Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \cos 2 \theta+\sin ^{2} \theta=1 $$

Step-by-Step Solution

Answer

\(\theta = 0^\circ, 180^\circ, 360^\circ \)

Step 1: Use Trigonometric Identity

Recall that the identity for cosine double angle is \( \cos 2\theta = 1 - 2\sin^2\theta \). Substitute this into the equation \( \cos 2\theta + \sin^2\theta = 1 \) to get \( 1 - 2\sin^2\theta + \sin^2\theta = 1 \).

Step 2: Simplify the Equation

Combine like terms to simplify the equation: \( 1 - \sin^2\theta = 1 \). Then subtract 1 from both sides to get \( -\sin^2\theta = 0 \).

Step 3: Solve for \(\sin\theta\)

The equation \( -\sin^2\theta = 0 \) simplifies to \( \sin^2\theta = 0 \). Taking the square root of both sides gives \( \sin\theta = 0 \).

Step 4: Determine \(\theta\) Values

The solutions for \( \sin\theta = 0 \) within the interval \( 0^\circ \leq \theta \leq 360^\circ \) are \( \theta = 0^\circ, 180^\circ, 360^\circ \), because sine is zero at these angles.

Key Concepts

Cosine Double Angle Identity

The cosine double angle identity is a useful trigonometric identity. It provides a relationship between the cosine of twice an angle and the sine of the angle. The formula for the cosine double angle is given by:

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• Using Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \), the formula also transforms to \( \cos 2\theta = 1 - 2\sin^2 \theta \) and \( \cos 2\theta = 2\cos^2 \theta - 1 \)

These variations can be used interchangeably based on the context of the problem you are solving.
In the exercise provided, the formula \( \cos 2\theta = 1 - 2\sin^2 \theta \) is particularly useful, because it allows us to directly substitute and simplify the given equation \( \cos 2\theta + \sin^2 \theta = 1 \).
Substituting gives us \( 1 - 2\sin^2 \theta + \sin^2 \theta = 1 \). This step is the foundation for solving the equation and finding the angle \( \theta \).

Sine Function Zeros

The sine function has its zeros at specific angles where the value of sine is exactly zero. When solving trigonometric equations, particularly ones involving the sine function, identifying where sine is zero can instantly provide solutions.
The sine function, \( \sin \theta \), is zero at multiples of \( 180^\circ \), particularly:

• \( \theta = 0^\circ \)
• \( \theta = 180^\circ \)
• \( \theta = 360^\circ \)

In the interval of \( 0^\circ \leq \theta \leq 360^\circ \), \( \sin\theta = 0 \) directly corresponds to these angle values.
In solving \( \cos 2\theta + \sin^2\theta = 1 \), we found through algebraic manipulation that \( \sin^2\theta = 0 \).
This leads to \( \sin \theta = 0 \), and the solutions are the zeros of the sine function within our specified interval, namely \( 0^\circ, 180^\circ, \text{and } 360^\circ \).

Solving Trigonometric Equations

Solving trigonometric equations involves using various identities and algebraic techniques to reveal angles that satisfy a given equation. The process usually involves:

• Identifying and applying relevant trigonometric identities (like \( \cos 2\theta = 1 - 2\sin^2\theta \))
• Simplifying the equation to make it more manageable
• Solving for the trigonometric function values (like \( \sin \theta = 0 \))
• Finding all angle solutions within the specified interval

In the example provided, the steps began with substituting the cosine double angle identity into \( \cos 2\theta + \sin^2\theta = 1 \).
This led us through simplifying and solving algebraically to find \( \sin^2\theta = 0 \).
Consequently, \( \sin \theta = 0 \) was determined, allowing us to identify the specific angles that solve the equation.
Each solution corresponds to a zero of the sine function within the interval, giving us the precise values \( 0^\circ, 180^\circ, \text{and } 360^\circ \).