Finding values of trig functions given a constraint

Question

Find the other trig functions if tan(thetha)=(3/4)  and sin(thetha) <0

 

I really need a detailed explanation on this. I really do not understand how to solve .

Answer 1

Draw a right triangle. Label one angle (other than the right angle) as theta. tan(theta) = opposite / adjacent = 3/4. Note that the sine function is negative and the tangent function is positive. This means that we are in the third quadrant. Tangent is positive in this quadrant, cosine is negative, and sine is negative. Back to the triangle: the opposite side is equal to 3 and the adjacent side is equal to 4. Therefore the hypotenuse of the right triangle is given by (3² + 4²)^1/2 = 5. Now we can easily see that the sin(theta) = opposite / hypotenuse = -3/5 (since we are in quadrant 3). The cos(theta) = adjacent / hypotenuse = -4/5 (3rd quadrant). The other trig functions are simple to find. sec(theta) = 1/cos(theta) = -5/4 ; csc(theta) = 1/sin(theta) = -5/3 ; cot(theta) = 1/tan(theta) = 4/3.