Problem 23 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \sin \left(x-\frac{\pi}{2}\right)=-\cos x $$

Step-by-Step Solution

Answer

The identity \( \sin(x-\frac{\pi}{2}) = -\cos x \) is proven using the sine difference identity and trigonometric values.

Step 1: Understand the Problem

We need to prove that \( \sin \left(x-\frac{\pi}{2}\right) = -\cos x \). This requires us to use the identity for the sine of a difference of angles and knowledge of trigonometric identities.

Step 2: Use the Sine Difference Identity

The sine of a difference identity is \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). In this case, we set \( a = x \) and \( b = \frac{\pi}{2} \). So, \( \sin(x - \frac{\pi}{2}) = \sin x \cos \left(\frac{\pi}{2}\right) - \cos x \sin \left(\frac{\pi}{2}\right) \).

Step 3: Apply Trigonometric Values

Recall \( \cos \left(\frac{\pi}{2}\right) = 0 \) and \( \sin \left(\frac{\pi}{2}\right) = 1 \). Substitute these values into the equation: \( \sin(x - \frac{\pi}{2}) = \sin x \cdot 0 - \cos x \cdot 1 \).

Step 4: Simplify the Expression

The expression simplifies to \( 0 - \cos x \), which is \( -\cos x \).

Step 5: Conclude the Proof

We have shown that \( \sin(x - \frac{\pi}{2}) = -\cos x \), hence the identity is proven.

Key Concepts

Sine Difference Identity

The sine difference identity is a key formula in trigonometry that helps us find the sine of the difference between two angles. The identity is expressed as follows:\[ \sin(a-b) = \sin a \cos b - \cos a \sin b \]Let's break this down:

• \(a\) and \(b\) are the angles involved.
• You're calculating how the sine function affects these angles when they are subtracted.
• The formula involves both the sine and cosine of each angle, which can be crucial in simplifying complex expressions or proving identities.

For instance, in the problem we are solving, using the sine difference identity allows us to convert \( \sin(x - \frac{\pi}{2}) \) into a formulation involving simple sine and cosine of \(x\) and \(\frac{\pi}{2}\). This step is essential in making it possible to simplify and thereby prove the identity at hand.

Cosine Values

To effectively use trigonometric identities, knowing the exact values of sine and cosine at key angles is essential. Here, the angles in focus are closely related to \(\frac{\pi}{2}\). The following key values are important:

• \( \cos(\frac{\pi}{2}) = 0 \)
• \( \sin(\frac{\pi}{2}) = 1 \)

These values largely determine the outcome when substituting into the sine difference identity formula. Since \( \cos(\frac{\pi}{2}) = 0 \), any component multiplied by it will effectively become zero. On the other hand, \( \sin(\frac{\pi}{2}) = 1 \) implies that elements multiplied by it remain essentially unchanged, except for their sign. Understanding these core values allows for quick and accurate simplification of trigonometric expressions like the one in this problem.

Angle Subtraction

Angle subtraction is a process within trigonometry that involves finding the trigonometric functions of one angle subtracted from another. In this situation, we deal with the expression \( x - \frac{\pi}{2} \). It presents a common scenario where angles are subtracted.Knowing how to handle angle subtraction can make simplifying expressions much more manageable. When substituting the angle subtraction into the sine difference identity, it enables a breakdown that leverages known sine and cosine values. For the specific case of \( x - \frac{\pi}{2} \), you effectively convert a sine function into a negative cosine function, like shown:

• \( \sin(x - \frac{\pi}{2}) = -\cos x \)

This conversion is possible because of specific angles chosen and the properties of the sine difference identity. By mastering angle subtraction, you can more easily navigate through trigonometric identities and proofs, ultimately enhancing your problem-solving arsenal.