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QuestionMathematics

What is sin^45∘?
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Answer

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Step 1:
I'll solve this step-by-step using precise LaTeX formatting:

Step 2:
: Recall the special angle values for sine

The sine of 45° is a well-known trigonometric value that can be derived geometrically or through the unit circle.

Step 3:
: Understand the 45° angle

A 45° angle is a special right angle where the two sides of the right triangle are equal. This creates an isosceles right triangle.

Step 4:
: Calculate the sine value

\sin(45°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
For a 45° angle in a right triangle:

Step 5:
: Simplify the expression

\sin(45°) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Final Answer

Key Insights: - This value is exactly \frac{\sqrt{2}}{2} - It represents the y-coordinate on the unit circle at 45° - The value is the same whether expressed in degrees or radians