Find cos 0 and sin 0, given tan 0 = 2/7 and cos 0 < 0.

Answer 1

Given that tan 0 = 2/7, we can use the Pythagorean identity to find the hypotenuse of the right triangle formed by the angle 0. Since tangent is positive and cosine is negative, we know that cosine is negative and sine is positive.

Let's assume that the adjacent side of the triangle is 2x and the opposite side is 7x. Therefore, tan 0 = 2/7 = opposite/adjacent = 7x/2x = 7/2.

Using the Pythagorean theorem:
(7x)^2 + (2x)^2 = h^2
49x^2 + 4x^2 = h^2
53x^2 = h^2
h = sqrt(53)x

Therefore, cos 0 = -2x/sqrt(53)x = -2/sqrt(53) = -2sqrt(53)/53
And sin 0 = 7x/sqrt(53)x = 7/sqrt(53) = 7sqrt(53)/53

So, cos 0 = -2sqrt(53)/53 and sin 0 = 7sqrt(53)/53.