Find the value of cos 15°?

To find the value of cos 15°, we can use a trigonometric identity that involves the cosine of the difference of two angles. Specifically, we can express 15° as the difference between 45° and 30°. The cosine difference formula states that:

Cosine of a Difference

The formula for the cosine of the difference of two angles is:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

In our case, let A = 45° and B = 30°. Therefore, we can write:

cos(15°) = cos(45° - 30°)

Values of Cosine and Sine

Next, we need the values of cos 45° and cos 30°, as well as sin 45° and sin 30°:

• cos 45° = √2/2
• sin 45° = √2/2
• cos 30° = √3/2
• sin 30° = 1/2

Applying the Values

Now, substituting these values into our cosine difference formula gives us:

cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)

Substituting the known values:

cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2)

Calculating the Expression

Now, let's simplify this expression step by step:

cos(15°) = (√6/4) + (√2/4)

Combining these fractions gives:

cos(15°) = (√6 + √2) / 4

Final Result

Thus, the value of cos 15° is:

cos(15°) = (√6 + √2) / 4

This result is exact and can be used in further calculations or applications in trigonometry. If you need a decimal approximation, you can calculate it using a calculator, which will yield approximately 0.9659.