College Algebra & Trigonometry, 2018a

10.3. TRIGONOMETRIC EQUATIONS 459
10.3 Trigonometric Equations
In the previous section on trigonometric identities we worked with equations
that would be true for all values of a particular angle θ. These are sort of like
the algebraic equations whose solution set is “all real numbers,” like 2x +10=
2(x +1)+8. In this section, we will solve trigonometric equations whose solution
set involves only certain values for the angle in question. Because of the cyclical
nature of the angles we’re working with, there will often be an infinite number of
solutions although not “all real numbers.”
Example 1
Here’s an example. Suppose that we consider the equation sin x =0.5. Whether
we use technology, a table or reasoning to solve this equation, it’s clear that one
solution is 30 ◦ . However, remember from the beginning of Chapter 2 that the
sine function is positive in Quadrant II. That means that a second quadrant angle
with a reference angle of 30 ◦ also has a sine equal to 0.5. Recall the ASTC diagram
from Chapter 2:
Sin
All
Tan
Cos