seamlessly to \( \sin^2 \alpha + \cos^2 \alpha = 1 \), affirming the Pythagorean identity's veracity. Understanding this identity allows you to juggle between sine and cosine values effectively, particularly under constraints like complementary angles.

Sine and Cosine Relationship

The sine and cosine relationship is intrinsically tied to the nature of angles, particularly complementary angles. For a given angle \( \alpha \), if another angle \( \beta \) is its complement, then \( \beta = 90^\circ - \alpha \). In such cases, the sine and cosine of those angles reveal a unique feature: \( \sin(90^\circ - \alpha) = \cos \alpha \). This relationship is pivotal because it links the trigonometric functions of the angle and its complement seamlessly. By employing this identity, one can convert problems involving complementary angles into simpler expressions. Furthermore, it highlights that the sine of one angle is equivalent to the cosine of its complementary angle, offering flexibility and a way to harness identities like the Pythagorean identity more effectively. This inherent connection is a staple in simplifying various trigonometric calculations and proofs.