Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The identity is proven using the sine difference formula and known trigonometric values.

Step 1: Apply the Sine Difference Identity Formula

The sine of a difference can be rewritten using the identity: \(\sin(a-b) = \sin a \cos b - \cos a \sin b\). In the context of our identity, we have \(a = x\) and \(b = \frac{\pi}{2}\), so apply this identity to rewrite \(\sin\left(x-\frac{\pi}{2}\right)\).

Step 2: Substitute Known Trigonometric Values

We know that \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\). Substitute these values into the rewritten identity from Step 1.

Step 3: Simplify the Expression

Substituting the values from Step 2 into the identity gives us: \[\sin\left(x-\frac{\pi}{2}\right) = \sin x \cdot 0 - \cos x \cdot 1 = -\cos x\].

Step 4: Conclude the Proof

The expression simplifies directly to \(-\cos x\), which matches the right side of the original identity. Thus, we have shown that \(\sin\left(x-\frac{\pi}{2}\right) = -\cos x\).

Key Concepts

Sine Difference Identity

The sine difference identity is a powerful tool in trigonometry. It helps express the sine of the difference between two angles in terms of the individual sines and cosines of these angles. This identity is written as:\[\sin(a-b) = \sin a \cos b - \cos a \sin b\]By using this identity, you can transform a complex trigonometric expression involving a difference into simpler, more manageable terms using basic sine and cosine functions. For example, let’s consider \(a = x\) and \(b = \frac{\pi}{2}\) to see how this identity plays out. If you substitute these values in the identity, it helps to break down the expression \(\sin\left(x-\frac{\pi}{2}\right)\), leading us to simplify the expression into a more comprehensible form. This is particularly useful when starting to prove more complicated identities.

Cosine Function

Understanding the cosine function is crucial for correctly applying trigonometric identities. The cosine function is one of the primary trigonometric functions and represents the x-coordinate of a point on the unit circle as the angle varies. The unit circle is incredibly helpful when visualizing and deriving important trigonometric values and properties.The key property to remember here is that at\(\frac{\pi}{2}\), the cosine function value is 0. That means \(\cos\left(\frac{\pi}{2}\right) = 0\), which directly affects how expressions involving this angle transform in identities like the sine difference identity. When applied to the sine difference identity, knowing \(\cos\left(\frac{\pi}{2}\right) = 0\) helps in reducing specific complex expressions to simpler ones.

Trigonometric Values

Knowing the exact trigonometric values at specific angles aids greatly in simplifying trigonometric expressions and identities. Let's consider two essential values that are used in this problem:

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

These particular angles on the unit circle are significant, as they form the basis for simplifying and proving many trigonometric identities. When simplifying the expression \(\sin\left(x-\frac{\pi}{2}\right)\) using the sine difference identity, substituting these known values allows us to see clearly that the expression reduces to \[-\cos x\]Understanding and recalling these fundamental trigonometric values can help break down complex trigonometric forms into their simplest expressions, making problem-solving more straightforward.