Use Pythagorean identities to find sin x if cot x = -√3/2 and cos x < 0

Question

use pythagorean identities to find sin help pls

Answer 1

The basic Pythagorean identity relating to trigonometric functions: sin²x + cos²x = 1.

Cotangent is cosine over sine. So cot x = cos x / sin x = -√3/2.

Multiply (sin x) to both sides of the equation "cos x / sin x = -√3/2", and we get cos x = -√3/2 sin x.

Now, we'll substitute cos x with -√3/2 sin x in sin²x + cos²x = 1.

The equation becomes sin²x + (3/4) sin²x = 1 if we do so.

Let's combine the sine terms to get (7/4) sin²x = 1,

then boil it down to (√7/2 sin x) = 1.

sin x is either 2/√7 or -(2/√7).

This is where we use the information "cos x < 0".

If cos x < 0 and cot x < 0, sin x must be positive. Therefore we can rule out -(2/√7).

Conclusion: sin x = 2/√7. = (2√7)/7.

(By the way, cos x would be -√(3/7), which is -√21 /7.