Find the value of cos75°.

To find the value of cos 75°, we can use the cosine addition formula. This formula states that cos(a + b) = cos a * cos b - sin a * sin b. For 75°, we can break it down into two angles that are easier to work with: 45° and 30°. So, we can express cos 75° as cos(45° + 30°).

Applying the Cosine Addition Formula

Using the formula, we have:

cos(75°) = cos(45° + 30°) = cos 45° * cos 30° - sin 45° * sin 30°.

Finding the Values of Cosine and Sine

Now, let's recall the exact values of the trigonometric functions for 45° and 30°:

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

Substituting the Values

Now we can substitute these values back into our equation:

cos(75°) = (√2/2) * (√3/2) - (√2/2) * (1/2).

Simplifying the Expression

Let's simplify this step by step:

• First term: (√2/2) * (√3/2) = (√6)/4
• Second term: (√2/2) * (1/2) = (√2)/4

Putting it all together, we have:

cos(75°) = (√6)/4 - (√2)/4.

Final Result

Now, we can combine the fractions:

cos(75°) = (√6 - √2)/4.

Thus, the value of cos 75° is (√6 - √2)/4. This result is exact and can be used in various applications, such as solving triangles or in calculus problems involving trigonometric functions.