3 cos(θ) - cos(θ) = 1
Understand the Problem
The question involves solving the equation 3 cos(θ) - cos(θ) = 1, which can be simplified and solved for the variable θ.
Answer
The solutions for $\theta$ are $\theta = 60^\circ + 360^\circ n$ and $\theta = 300^\circ + 360^\circ n$.
Answer for screen readers
The solutions for $\theta$ are: $$ \theta = 60^\circ + 360^\circ n \quad \text{and} \quad \theta = 300^\circ + 360^\circ n $$
Steps to Solve
-
Combine Like Terms Start by simplifying the left side of the equation. Combine the cosine terms: $$ 3 \cos(\theta) - \cos(\theta) = 2 \cos(\theta) $$ So, the equation becomes: $$ 2 \cos(\theta) = 1 $$
-
Isolate the Cosine Term Next, divide both sides of the equation by 2 to isolate $\cos(\theta)$: $$ \cos(\theta) = \frac{1}{2} $$
-
Find the Angle θ Identify the angles where the cosine value is $\frac{1}{2}$. This occurs at: $$ \theta = 60^\circ + 360^\circ n \quad \text{and} \quad \theta = 300^\circ + 360^\circ n $$ where $n$ is any integer.
The solutions for $\theta$ are: $$ \theta = 60^\circ + 360^\circ n \quad \text{and} \quad \theta = 300^\circ + 360^\circ n $$
More Information
The angles where the cosine equals $\frac{1}{2}$ correspond to $60^\circ$ and $300^\circ$ within one full rotation (0° to 360°). These angles are part of the unit circle representation of trigonometric functions.
Tips
- Overlooking Periodicity: A common mistake is not accounting for the periodic nature of cosine, which repeats every 360°.
- Wrong Quadrants: Some might forget that cosine is positive in the first and fourth quadrants, leading to missing solutions.
AI-generated content may contain errors. Please verify critical information
