Simple Trig Equations Dr. Shildneck

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.

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

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

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?

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 3 3 ? 𝑡𝑎𝑛 2 𝑥= 1 3 x = 𝜋 6 and 7𝜋 6 𝑡𝑎𝑛𝑥=± 1 3 When is the tangent equal to negative 3 3 ? x = 5𝜋 6 and 11𝜋 6 𝑡𝑎𝑛𝑥=± 3 3