Problem 15 - Contemporary Precalculus

Question

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\cos \left(x-\frac{3 \pi}{2}\right)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\cos \left(x-\frac{3 \pi}{2}\right)\) in terms of sin(x) and cos(x). Answer: \(-\sin x\)

Step 1: Recall and apply angle subtraction formula for cosine

The angle subtraction formula for cosine states that for any angles x and y, \(\cos(x - y) = \cos x \cdot \cos y + \sin x \cdot \sin y\). In this case, we have \(x\) and \(\frac{3\pi}{2}\) as our angles. Substitute these angles into the formula: $$\cos \left(x-\frac{3 \pi}{2}\right) = \cos x \cdot \cos \left(\frac{3\pi}{2}\right) + \sin x \cdot \sin \left(\frac{3\pi}{2}\right)$$

Step 2: Evaluate the sine and cosine of \(\frac{3\pi}{2}\)

Recall that \(\cos(\frac{3\pi}{2}) = 0\) and \(\sin(\frac{3\pi}{2}) = -1\). Substitute these values into the expression from Step 1: $$\cos \left(x-\frac{3 \pi}{2}\right) = \cos x \cdot 0 + \sin x \cdot (-1)$$

Step 3: Simplify the expression

As the cosine term is multiplied by 0, it becomes 0 and only the sine term remains, making the expression: $$\cos \left(x-\frac{3 \pi}{2}\right) = -\sin x$$ The given expression \(\cos \left(x-\frac{3 \pi}{2}\right)\) is now rewritten in terms of \(\sin x\) and \(\cos x\), as \(-\sin x\)

Key Concepts

Angle Subtraction Formula

To understand the problem better, let's delve into the angle subtraction formula for cosine. This formula allows us to handle trigonometric expressions involving differences of angles. It's incredibly useful in transforming and simplifying expressions.

The formula is given by:

• \( \cos(x - y) = \cos x \cdot \cos y + \sin x \cdot \sin y \)

This tells us we can break down the cosine of a difference into separate cosine and sine components of each angle. In our case, we need to rewrite the expression \( \cos \left(x-\frac{3 \pi}{2}\right) \). By applying the formula directly:

• \( \cos(x) \cdot \cos \left(\frac{3\pi}{2}\right) + \sin(x) \cdot \sin \left(\frac{3\pi}{2}\right) \)

This step is crucial as it sets the ground for further simplification.

Sine and Cosine Values

Now, let us evaluate the sine and cosine at specific angles, which often appear in subtraction formulas. Specifically, we look at \( \frac{3\pi}{2} \), also known as 270 degrees, an important angle on the unit circle.

To understand this step:

• We've found \( \cos(\frac{3\pi}{2}) = 0 \)
• Also \( \sin(\frac{3\pi}{2}) = -1 \)

This is straightforward if you visualize the unit circle, where these values are derived from the positions of sine and cosine on the x and y-axes. Knowing these specific values helps us plug them back into the trigonometric formula we've set up earlier.

Expression Simplification

Expression simplification is the final stage in transforming the given trigonometric expression into a simpler form. Using the results from evaluating \( \cos \left(\frac{3\pi}{2}\right) \) and \( \sin \left(\frac{3\pi}{2}\right) \), we simplify our expression further.

Here's how it's done:

• Replace \( \cos \left(\frac{3\pi}{2}\right) = 0 \) in the original expression, resulting in the term \( \cos x \cdot 0 \). This obviously results in 0.
• Next, \( \sin x \cdot (-1) \) becomes \( -\sin x \).

Combining these results, the entire expression simplifies down to \( -\sin x \). Thus, \( \cos \left(x-\frac{3 \pi}{2}\right) \) is efficiently rewritten, showcasing the power of trigonometric identities to make expressions simpler.