Question

Find the exact value of cos 75 degrees by using 70 degrees = 45 degrees + 30 degrees. Utilize the given angle relationships to determine the cosine value.

Answer 1

The exact value of cos 75 degrees, which can be found using the cosine angle sum identity, is (√6 - √2) / 4.

Step-by-step explanation:

To find the exact value of cos 75 degrees, we can use the angle sum identity for cosine which states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b). Since 75 degrees can be expressed as the sum of 45 degrees and 30 degrees, we will apply this identity using cos(45 degrees) and cos(30 degrees) which are known values from the unit circle.

The cosine of 45 degrees equals √2/2 and the cosine of 30 degrees equals √3/2. The sine of 45 degrees is also √2/2 and the sine of 30 degrees is 1/2.

So, cos(75 degrees) can be calculated as follows:

1. cos(75 degrees) = cos(45 degrees + 30 degrees)
2. cos(75 degrees) = cos(45 degrees) * cos(30 degrees) - sin(45 degrees) * sin(30 degrees)
3. cos(75 degrees) = (√2/2) * (√3/2) - (√2/2) * (1/2)
4. cos(75 degrees) = √6/4 - √2/4
5. cos(75 degrees) = (√6 - √2) / 4

Therefore, the exact value of cos 75 degrees is ∖(√6 - √2) / 4.