Question

Find sin theta if cot theta =-4 and cos theta < 0

Answer 1

Answer:

sin (Q) = 1 / sqrt(17)

Step-by-step explanation:

Given:

                                 cot (Q) = -4

Find:

- find sin theta if cos theta < 0

Solution:

- We will construct a right angle triangle first, mark one of the angle as Q.

- We know from trigonometric relations that:

                                 cot ( Q ) = 1 / tan (Q)

- Hence, according to tan (Q) your opposite side is -4 and base is 1. However, since cot (Q) is a reciprocal of that we will mark -4 as base and +1 as opposite.

- Now to compute sin(Q), we will have to find the hypotenuse first:

                                 H^2 = P^2 + B^2

                                 H^2 = (-4)^2 + (1)^2

- we get,                   H^2 = 17

                                 H = sqrt (17)

- so from hypotenuse we can determine sin (Q) as follows:

                                 sin(Q) = P / H

                                 sin (Q) = 1 / sqrt(17)

- since, cos (Q) < 0, then sin (Q) must be > 0.        

Answer 2

Imagine the unit circle. The cot(theta) is a line from (0,1) to (-4,1). Imagine it is part of a triangle with the origin (draw it!).

Then the hypotenuse length is √(1+4²) = √17.

The sine rule says that sin(90)/√17 must equal sin(theta)/4, and sin(90)=1, so

$sin(\theta) = \frac4{\sqrt{17}}$