Solve the equation 3sin(θ) - cos(θ) = 1 for 0° ≤ θ ≤ 360°.
Understand the Problem
The question involves solving a trigonometric equation involving sine and cosine functions. Specifically, it's asking to solve the equation 3sin(θ) - cos(θ) = 1 for angles θ in the specified range. The problem also provides a trigonometric identity involving cosecant.
Answer
The angles are approximately $23.6°$ and $203.6°$.
Answer for screen readers
The solutions for the equation $3\sin(\theta) - \cos(\theta) = 1$ in the interval $0° \leq \theta \leq 360°$ are approximately $\theta = 23.6°$ and $\theta = 203.6°$.
Steps to Solve
-
Rearranging the Equation We begin with the equation: $$ 3\sin(\theta) - \cos(\theta) = 1 $$ Rearranging gives us: $$ 3\sin(\theta) = \cos(\theta) + 1 $$
-
Dividing by Cosine Next, we divide both sides by $\cos(\theta)$ to use the tangent function: $$ \tan(\theta) = \frac{\cos(\theta) + 1}{3} $$
-
Using Values of Sine and Cosine From the previous steps, we know that if $\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$. We can find $\sin(\theta)$ and $\cos(\theta)$.
-
Finding Reference Angles Use inverse tangent to find the reference angle: $$ \theta = \tan^{-1}\left(\frac{\cos(\theta) + 1}{3}\right) $$
-
Calculating Actual Angles Evaluate possible angles in the given range, taking into account all quadrants where $\tan(\theta)$ is positive.
-
Final Solutions Check Check if the angles calculated lie in the interval $[0°, 360°]$.
The solutions for the equation $3\sin(\theta) - \cos(\theta) = 1$ in the interval $0° \leq \theta \leq 360°$ are approximately $\theta = 23.6°$ and $\theta = 203.6°$.
More Information
The solutions represent the angles where the equation holds true within the specified range. This problem illustrates the application of trigonometric identities and the usage of the tangent function to solve more complex trigonometric equations.
Tips
- Forgetting Quadrants: One common mistake is neglecting to find solutions in all applicable quadrants. Remember, the tangent function is positive in both the first and third quadrants.
- Incorrect Inverse Function Use: Misapplying the inverse tangent function can lead to wrong reference angles. Always check the angle range.
AI-generated content may contain errors. Please verify critical information
