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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. 2 sin( x) 1 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 2 sin( x) 1 1 sin( x) 2 Now figure out where sin = -1/2 on the unit circle. 1 7 11 sin at and 2 6 6 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 coterminal angle with the equivalent trigonometric ratio. Solutions : 7 2 k 6 x 11 2k 6 Where k is an integer and gives all the coterminal angles of the solution. Practice • Solve the equation. Find the general solutions 3 csc 2 0 3 csc 2 2 csc 3 3 which means that sin 2 2 2 k , 2 k 3 3 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. Solve : 4sin 2 ( x) 1 0 over the int erval [0, 2 ) This is a difference of squares and can factor (2sin x 1)(2sin x 1) 0 Solve each factor and you should end up with 4 solutions 1 1 sin x and sin x 2 2 x 5 7 11 6 , 6 , 6 , 6 Practice Find the general solutions for tan x 2 tan x 1 2 tan x 2 tan x 1 0 2 (tan x 1)(tan x 1) 0 tan x 1 3 7 x k , k 4 4 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 2 cos x sin x 1 0 2 Replace cos2 with 1-sin2 2(1 sin 2 x) sin x 1 0 2 2sin 2 x sin x 1 0 2sin 2 x sin x 1 0 2sin 2 x sin x 1 0 (2sin x 1)(sin x 1) 0 1 sin x and sin x 1 2 7 11 x 2k , x 2k , x 2 k 6 6 2 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 sin3x 1 3 3x 2 3 2 k 2 x 3 3 2 x k 2 3 Practice Solve 2 cos 4 x 3 0 3 cos 4 x 2 5 7 4x and 4 x 6 6 5 k 7 k x , x 24 2 24 2