If tan(x)= -(3/4) and x is obtuse, evaluate sin(2x).

Question

How do I get started on this question?

Answer 1

This angle x is obtuse, between 90 and 180 degrees. Therefore 2x is between 180 and 360.
tan x = -3/4
height = 3
base = 4
hypotenuse = 5 (3,4, 5 right triangle)
so
sin x = 3/5
cos x = -4/5
and we know sin 2x = 2 sin x cos x
so
sin 2x = 2(3/5)(-4/5)
sin 2x = -24/25

Answer 2

Angles with negative values of the tangent are in the second or fourth quadrant. If it is obtuse (between 90 and 180 degrees), it is in the second quadrant. The supplement of the angle is sin^-1 (3/5)= 36.87 degrees. (Think of a 3,4,5 right reference triangle). The angle x is therefore 143.13 degrees, and its sine is 3/5. Its cosine is -4/5. Use sin 2x = 2 sin x cos x to get the sine of twice the angle.

Sin (2x) = 2(3/5)(-4/5) = -24/25 = -0.96. It is in the fourth quadrant.