Problem 41 - Thomas Calculus

Question

In Exercises \(39-42,\) express the given quantity in terms of \(\sin x\) and \(\cos x\). $$\sin \left(\frac{3 \pi}{2}-x\right)$$

Step-by-Step Solution

Answer

The expression is \(-\cos x\).

Step 1: Recall the Relevant Identity

To solve the problem, recall the identity for sine of an angle difference: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). In our case, we have \( a = \frac{3\pi}{2} \) and \( b = x \).

Step 2: Substitute the Known Values

Substitute \( a = \frac{3\pi}{2} \) and \( b = x \) into the identity: \( \sin\left(\frac{3\pi}{2} - x\right) = \sin\left(\frac{3\pi}{2}\right)\cos x - \cos\left(\frac{3\pi}{2}\right)\sin x \).

Step 3: Calculate the Trigonometric Constants

From trigonometric values, we know \( \sin\left(\frac{3\pi}{2}\right) = -1 \) and \( \cos\left(\frac{3\pi}{2}\right) = 0 \).

Step 4: Simplify the Expression

Substitute the calculated values into the expression: \( \sin\left(\frac{3\pi}{2} - x\right) = (-1)\cos x - 0\sin x = -\cos x \).

Step 5: Express in Terms of \( \sin x \) and \( \cos x \)

The simplified expression is now \( \sin\left(\frac{3\pi}{2} - x\right) = -\cos x \), which is already expressed using \( \cos x \).

Key Concepts

Angle Difference Identity

The Angle Difference Identity is an essential concept in trigonometry that allows us to find the sine or cosine of the difference between two angles. It is represented by the formula:

• For sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)
• For cosine: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

When applying this identity, it is important to correctly identify the two angles (often denoted as \( a \) and \( b \)) and substitute their values into the formula. In the given exercise, the angles we work with are \( a = \frac{3\pi}{2} \) and \( b = x \). By substituting these into the sine angle difference formula, we find the expression needed for further simplification.

Trigonometric Simplification

Trigonometric Simplification involves making a complex trigonometric expression simpler or transforming it into a more manageable form. Once the angle difference identity is applied, simplification typically uses known trigonometric constants. This is crucial for getting the expression in the simplest possible form.In this exercise, after substituting the angles into the identity, you can utilize the known values of sine and cosine of special angles. For \( \frac{3\pi}{2} \), these values are:

• \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• \( \cos\left(\frac{3\pi}{2}\right) = 0 \)

Plugging these into the expression \( \sin\left(\frac{3\pi}{2} - x\right) = \sin\left(\frac{3\pi}{2}\right)\cos x - \cos\left(\frac{3\pi}{2}\right)\sin x \), results in \( -\cos x \). Trigonometric simplification, therefore, reduces the expression into a much neater form.

Trigonometric Constants

Trigonometric Constants are specific values of sine, cosine, and other trigonometric functions for particular angles. These constants are critical in solving many trigonometric problems, as they provide exact values that can simplify calculations. Some well-known trigonometric constants include:

• \( \sin(0) = 0 \)
• \( \cos(0) = 1 \)
• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \cos\left(\frac{\pi}{2}\right) = 0 \)
• \( \sin\left(\frac{\pi}\right) = 0 \)
• \( \cos\left(\frac{\pi}\right) = -1 \)
• \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• \( \cos\left(\frac{3\pi}{2}\right) = 0 \)

These constants assist in converting trigonometric expressions to simpler forms, allowing us to easily solve or evaluate them. In the original exercise, understanding that \( \sin\left(\frac{3\pi}{2}\right) = -1 \) and \( \cos\left(\frac{3\pi}{2}\right) = 0 \) enables us to effectively use the angle difference identity for simplification.