arcsin(sqrt(2)/2)
Understand the Problem
The question is asking for the angle whose sine value is sqrt(2)/2. This will require knowledge of trigonometry to find the angles from the sine function.
Answer
The angles are $45^\circ$ and $135^\circ$.
Answer for screen readers
The angles whose sine value is $\frac{\sqrt{2}}{2}$ are $45^\circ$ and $135^\circ$.
Steps to Solve
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Identify the sine value The problem states that we need to find the angle whose sine value is $\frac{\sqrt{2}}{2}$.
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Recall common sine values From trigonometric knowledge, we recall that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$. Since the sine function is positive in the first and second quadrants, there could be another angle.
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Find the second angle using reference Since sine is also positive in the second quadrant, we can find the second angle using:
$$ 180^\circ - 45^\circ = 135^\circ $$
- List all angles The angles that satisfy $\sin(\theta) = \frac{\sqrt{2}}{2}$ are:
$$ \theta = 45^\circ \text{ and } \theta = 135^\circ $$
The angles whose sine value is $\frac{\sqrt{2}}{2}$ are $45^\circ$ and $135^\circ$.
More Information
The values found are based on the unit circle, and these angles are commonly used in various applications in trigonometry, especially in solving triangle problems.
Tips
- Forgetting that sine is positive in two quadrants can lead to missing the second angle.
- Not recalling the standard angles associated with sine values.
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