cos15度等于多少

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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