Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \sin \left(\theta-\frac{3 \pi}{2}\right)=\cos \theta $$

Step-by-Step Solution

Answer

The identity is verified as \( \sin(\theta - \frac{3\pi}{2}) = \cos \theta \).

Step 1: Understanding the Formula

We need to verify that \(\sin\left(\theta-\frac{3\pi}{2}\right) = \cos \theta\). This involves understanding the properties of trigonometric functions, particularly the angle subtraction and addition formulas.

Step 2: Recall Trigonometric Identity

Recall the angle subtraction identity for sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). This identity will be essential in transforming the given expression.

Step 3: Apply the Angle Subtraction Formula

Set \( a = \theta \) and \( b = \frac{3\pi}{2} \). Thus, \( \sin\left(\theta - \frac{3\pi}{2}\right) = \sin \theta \cos \frac{3\pi}{2} - \cos \theta \sin \frac{3\pi}{2} \).

Step 4: Simplify Using Trigonometric Values

Calculate \( \cos \frac{3\pi}{2} \) and \( \sin \frac{3\pi}{2} \). From the unit circle, \( \cos \frac{3\pi}{2} = 0 \) and \( \sin \frac{3\pi}{2} = -1 \). Substitute these values: \( \sin \theta (0) - \cos \theta (-1) = \cos \theta \).

Step 5: Verification

The expression simplifies to \( \cos \theta \), which matches the right side of the original equation. Thus, we have verified that \( \sin \left(\theta-\frac{3 \pi}{2}\right)=\cos \theta \).

Key Concepts

Angle Subtraction Identity

The angle subtraction identity for sine is a crucial tool in trigonometry. It allows us to expand the sine of the difference of two angles using the known sine and cosine values of each angle. Mathematically, it is expressed as: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] This identity is invaluable when dealing with complex trigonometric expressions. In our exercise, by setting \(a = \theta\) and \(b = \frac{3\pi}{2}\), we can use the subtraction formula to simplify \(\sin(\theta - \frac{3\pi}{2})\).

• Expand the expression using the identity.
• Simplify by substituting known trigonometric values.
• Verify the result corresponds with the right side of the equation.

By understanding and applying this identity, students can transform angles into more familiar forms, often leading to simpler expressions.

Unit Circle

The unit circle is a very helpful visual tool in trigonometry. When we talk about trigonometric functions, the unit circle helps to understand the values of sine, cosine, and tangent for any given angle. A unit circle is centered at the origin (0, 0) of a coordinate plane with a radius of 1. For any angle \(\theta\), the coordinates of the point on the unit circle are \((\cos \theta, \sin \theta)\). For more complex angles, like \(\frac{3\pi}{2}\), we can still find their sine and cosine values by looking at their positions on the unit circle.

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

Understanding the unit circle can help greatly in visualizing how the sine and cosine functions relate to angles geometrically, and it simplifies the calculation of these trigonometric values across the full range of angles.

Trigonometric Values

Trigonometric values for specific angles are fundamental in solving problems like our original exercise. For angles like \(\frac{3\pi}{2}\), these values are often memorized or quickly referenced using the unit circle. Knowing these values allows us to break down complex expressions into simpler parts. Key angles and their sine and cosine values:

• \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\): Simple base angles used frequently in calculations.
• \(\cos 0 = 1, \sin 0 = 0\)
• \(\cos \frac{\pi}{2} = 0, \sin \frac{\pi}{2} = 1\)
• \(\cos \pi = -1, \sin \pi = 0\)
• \(\cos \frac{3\pi}{2} = 0, \sin \frac{3\pi}{2} = -1\)

Recognizing these trigonometric values shortcuts many calculations and allows us to verify trigonometric identities accurately. Understanding where these values come from and their applications is crucial for anyone studying trigonometry.