Use the adjacent figure to find the exact value of the following trigonometric function. cos(α/2)
Use the adjacent figure to find the exact value of the following trigonometric function.
cos(α/2)
cos(α/2)
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info@snapandsolve.comTo find cos(α/2), we can use the half-angle identity for cosine:
cos(α/2) = ±√((1 + cos(α))/2)
First, we need to find cos(α) using the triangle. The adjacent side is 7, and the hypotenuse can be found using the Pythagorean theorem:
hypotenuse = √(7² + 24²) = √(49 + 576) = √625 = 25
So, cos(α) = adjacent/hypotenuse = 7/25.
Now, substitute into the half-angle identity:
cos(α/2) = ±√((1 + 7/25)/2) = ±√((32/25)/2) = ±√(16/25) = ±4/5
The sign depends on the quadrant where α/2 lies. Assuming α is in the first quadrant, α/2 is also in the first quadrant, so the cosine is positive.
Thus, cos(α/2) = 4/5.
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