If sin(a) = -3/5 and tan a< 0, find the exact value of a) tan(a-pi/4 ) and b) cos(2a)

Question

Hehe I am so confused someone help

Answer 1

Once you know the value of one trig function for a given angle, you know them all. You can always build a right triangle using the principal angle in the first quadrant from which you can pick off all of the trig ratios taking care of the signs as you go:

Sin(a)= -3/5, implies that the angle is between pi and 2pi or 3rd or 4th quadrant. The sine value is negative when the y component is negative. The principal right triangle you can build will have an opposite of 3, a hypotenuse of 5, and you can work out the other side by rearranging the Pythagorean Theorem. (Or, in this case recognize a 3-4-5 right triangle

tan(a) < 0 implies that it must be the 4th quadrant because tan is negative when x is + and y is -, or the 2nd and 4th quadrant. Tan > 0 in 1st and 3rd.

So we know that we have a corresponding angle in the 4th quadrant. (Easiest to express as a=-Θ where Θ is the principal angle between 0 and pi/2 that has a sin = +3/5 (sin(-Θ) = -sin(Θ)

a) tan(A-B) = (tan(A) - tan(B))/(1+ tan(A)tan(B)) - known identity

You know tan(a) = -3/4 (opp/adj or sin/cos and <0) and that tan(pi/4) = 1

b) cos(2x) = cos²(x) - sin²(x) - known identity with cos(a) = 4/5 (adj/hyp) and 4th quadrant >0

Good luck!