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Solving Trig Equations Finding the angle when the trig function value is known. I. Isolate the trig function: sin 0.23, cos 0.567, tan 2.367, etc. II. Check the domain: 00 3600 or 0 2 , is the usual case for now. Note: this means you will probably have two answers, since each tirg function is in two quads and in two quads. III. Identify which two quadrants the terminal side lies in. IV. Find the reference angle, it is always positive: 00 ref angle 900 V. Calculate the measure of the angle in standard position! NOTE: we can use all the algebraic techniques to solve trig equations that we use to solve algebraic equations. For example: 1. If degree 2, make one side = zero and factor, 2. If you can't factor then use the quad formula, 3. If degree 3, then you should be able to factor, or possibly use the polynomial techniques to solve degree 3 or higher equations, ie. rational zeros theorem In all of these cases, remember that when you get to isolating the trig function: sin 0.567, or cos 0.126, or tan 5.483, there are usually two answers for each value! If we are solving over the set of real numbers, then there are an infinite number of solutions (coterminal angles). Solve the trigonometric equation: 2cos x 2 3 3, over: a) 0 x 2 , b) , , over the reals, find exact values! Step I. Isolate the trig function. 3 2cos x 3, cos x , x lies in quads II & III, 2 3 1 0 Step II. Find the reference angle: ref cos or 30 2 6 Step III. Calculate the angle in standard position: Sketch the angles: Indicate the angles in standard position: Now we can calculate the angle in standard position part a: 5 Quadrant II: 6 6 7 Quadrant III: 6 6 7 6 6 6 5 7 Therefore the solutions are: , 6 6 5 6 Part b: over the set of real numbers. 5 7 This means any angles coterminal with & , 6 6 since the period of the cosine function is 2 5 7 Solution is: 2n , 2n , n Integers 6 6 Solve the trigonometric equation: 2sin 2 x 1 2, over: a) 0 x 2 , b) , , over the reals, 1 2sin x 1, sin x , 2 x lies in quads I,II, III & IV, 2 2 1 reference sin 4 2 1 find exact values! 1 sin x 2 45 0 Sketch the reference angles in all four quadrants: Now we can calculate the angles in standard position: Quad I ref in standard position 3 Quad II 4 4 5 Quad III 4 4 7 Quad IV 2 4 4 4 3 4 4 5 4 7 4 3 5 7 Therefore the solutions are: , , , & 4 4 4 4 Solve the trigonometric equation: 3tan 2 x 2 tan x 3, over: a) 0 x 2 , b) , , over the reals, answers accurate to four decimals! Since degree two, write in standard form and see if we can factor. 3tan 2 x 2 tan x 3 0, can't factor, so use the quad formula, a 3, b 2, c 3 b b 4ac tan x 2a 2 2 2 4 3 3 2 2 3 2 4 36 6 2 40 2 2 10 1 10 tan x 6 6 3 Since tan x will be + & , we can expect four answers from 0 x 2 Now find the reference angle! 1 10 ref tan 0.624522886 1 1 10 ref tan 0.94627344 3 3 1 Sketch the reference angles in all four quadrants: In quad I, ref in standard position, x 0.9463 In quad II, x 0.624522886 x 2.5171 In quad III, x 0.94627344 x 4.0879 In quad IV, x 2 0.624522886 x 5.6587 Therefore the solutions for part a) x 0.9463, 2.5171,4.0879, 5.6587 Therefore the solutions for part a) x 0.9463, 2.5171, 4.0879, 5.6587 part b) x 0.9463 n x 2.5171 n x 0.9463 n or n Integers x 4.0879 n x 2.5171 n x 5.6587 n The shorter version at the right is because the period of the tangent function is radians or 180 0 All the positive answers differ by radians, and all the negative answers differ by radians! Solving trig equations involving a phase shift and/or multiple angles! Solve the following trig equation over: a) 0 2 , b) , . 2sin 3 0 6 Step I: isolate the trig function. 3 sin 6 2 3 Step II: let x , so we have: sin x 6 2 Step III: Find the reference angle, Angle x is in quadrants III & IV (since negative), 3 reference sin 2 3 1 Sketch the reference angles in quad III & IV: 4 Quad III angle: 3 3 5 Quad IV angle: 2 3 3 Now these two angles have been shifted left. 6 4 5 So x , 6 3 3 4 5 , 3 6 3 6 7 9 3 , 6 6 2 4 3 5 3 Therefore the solutions for 0 2 are: 7 3 , 6 2 Therefore the solutions for , are: 7 3 2n , 2n , n Integers 6 2 Solve a trig function involving multiple or half angles: Solve the following trig equation over: a) 0 2 , b) , . 3 tan 2 x 1 0 1 Step I: isolate the trig function: tan 2 x 3 1 Step II : let 2 x tan 3 is in quads II & IV, now find the reference angle: 1 Step III : ref tan 6 3 1 Sketch the reference angles in quad II & IV: 5 Quad II angle : 6 6 11 Quad IV angle : 2 6 6 5 11 17 23 So now we have : 2 x , , , 6 6 6 6 5 11 17 23 x , , , 12 12 12 12 5 6 11 6 Over the set of real numbers: since period is 5 n solution is: , n Integers 12 2 This is the graph of y tan 2 x 1 For y 3 2 ,