Problem 35 - Algebra 2

Question

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in radians. \(\cos 2 \theta=1-\sin \theta\)

Step-by-Step Solution

Answer

The solutions are \( \theta = n\pi \), \( \theta = \frac{\pi}{6} + 2n\pi \), and \( \theta = \frac{5\pi}{6} + 2n\pi \) for integer \( n \).

Step 1: Use the Double Angle Identity for Cosine

The double angle identity for cosine states that \( \cos 2\theta = 1 - 2\sin^2 \theta \). Substitute it in the equation to get: \( 1 - 2\sin^2 \theta = 1 - \sin \theta \).

Step 2: Simplify the Equation

Cancel the \(1\) from both sides of the equation to simplify it to: \( -2\sin^2 \theta = -\sin \theta \).

Step 3: Factor the Equation

Factor out \(-1\) and simplify the equation: \( 2\sin^2 \theta - \sin \theta = 0 \). Factor by pulling out \( \sin \theta \): \( \sin \theta (2\sin \theta - 1) = 0 \).

Step 4: Solve for Solutions

Set each factor equal to zero to solve for \( \theta \). First, set \( \sin \theta = 0 \), giving solutions \( \theta = 0, \pi, 2\pi \). Secondly, set \( 2\sin \theta - 1 = 0 \), leading to \( \sin \theta = \frac{1}{2} \). This gives solutions \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \) plus additional solutions at intervals of \( 2\pi \).

Step 5: Include All General Solutions

For \( \sin \theta = 0 \), include all general solutions \( \theta = n\pi \). For \( \sin \theta = \frac{1}{2} \), solutions are \( \theta = \frac{\pi}{6} + 2n\pi \) and \( \theta = \frac{5\pi}{6} + 2n\pi \), where \( n \) is any integer.

Key Concepts

Double Angle Identities

Trigonometric identities are essential tools in trigonometry that help simplify complex equations. One of the most commonly used in such cases is the double angle identity for cosine. This identity helps express the cosine of a double angle, or twice a given angle, in terms of the sine of that angle.
The double angle identity for cosine is:

• \( \cos 2\theta = 1 - 2\sin^2 \theta \)

This formula comes in handy when needing to transform a trigonometric equation into a more manageable form.
By substitution, it aids in eliminating one trigonometric function in exchange for another, making it easier to solve equations.
Such identities provide a path to link different trigonometric functions, like sine and cosine, using algebraic methods.

Solving Trigonometric Equations

Solving trigonometric equations often involves a combination of algebraic techniques and trigonometric identities. The goal is to find all possible angles that make the equation true.
Here is a brief walkthrough:

• Substitution: Use known identities to replace parts of your equation. With \( \cos 2\theta = 1 - \sin \theta \), substitution is a starting point, using the double angle identity to simplify.
• Simplification: Simplify the equation after substitution by canceling terms or combining like terms, reducing complexity.
• Factoring: Look for opportunities to factor the equation, which often reveals simple roots or solutions.
• Solving factors: Set each factor equal to zero, since any product equals zero if any of its factors are zero. Solve for the angle \( \theta \) that satisfies these conditions.

Each step requires carefully considering the domain of the function and the periodic nature of the trigonometric functions involved.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are vital in solving equations in trigonometry. Each function oscillates between specific values as angles change, contributing to their unique properties.
Here's a quick rundown on these functions:

• **Sine (\(\sin \theta\))**: Sine represents the y-coordinate on the unit circle for any angle. It has a range of \([-1, 1]\) and repeats every \(2\pi\).
• **Cosine (\(\cos \theta\))**: Cosine gives the x-coordinate on the unit circle. Like sine, its range is \([-1, 1]\) and also has a period of \(2\pi\).
• Understanding these functions' ranges and periods is crucial for predicting behavior and finding solutions.

The interplay of these functions often shapes how trigonometric equations can be solved.

Radian Measure

Radian measure is an alternative to degree measure when working with angles, providing a natural way to understand angles in terms of circles. In radians, angles are measured by the length of the arc they subtend on the unit circle.
Some important notes about radians:

• **Conversion**: 1 full circle is \(2\pi\) radians, equivalent to 360 degrees. Thus, \(\pi\) radians equal 180 degrees.
• **Radian advantage**: Using radians simplifies many mathematical computations, particularly calculus, where they are preferred due to their natural relations in circle geometry.
• **Equations with radians**: Most trigonometric equations in higher mathematics default to radians because of their direct relation to the properties of circles and cyclic behaviors.

Being familiar with radians is fundamental for deeper exploration into trigonometric applications and advanced mathematics.