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cosine of 15 degrees, denoted as cos(15°), can be calculated using trigonometric principles. The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle, where the angle of interest is 15 degrees.
To calculate cos(15°), we can utilize the trigonometric identity or reference angles to find its exact value. One common approach is to express 15 degrees as a combination of angles for which the cosine values are known, such as 30 degrees and 45 degrees.
Here's how we can calculate cos(15°) using trigonometric identities:
1. Express 15 degrees as the sum of two angles:
15 degrees = 45 degrees - 30 degrees.
2. Apply the cosine of the difference formula:
The formula for cos(A - B) is cos(A)cos(B) + sin(A)sin(B).
3. Substitute known values:
Let A = 45 degrees and B = 30 degrees:
cos(15°) = cos(45° - 30°).
4. Calculate the cosine values:
Using known values:
cos(45°) = sqrt(2) / 2
cos(30°) = sqrt(3) / 2
sin(45°) = sqrt(2) / 2
sin(30°) = 1 / 2
5. Substitute values into the formula:
cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)
= (sqrt(2) / 2)(sqrt(3) / 2) + (sqrt(2) / 2)(1 / 2)
= (sqrt(6) / 4) + (sqrt(2) / 4)
= (sqrt(6) + sqrt(2)) / 4
So, the exact value of cos(15°) is (sqrt(6) + sqrt(2)) / 4, which is approximately 0.96592582628 when rounded to six decimal places.
Another approach to finding cos(15°) involves using the half-angle identity for cosine:
1. Express 15 degrees as half of 30 degrees:
15 degrees = 1/2 * 30 degrees.
2. Apply the half-angle identity for cosine:
cos(15°) = sqrt((1 + cos(30°)) / 2).
3. Substitute the value of cos(30°):
cos(30°) = sqrt(3) / 2.
4. Calculate:
cos(15°) = sqrt((1 + sqrt(3) / 2) / 2)
= sqrt((2 + sqrt(3)) / 4).
So, using the half-angle identity, cos(15°) equals sqrt((2 + sqrt(3)) / 4), which is approximately 0.96592582628 when rounded to six decimal places.
Both methods yield the same result, which is approximately 0.96592582628.
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