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Magazine Chapter 7: Problem 62
Show that if \(\beta-\alpha=\pi / 2,\) then $$ \sin (x+\alpha)+\cos (x+\beta)=0 $$
Short Answer
Expert verified
If \(\beta - \alpha = \frac{\pi}{2}\), then \(\sin(x + \alpha) + \cos(x + \beta) = 0\).
Step by step solution
01
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.
02
Use angle subtraction
Given \(\beta - \alpha = \frac{\pi}{2}\), rewrite \(\cos(x + \beta)\) using \(\beta = \alpha + \frac{\pi}{2}\). This simplifies to \(\cos(x + \alpha + \frac{\pi}{2})\).
03
Apply trigonometric identities
Recall the identity for cosine with a shifted angle: \(\cos(x + \frac{\pi}{2}) = -\sin(x)\). Thus, \(\cos(x + \alpha + \frac{\pi}{2}) = -\sin(x + \alpha)\).
04
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.
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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:
\[\cos(x + \beta) = \cos(x + \alpha + \frac{\pi}{2})\]
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.
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:
\[\cos(x + \frac{\pi}{2}) = -\sin(x)\]
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.
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:
\[\sin(x + \alpha) + \cos(x + \beta) = 0\]
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.
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:
\[\sin(x + \alpha) - \sin(x + \alpha) = 0\]
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.
