Find the related acute angle for csc(θ) - 2 = 0.
Understand the Problem
The question is asking to find the acute angle related to the equation csc(θ) - 2 = 0. This involves solving for the angle θ where the cosecant function has a specific value. To solve this, we can isolate csc(θ) and then find the corresponding angle using the relationship between cosecant and sine.
Answer
The acute angle \( \theta \) is \( 30^\circ \).
Answer for screen readers
The acute angle ( \theta ) is ( 30^\circ ).
Steps to Solve
- Isolate csc(θ)
To solve the equation ( \text{csc}(\theta) - 2 = 0 ), we first isolate ( \text{csc}(\theta) ).
This can be done by adding 2 to both sides:
$$ \text{csc}(\theta) = 2 $$
- Convert to sine
Next, we use the identity that relates cosecant to sine. The cosecant function is the reciprocal of the sine function:
$$ \text{csc}(\theta) = \frac{1}{\sin(\theta)} $$
This gives us the equation:
$$ \frac{1}{\sin(\theta)} = 2 $$
- Solve for sin(θ)
Now, we solve for ( \sin(\theta) ) by taking the reciprocal of both sides:
$$ \sin(\theta) = \frac{1}{2} $$
- Find the angle(s)
The sine function equals ( \frac{1}{2} ) at specific angles. In the first and fourth quadrants (since we are looking for the acute angle):
$$ \theta = 30^\circ $$
- Verify the range
Since we are specifically looking for the acute angle, ( \theta ) must be between ( 0^\circ ) and ( 90^\circ ). Thus, we confirm:
$$ \theta = 30^\circ $$ is valid.
The acute angle ( \theta ) is ( 30^\circ ).
More Information
The sine of ( 30^\circ ) is a classic value in trigonometry, and it is often used to simplify problems involving angles in the unit circle and right triangles.
Tips
- Confusing cosecant with sine: Remember that ( \text{csc}(\theta) ) is the reciprocal of ( \sin(\theta) ).
- Forgetting to consider the range of angles when looking for acute angles.
AI-generated content may contain errors. Please verify critical information
