Kara Z. answered • 07/02/19
Master's in Chemical Engineering with 15+ Years Tutoring Experience
There are a couple different ways you can approach this problem: either using the unit circle or the special right triangles.
Using the Unit Circle
Since we want to evaluate tanθ = -1 and we know that tanθ = sinθ/cosθ, that means we want to find the points on the unit circle where sin and cos have the same numerical value but opposite signs.
Looking at the unit circle, this occurs at 3π/4 and 7π/4 so those would be our answers. θ = 3π/4, 7π/4
Using the Special Right Triangles
If you don't have the unit circle memorized and don't have access to it, you can always solve trig equations using the special right triangles. They first appear in Geometry and are called the 30-60-90 and 45-45-90 triangles.
We want to evaluate tanθ = -1, but for now we just want to determine where tanθ = 1. The ratio for tan is opposite/adjacent, and looking at the two triangles, we can see that opposite/adjacent = 1 in the 45-45-90 triangle at 45º (or π/4 in radians). That means that the reference angle we're going to be working with is π/4 ( θref = π/4 ). Now we want to account for the negative sign.
When we think about the unit circle, certain trig functions are positive in different quadrants. This is sometimes referred to as the "All Students Take Calculus" rule. In the first quadrant, All trig functions are positive. In the second quadrant, Sin (and csc) is positive. In the third quadrant, Tan (and cot) are positive. In the fourth quadrant, Cos (and sec) are positive.
Since we want tanθ = -1, we're going to be looking at the quadrants where tangent is negative: Quadrants 2 and 4.
To find an angle in Quadrant 2, we subtract the reference angle from π (when working in radians).
θ = π - θref = π - π/4 = 3π/4
To find an angle in Quadrant 4, we subtract the reference angle from 2π.
θ = 2π - θref = 2π - π/4 = 7π/4
Therefore, our two answers are θ = 3π/4, 7π/4
