Problem 62 - 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 equation holds true and simplifies to 0 as required.

Step 1: Substitute \\( \beta = \alpha + \frac{\pi}{2} \\\)

Start by substituting \(\beta = \alpha + \frac{\pi}{2}\) into the given equation: \[\sin(x + \alpha) + \cos(x + \beta) = \sin(x + \alpha) + \cos\left(x + \alpha + \frac{\pi}{2}\right).\]

Step 2: Use Cosine Angle Sum Identity

Utilize the identity for cosine: \[\cos\left(x + \alpha + \frac{\pi}{2}\right) = -\sin(x + \alpha)\] since \( \cos(\theta + \frac{\pi}{2}) = -\sin(\theta) \).

Step 3: Simplify the Expression

Substitute this identity back into the equation:\[\sin(x + \alpha) + (-\sin(x + \alpha)) = \sin(x + \alpha) - \sin(x + \alpha) = 0.\] Therefore, the left-hand side simplifies to zero.

Key Concepts

Angle Sum Identity

When working with trigonometric identities, one of the key concepts is the angle sum identity. This identity helps us to find the sine or cosine of an angle that is a sum or difference of two angles. For instance, the cosine angle sum identity states that:

• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)

In the original exercise, the cosine identity was adapted because of the "+\frac{\pi}{2}" offset. Specifically, it's crucial to know that shifting an angle by \( \frac{\pi}{2} \) transforms the cosine function. This is used to convert between sine and cosine due to their complementary nature.
Using the identity for \( \cos(\theta + \frac{\pi}{2}) = -\sin(\theta) \) helps simplify expressions involving angle shifts. Recognizing these relationships enables students to manipulate and prove trigonometric equations effectively. Understanding and applying the angle sum identities are foundational skills needed for deeper exploration in trigonometry.

Trigonometric Equations

Trigonometric equations can initially appear complex, but they often become more manageable with the right approach. When solving these equations, such as the one in the given exercise, substitution is a valuable technique. For example, recognizing that \( \beta = \alpha + \frac{\pi}{2} \) allows us to replace and simplify terms.

• By replacing \( \beta \) as such, it turns the equation into one involving a known identity.
• This kind of substitution is essential, especially in more complicated trigonometric problems.

Strategies like substitution turn seemingly complex equations into simpler ones, where identities can be easily applied to find a solution.
Remember, when dealing with trigonometric equations:

• Identify relationships between angles.
• Use identities wisely to simplify.
• Double-check the applicability of the identities to the given terms.

Developing these strategies helps in building confidence to handle more advanced problems later on.

Proof in Trigonometry

Proving a theorem or identity within trigonometry involves a clear understanding of the relationships and transformations between angles and their trigonometric functions. In the given exercise, the goal was to demonstrate that a particular equation resolved to zero.
When engaging in a proof, it’s crucial to:

• Start with the hypotheses or given information, such as \( \beta - \alpha = \frac{\pi}{2} \).
• Substitute these into the equation sensibly, transforming the given terms where necessary.
• Use fundamental identities, as evidenced by employing \( \cos(\theta + \frac{\pi}{2}) = -\sin(\theta) \) in this case.
• Simplify step by step to reach the desired conclusion logically.

The process of proof in trigonometry allows one to verify and authenticate mathematical statements rigorously. It enhances both analytical and problem-solving skills, giving students a stronger grasp of mathematical reasoning.