Problem 62 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

If \(\beta - \alpha = \frac{\pi}{2}\), then \(\sin(x + \alpha) + \cos(x + \beta) = 0\).

Step 1: Understand the problem statement

We need to show that if \(\beta - \alpha = \frac{\pi}{2}\), then the expression \(\sin(x + \alpha) + \cos(x + \beta) = 0\) holds true.

Step 2: Use angle subtraction

Given \(\beta - \alpha = \frac{\pi}{2}\), rewrite \(\cos(x + \beta)\) using \(\beta = \alpha + \frac{\pi}{2}\). This simplifies to \(\cos(x + \alpha + \frac{\pi}{2})\).

Step 3: Apply trigonometric identities

Recall the identity for cosine with a shifted angle: \(\cos(x + \frac{\pi}{2}) = -\sin(x)\). Thus, \(\cos(x + \alpha + \frac{\pi}{2}) = -\sin(x + \alpha)\).

Step 4: Substitute back into the equation

Substitute \(\cos(x + \beta) = -\sin(x + \alpha)\) into the original equation:\[\sin(x + \alpha) - \sin(x + \alpha) = 0\] which simplifies to 0.

Step 5: Conclude the proof

The expression \(\sin(x + \alpha) + \cos(x + \beta) = 0\) simplifies to zero, thus confirming the original statement. The solution is verified based on the identity transformation.

Key Concepts

Angle Subtraction

When tackling trigonometric identities, angle subtraction is a powerful tool for simplifying expressions. In this exercise, we have \(\beta - \alpha = \frac{\pi}{2}\), meaning there is a specific relationship between the angles \(\beta\) and \(\alpha\).
This helps us to leverage known identities that relate the trigonometric functions of certain angle differences. By examining the difference of \(\beta - \alpha\), we can transform or substitute expressions involving these angles.
Understanding angle subtraction allows us to rewrite terms in a way that simplifies the problem significantly:

• We rewrite the cosine term with this specific angle subtraction.
• This is the foundation for employing other identities effectively.

Cosine Transformation

Cosine transformation is a key step in this proof. Given that \(\beta = \alpha + \frac{\pi}{2}\), we explore the transformation:
\[\cos(x + \beta) = \cos(x + \alpha + \frac{\pi}{2})\]
The identity \(\cos(x + \frac{\pi}{2}) = -\sin(x)\) tells us that adding \(\frac{\pi}{2}\) to an angle in cosine inversely transforms it into a sine function. This powerful transformation:

• Turns cosine expressions into sine expressions with a negative sign.
• Simplifies complex expressions for easier manipulation.

Being familiar with such transformations is incredibly useful when dealing with trigonometric identities. Through transforming, we can switch between trigonometric functions seamlessly.

Sine and Cosine Relationship

The interconnected relationship between sine and cosine is crucial here. Recognizing the identity:
\[\cos(x + \frac{\pi}{2}) = -\sin(x)\]
shows how a shift by \(\frac{\pi}{2}\) turns cosine into a negative sine function. This identity affects how terms can be replaced or substituted within expressions to prove equations.

Other relationships include:

• Sine and cosine functions are co-functions, where moving an angle by \(\pi/2\) maps one to the other.
• The negative sign is important when transforming cosine to sine, affecting the overall sign of the expression.

Understanding these relations allows us to handle expressions deftly and break down complex equations into manageable components.

Proof Strategy

A solid proof strategy often involves step-by-step transformations using known identities. In proving:
\[\sin(x + \alpha) + \cos(x + \beta) = 0\]
we utilize a strategic approach by substituting \(\cos(x + \beta) = -\sin(x + \alpha)\).
The strategy involves:

• Start with simplifying known expressions using angle subtraction and identities.
• Use logical substitutions and transformations.
• Verify each step logically aligns with prior steps.

This systematic strategy helps ensure correctness at each stage and leads to a successful proof completion.

Identity Verification

Verifying a trigonometric identity involves ensuring all transformations and substitutions lead to a true statement. Starting from:
\[\sin(x + \alpha) - \sin(x + \alpha) = 0\]
The proof concludes since the left side simplifies directly to zero, confirming that our transformations were correct.
Verification process involves:

• Checking each transformation or simplification within the context of known identities.
• Ensuring that no algebraic step contradicts established mathematical principles.
• The final expression aligns with the goal of the proof.

This thorough process ensures your proof stands robust against scrutiny.