To find the value of cos(15°), we can utilize the cosine subtraction identity, which is given by:
cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
In this case, we can express 15° as the difference of two angles: 45° and 30°.
So, we set:
- a = 45°
- b = 30°
Now substituting into the cosine subtraction identity:
cos(15°) = cos(45° – 30°)
Using the identity, we get:
cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)
Next, we need to find the trigonometric values for cos(45°), cos(30°), sin(45°), and sin(30°):
- cos(45°) = √2/2
- cos(30°) = √3/2
- sin(45°) = √2/2
- sin(30°) = 1/2
Now we can substitute these values back into our equation:
cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2)
This simplifies to:
cos(15°) = (√6/4) + (√2/4)
Combining the fractions, we get:
cos(15°) = (√6 + √2) / 4
Thus, the value of cos(15°) is:
cos(15°) = (√6 + √2) / 4
This representation is often left in its exact form, which is both precise and useful for further calculations.