1. Simple Trig Equations
Dr. Shildneck

2. General Procedure Solve the equation for the trigonometric function.
This means get it down to “function = ratio.”
Ask yourself, “When does this trig function have this value?”
Determine the angle(s) for which the trig function has that value.
Typically there are two answers between 0 and 2π.

3. Example 1. Solve the equation on the interval [0,2π).
sin 𝑥 =− 1 2
When is the sine negative?
“on the bottom”
“at the 𝜋 6 ′ 𝑠”
When is the sine have ratios of ½?
So the answers are:
x = 7𝜋 6 and 11𝜋 6

4. Example 2. Solve the equation on the interval [0,2π).
When is the secant positive?
“on the right”
When does the secant have ratios of 2?
“when the cosine is ½ ”
So the answers are:
x = 𝜋 3 and 5𝜋 3

5. Example 3. Solve the equation on the interval [0,2π).
𝑠𝑖𝑛 2 𝑥−𝑠𝑖𝑛𝑥=2
𝑠𝑖𝑛 2 𝑥−𝑠𝑖𝑛𝑥−2=0
(𝑠𝑖𝑛𝑥−2)(𝑠𝑖𝑛𝑥+1)=0
𝑠𝑖𝑛𝑥−2=0 𝑜𝑟 𝑠𝑖𝑛𝑥+1=0
𝑠𝑖𝑛𝑥=2 𝑜𝑟 𝑠𝑖𝑛𝑥=−1
When is the sine equal to positive 2?
x = 3𝜋 2
x = 𝐷𝑁𝐸
When is the sine equal to negative 1?

6. Example 4. Solve the equation on the interval [0,2π).
3𝑡𝑎𝑛 2 𝑥−1=0
3𝑡𝑎𝑛 2 𝑥=1
When is the tangent equal to positive ?
𝑡𝑎𝑛 2 𝑥= 1 3
x = 𝜋 6 and 7𝜋 6
𝑡𝑎𝑛𝑥=± 1 3
When is the tangent equal to negative ?
x = 5𝜋 6 and 11𝜋 6
𝑡𝑎𝑛𝑥=±