How can we use the Pythagorean identity to find cos(8) given that sin(8) is 15/17?

The Pythagorean identity in trigonometry states that for any angle θ, the following relationship holds:

sin²(θ) + cos²(θ) = 1

In this case, we know that sin(8) = 15/17. To find cos(8), we first need to calculate sin²(8).

Calculating sin²(8):

sin2(8) = (15/17)2 = 225/289

Now we can substitute this back into the Pythagorean identity:

sin2(8) + cos2(8) = 1

Substituting the value of sin²(8):

225/289 + cos2(8) = 1

To isolate cos²(8), we subtract 225/289 from both sides:

cos2(8) = 1 - 225/289

Since 1 can be expressed as 289/289, we have:

cos2(8) = 289/289 - 225/289 = (289 - 225)/289 = 64/289

Now we take the square root to find cos(8):

cos(8) = ±√(64/289) = ±(8/17)

Since the angle 8 is not specified to be in any particular quadrant, both the positive and negative values are valid depending on the specific scenario. Thus:

cos(8) = 8/17 or cos(8) = -8/17

In conclusion, using the Pythagorean identity, we found that cos(8) can be either 8/17 or -8/17 depending on the context of the angle.