Problem 24 - Algebra 2

Question

Solve each equation for all values of \(\theta\). \(\sin ^{2} \theta+\cos 2 \theta-\cos \theta=0\)

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{2} + n\pi \) or \(\theta = 2n\pi\).

Step 1: Use a Trigonometric Identity for Simplification

Recall the trigonometric identity \( \cos 2\theta = 1 - 2\sin^2 \theta \). Substitute this identity into the original equation:\[ \sin^2 \theta + (1 - 2 \sin^2 \theta) - \cos \theta = 0 \] Simplify the expression:\[ 1 - \sin^2 \theta - \cos \theta = 0 \]

Step 2: Express Everything in Terms of Sine or Cosine

Using the Pythagorean identity, \( \sin^2 \theta = 1 - \cos^2 \theta \), substitute the value of \( \sin^2 \theta \) in the equation:\[ 1 - (1 - \cos^2 \theta) - \cos \theta = 0 \]This simplifies to:\[ \cos^2 \theta - \cos \theta = 0 \]

Step 3: Factor the Equation

Factor the quadratic equation \( \cos^2 \theta - \cos \theta = 0 \):\[ \cos \theta (\cos \theta - 1) = 0 \]This gives the solutions \( \cos \theta = 0 \) or \( \cos \theta = 1 \).

Step 4: Solve for \(\theta\)

Determine the values of \( \theta \) for each factor:1. For \( \cos \theta = 0 \), \( \theta = \frac{\pi}{2} + n\pi \) for integer \(n\), where \( \theta = 90^\circ + 180^\circ n\).2. For \( \cos \theta = 1 \), \( \theta = 2n\pi \), where \( \theta = 0^\circ + 360^\circ n\).

Step 5: Combine Solutions

Combine the solutions from both factors:- \( \theta = \frac{\pi}{2} + n\pi \)- \( \theta = 2n\pi \)These represent the general solutions for the original equation.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in mathematics used to simplify equations and expressions involving trigonometric functions like sine, cosine, and tangent.
They are relationships that hold true for any angle, making them essential for solving trigonometric equations. In solving the given equation, we use the identity for the double angle of cosine:

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

This identity allows us to express \( \cos 2\theta \) in terms of \( \sin \theta \), helping to simplify our original equation.
By substituting this identity, we make the equation easier to handle, paving the way for further simplification using other identities or algebraic techniques.

Pythagorean Identity

The Pythagorean identity is another cornerstone in trigonometry, expressing a fundamental relationship between sine and cosine functions:

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

In the context of our exercise, it allows us to interchange sine and cosine functions.
By rearranging the identity, we can express \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \), enabling us to substitute this into the modified equation:

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

Doing so further simplifies the equation, transforming it into a form that is ready for factoring.
This highlights the utility of the Pythagorean identity in manipulating and resolving trigonometric equations.

Factoring Equations

Once we have an equation expressed in simpler terms, like the quadratic equation \( \cos^2 \theta - \cos \theta = 0 \), factoring becomes a powerful solving method.
Factoring is the process of breaking down an equation into simpler components that, when multiplied together, return the original equation. In our case:

• \( \cos^2 \theta - \cos \theta = 0 \) turns into \( \cos \theta (\cos \theta - 1) = 0 \)

From here, we can explore solutions for each factor separately. This gives us two simpler equations to solve:

\( \cos \theta = 0 \)

\( \cos \theta = 1 \)

The solutions to these are much easier to find and interpret, providing the specific values for \( \theta \) that satisfy the original equation.

General Solutions

Finding the general solutions of a trigonometric equation involves identifying all possible values of \( \theta \) that satisfy the equation for any integer value of \( n \).
For our factors, the solutions are:

• \( \cos \theta = 0 \) gives \( \theta = \frac{\pi}{2} + n\pi \)
• \( \cos \theta = 1 \) gives \( \theta = 2n\pi \)

These solutions account for the periodic nature of trigonometric functions, reflecting the fact that such functions repeat their values in a regular pattern.
The variable \( n \) represents any integer, indicating that there are infinitely many solutions corresponding to different rotations around the circle.
Combining these solutions provides a comprehensive answer to the original problem, encapsulating every possible angle for \( \theta \) that satisfies the equation.