Solving Linear Trig Equations (for angles NOT on the unit circle)
When solving a trig equation for solutions outside the first quadrant it is important to review how to
find the “matching” angle in the other 3 quadrants. Remember that the angle in the first quadrant is
often referred to as the “reference” angle. Sometimes the reference angle is one of your solutions,
however, other times you simply use it as a “reference” to find the solutions that satisfy the equations.
How to use a reference angle to find a matching angle:
Quadrant II
In degrees: 180 – reference angle
In radians: π – reference angle
Quadrant III
In degrees: 180 + reference angle
In radians: π + reference angle
Quadrant I
Angles in the first quadrant are sometimes one of
the solutions, but sometimes used as a reference
to find the solutions. You must decide depending
upon the value (positive or negative) of the trig
relationship you are working with.
Quadrant IV
In degrees: 360 – reference angle
In radians: 2π – reference angle
Examples:
1) Solve for 0 < x < 360:
tan(x) = ¼
In Degrees:
First, I notice that ¼ is not a value on my unit circle, so this equation must be
solved using the inverse tangent feature on my calculator.
Tan-1(1/4) ≈ 14.04
Since Tangent values are positive in the first and 3rd quadrants this reference
angle is also one of my solutions. To find the other solution in the 3rd quadrant, I must add 180o
180 + 14.04 = 194.04
Final answer:
x ≈14.04o and x ≈ 194.04o
2) Solve for all possible solution:
cos (x) =
In Radians:
Whenever solving a trig equation set equal to a negative value, we first find our reference angle
by setting the equation equal to the same value, only positive. Then use the inverse function to solve for
your reference angle:
Cos-1 (
) ≈ 1.14
Now we must reconsider the original problem, and ask “in what quadrants will cosine values be
negative?” Answer: 2nd and 3rd. To find the solution in the 2nd quadrant we subtract our reference angle
from pi, to find our solution in the 3rd quadrant we add our reference angle to pi.
x = 3.14 – 1.14 = 2.00 radians
Final answer:
x = 3.14 + 1.14 = 3.28 radians
x ≈ 2.00, 3.28, 2.00 + 2πn, and 3.28 + 2πn
(Since the directions said to list all possible solutions we must put “+ 2πn” on each of our answer to
represent that there will be another solution every 2π radians after our original solutions, because
cosine y-values repeat every 2π radians.)