Problem 47 - Algebra and Trigonometry

Question

Show that if \(\beta-\alpha=\pi / 2\) , then $$ \sin (x+\alpha)+\cos (x+\beta)=0 $$

Step-by-Step Solution

Answer

The expression is true when \( \beta - \alpha = \frac{\pi}{2} \), as shown by simplifying using trigonometric identities.

Step 1: Understand the Problem Statement

We are given that \( \beta - \alpha = \frac{\pi}{2} \) and we need to show that the expression \( \sin(x+\alpha) + \cos(x+\beta) = 0 \).

Step 2: Use Trigonometric Identities

We know that \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \). Let's rewrite \( \cos(x+\beta) \) using this identity. Let \( \gamma = x+\beta \), then \( \cos(x+\beta) = \sin(\frac{\pi}{2} - (x+\beta)) \).

Step 3: Substitute Values for Simplification

Since \( \beta - \alpha = \frac{\pi}{2} \), we have \( \alpha = \beta - \frac{\pi}{2} \). Substitute \( \alpha = \beta - \frac{\pi}{2} \) into the expression \( \sin(x+\alpha) \). \( \sin(x+\alpha) = \sin(x + \beta - \frac{\pi}{2}) \).

Step 4: Use Sine Complementary Angle Identity

Apply the identity \( \sin(A - \frac{\pi}{2}) = -\cos(A) \) to \( \sin(x + \beta - \frac{\pi}{2}) \), giving \( \sin(x + \beta - \frac{\pi}{2}) = -\cos(x+\beta) \).

Step 5: Simplify the Expression

Substitute \( \sin(x + \beta - \frac{\pi}{2}) = -\cos(x+\beta) \) into the original expression, replacing \( \sin(x+\alpha) \). The expression becomes \( -\cos(x+\beta) + \cos(x+\beta) = 0 \), which simplifies to \( 0 = 0 \).

Step 6: Conclusion

The expression \( \sin(x+\alpha) + \cos(x+\beta) = 0 \) holds true given the condition \( \beta - \alpha = \frac{\pi}{2} \). This validates our proof.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to \( \frac{\pi}{2} \) radians or 90 degrees. When you come across trigonometric expressions involving complementary angles, a useful relationship emerges.
- For example, if you have two angles, \( \alpha \) and \( \beta \), such that \( \beta - \alpha = \frac{\pi}{2} \), they are complementary.
- This specific relationship helps in transforming trigonometric expressions through identities that switch angles from sine to cosine and vice versa, owing to their complementary nature.
Understanding complementary angles is crucial for simplifying and proving trigonometric identities because they often allow substitution of one function with another, facilitating simplification.

Sine and Cosine Relationship

The relationship between sine and cosine, especially when dealing with complementary angles, is a fundamental aspect of trigonometry. According to the complementary angle identity, \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \). This identity shows how the cosine of an angle is equal to the sine of its complement.
This relationship allows us to rewrite expressions in terms of either sine or cosine. For instance, if you know one angle measure, you can easily find the trigonometric function of its complement using this relationship.
- In the given problem, we used a similar identity for cosine, replacing \( \cos(x+\beta) \) with \( \sin(\frac{\pi}{2} - (x+\beta)) \).
- Additionally, by recognizing \( \beta - \alpha = \frac{\pi}{2} \), we substituted \( \alpha = \beta - \frac{\pi}{2} \) to transform the expression further.
Utilizing these identities simplifies expressions and aids in proving trigonometric equations, as seen in the problem example.

Expression Simplification

Simplifying expressions involves replacing complex expressions with simpler ones without changing the expression's value. In trigonometry, simplification often utilizes identities and relationships.
In the problem, - We started by substituting known identities related to complementary angles and their sine-cosine relationships.
- The key simplification step was using the identity \( \sin(A - \frac{\pi}{2}) = -\cos(A) \) to rewrite \( \sin(x+\beta - \frac{\pi}{2}) \) as \(-\cos(x+\beta) \).
This turned the original expression \( \sin(x+\alpha) + \cos(x+\beta) \) into a simpler form \(-\cos(x+\beta) + \cos(x+\beta) = 0 \).
Expression simplification is crucial in mathematics because it makes complex problems more manageable, ensuring we can easily prove or derive important results with minimal effort. Understanding and applying identities correctly is the cornerstone of effective simplification.