Solving Trigonometric Equations

First Degree Trigonometric Equations: These are equations where there is one kind of trig function in the equation and that function is raised to the first power.

Steps for Solving: Isolate the Trigonometric function. Use exact values to solve and put answers in terms of radians. If the answer is not an exact value, then use inverse functions on your calculator to get answers

Now figure out where sin = -1/2 on the unit circle.

Complete the List of Solutions: If you are not restricted to a specific interval and are asked to give the general solutions then remember that adding on any integer multiple of 2π represents a co-terminal angle with the equivalent trigonometric ratio.

Where k is an integer and gives all the coterminal angles of the solution.

Practice Solve the equation. Find the general solutions

Second Degree Trigonometric Equations: These are equations that have one kind of Trigonometric function that is squared in the problem. We treat these like quadratic equations and attempt to factor or we can use the quadratic formula.

This is a difference of squares and can factor Solve each factor and you should end up with 4 solutions

Drill  

Practice Find the general solutions for

Writing in terms of 1 trig fnc If there is more than one trig function involved in the problem, then use your identities. Replace one of the trig functions with an identity so there is only one trig function being used

Solve the following Replace cos2x with 1 - sin2x

Extra Practice  

Solving for Multiple Angles Multiple angle problems will now have a coefficient on the x, such as sin2x=1 Solve the same way as previous problems, but divide answers by the coefficient For general solutions divide 2 by the coefficient for sin and cos. Divide by the coefficient for tan and cot.

Find the general solutions for sin 3x + 2 = 1

Practice