Precalculus Question

Question

Find the value of sin 2θ, cos 2θ and tan 2θ in the interval (pi/2,pi) if cosθ=-4/5

Answer 1

Imagine drawing the angle on the coordinate plane (a graph) and creating a triangle by drawing a line perpendicular to the x-axis. The angle near the origin is θ. The adjacent side of θ is always the side on the x-axis. cos(θ) = adjacent side / hypotenuse. sin(θ) = opposite / hypotenuse.

Since cos(θ) is negative and the hypotenuse is always positive, then the adjacent side must be negative. This means the ray of the angle lies in the second or third quadrant (the two left quadrants). Because an interval is given from π/2 to π, this tells you it must be the second quadrant. (Remember pi is 180˚ and it is the line that points to the left on a coordinate plane).

cos(θ) = -4/5 tells you one side is 4 and the hypotenuse is 5. Using the Pythagorean theorem, you can find the third side.

a² + b² = c²

4² + b² = 5²

16 + b² = 25

b² = 25 - 16

b² = 9

b = 3

Then you can find sin(θ) = 3/5. It is important to remember where your angle is drawn on the graph because in the second quadrant, sine is positive. With this, you can use trig identities to find sin(2θ), cos(2θ) and tan(2θ).

sin(2θ) = 2*sin(θ)*cos(θ)

sin(2θ) = 2*(3/5)*(-4/5) = -24/25

cos(2θ) = cos²(θ) - sin²(θ)

cos(2θ) = (-4/5)² - (3/5)² = (16/25) - (9/25) = 7/25

tan(2θ) = sin(2θ) / cos(2θ) =

tan(2θ) = (-24/25) / (7/25) = -24/7