If sin(8) = 21 over 29, can we use the Pythagorean identity to find cos(8)?

To determine cos(8) given that sin(8) = 21/29, we will utilize the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

This identity states that for any angle θ, the square of the sine of the angle plus the square of the cosine of that same angle is equal to 1. Now, let’s calculate:

sin²(8) = (21/29)² = 441/841

Now, we apply this result into the Pythagorean identity:

cos²(8) = 1 – sin²(8)

Substituting the value of sin²(8):

cos²(8) = 1 – 441/841

To rewrite 1 in terms of the same denominator, we have:

1 = 841/841

Therefore:

cos²(8) = 841/841 – 441/841 = 400/841

To find cos(8), we take the square root of both sides:

cos(8) = ±√(400/841)

Which simplifies to:

cos(8) = ±(20/29)

Since cosine of an angle is positive in the first quadrant and we assume that 8 degrees is in the first quadrant, we have:

cos(8) = 20/29

In summary, using the Pythagorean identity, we find that cos(8) = 20/29.