Problem 31 - Algebra 2

Question

Verify that each of the following is an identity. \(\sin \left(\theta+\frac{3 \pi}{2}\right)=-\cos \theta\)

Step-by-Step Solution

Answer

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

Step 1: Use sine addition formula

The sine addition formula is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Let's apply this formula to \( \sin \left(\theta + \frac{3\pi}{2}\right) \).

Step 2: Substitute values

Set \( a = \theta \) and \( b = \frac{3\pi}{2} \) into our formula, leading to \( \sin(\theta)\cos\left(\frac{3\pi}{2}\right) + \cos(\theta)\sin\left(\frac{3\pi}{2}\right) \).

Step 3: Evaluate trigonometric values

Using the unit circle, recall that \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \). Substitute these values into our expression.

Step 4: Simplify the expression

Substituting the values from the previous step, we have \( \sin(\theta) \cdot 0 + \cos(\theta) \cdot (-1) \), which simplifies to \( -\cos(\theta) \).

Step 5: Confirm identity

Our simplified expression \( -\cos(\theta) \) is the right side of the original identity \( \sin\left(\theta+\frac{3 \pi}{2}\right) = -\cos \theta \). This confirms the identity is correct.

Key Concepts

Sine Addition Formula

The sine addition formula is a crucial tool in trigonometry. It allows us to find the sine of the sum of two angles. Mathematically, it is represented as:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

This formula is particularly helpful when dealing with compound angles, as it breaks down a complex expression into simpler components. In the exercise above, we used this formula to verify the identity \( \sin \left(\theta+\frac{3 \pi}{2}\right) = -\cos \theta \). By setting \( a = \theta \) and \( b = \frac{3\pi}{2} \), we systematically applied this formula to simplify and solve the equation step by step. Understanding the sine addition formula is essential for manipulating and simplifying expressions involving sine functions.

Unit Circle

The unit circle is a fundamental concept in trigonometry, acting as a tool for understanding the values of sine, cosine, and tangent for different angles. It is a circle with a radius of 1, centered at the origin of a coordinate system.

• The x-coordinate corresponds to the cosine of an angle, while the y-coordinate represents the sine.
• Key angles, such as \( \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \), have specific sine and cosine values that are pivotal in calculations.

In our exercise, the unit circle helps us recall that \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \). These values were substituted into the expression derived from the sine addition formula to simplify and verify the trigonometric identity. Hence, the unit circle is indispensable for evaluating trigonometric functions at these critical angles.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is an important skill in mathematics, easing the process of solving equations and proving identities. The key is to break down and streamline complex expressions using known identities and properties.
Use strategies such as:

• Applying trigonometric identities (like the sine addition formula).
• Substituting known values from the unit circle.
• Combining like terms and eliminating terms that equal zero.

In the given problem, simplification began with applying the sine addition formula and substituting values from the unit circle. This resulted in an expression that was greatly condensed, \( \sin(\theta) \cdot 0 + \cos(\theta) \cdot (-1) \), which simplifies directly to \( -\cos(\theta) \). This demonstrates how effective simplification can validate complex trigonometric identities.