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5.3 SOLVING TRIG EQUATIONS SOLVING TRIG EQUATIONS Solve the following equation for x: Sin x = ½ SOLVING TRIG EQUATIONS In this section, we will be solving various types of trig equations You will need to use all the procedures learned last year in Algebra II All of your answers should be angles. Note the difference between finding all solutions and finding all solutions in the domain [0, 2π) SOLVING TRIG EQUATIONS Guidelines to solving trig equations: 1) Isolate the trig function 2) Find the reference angle 3) Put the reference angle in the proper quadrant(s) 4) Create a formula for all possible answers (if necessary) SOLVING TRIG EQUATIONS 1- 2 Cos x = 0 1) Isolate the trig function 1- 2 Cos x = 0 + 2 Cos x = + 2 Cos x 1= 2 Cos x 2 2 Cos x = ½ SOLVING TRIG EQUATIONS Cos x = ½ 2) Find the reference angle x= 3 3) Put the reference angle in the proper quadrant(s) 5 I= IV = 3 3 SOLVING TRIG EQUATIONS Cos x = ½ 4) Create a formula if necessary x = 2n 3 5 2n x= 3 SOLVING TRIG EQUATIONS Find all solutions to the following equation: Sin x + 1 = - Sin x + Sin x + Sin x → 2 Sin x + 1 = 0 -1 -1 → 2 Sin x = -1 → Sin x = - ½ SOLVING TRIG EQUATIONS Sin x = - ½ Ref. Angle: 6 7 2n III: 6 11 2n Iv: 6 Quad.: III, IV SOLVING TRIG EQUATIONS Find the solutions in the interval [0, 2π) for the following equation: Tan²x – 3 = 0 Tan²x = 3 Tan x = 3 SOLVING TRIG EQUATIONS Tan x = 3 Ref. Angle: 3 I: 3 2 II: 3 Quad.: I, II, III, IV 4 III: 3 2 4 5 x= 3, 3 , 3 , 3 5 IV: 3 SOLVING TRIG EQUATIONS Solve the following equations for all real values of x. a) Sin x + 2 = - Sin x b) 3Tan² x – 1 = 0 c) Cot x Cos² x = 2 Cot x SOLVING TRIG EQUATIONS Find all solutions to the following equation: Sin x + 2 = - Sin x 2 Sin x = - 2 2 Sin x = 2 5 x = 4 2n 7 x = 4 2n SOLVING TRIG EQUATIONS 3Tan² x – 1 = 0 1 Tan² x = 3 1 Tan x = 3 x = 6 2n 5 2n x= 6 7 2n x= 6 11 2n x= 6 SOLVING TRIG EQUATIONS Cot x Cos² x = 2 Cot x Cot x Cos² x – 2 Cot x = 0 Cot x (Cos² x – 2) = 0 Cot x = 0 Cos² x – 2 = 0 Cos x = 0 Cos² x – 2 = 0 x = 2n Cos x = 2 2 3 2n x= 2 No Solution SOLVING TRIG EQUATIONS 2 5.3 SOLVING TRIG EQUATIONS SOLVING TRIG EQUATIONS Find all solutions to the following equation. 4 Tan²x – 4 = 0 Tan²x = 1 Tan x = ±1 Ref. Angle = 4 x = 4 n 3 n x= 4 SOLVING TRIG EQUATIONS Equations of the Quadratic Type Many trig equations are of the quadratic type: 2Sin²x – Sin x – 1 = 0 2Cos²x + 3Sin x – 3 = 0 To solve such equations, factor the quadratic or, if that is not possible, use the quadratic formula SOLVING TRIG EQUATIONS Solve the following on the interval [0, 2π) 2Cos²x + Cos x – 1 = 0 2x² + x - 1 If possible, factor the equation into two binomials. (2Cos x – 1) (Cos x + 1) = 0 Now set each factor equal to zero SOLVING TRIG EQUATIONS 2Cos x – 1 = 0 Cos x = ½ Ref. Angle: 3 Quad: I, IV 5 x= , 3 3 Cos x + 1 = 0 Cos x = -1 x= SOLVING TRIG EQUATIONS Solve the following on the interval [0, 2π) 2Sin²x - Sin x – 1 = 0 (2Sin x + 1) (Sin x - 1) = 0 SOLVING TRIG EQUATIONS 2Sin x + 1 = 0 Sin x = - ½ Ref. Angle: 6 Quad: III, IV 7 11 x= , 6 6 Sin x - 1 = 0 Sin x = 1 x= 2 SOLVING TRIG EQUATIONS Solve the following on the interval [0, 2π) 2Cos²x + 3Sin x – 3 = 0 Convert all expressions to one trig function 2 (1 – Sin²x) + 3Sin x – 3 = 0 2 – 2Sin²x + 3Sin x – 3 = 0 0 = 2Sin²x – 3Sin x + 1 SOLVING TRIG EQUATIONS 0 = 2Sin²x – 3Sin x + 1 0 = (2Sin x – 1) (Sin x – 1) 2Sin x - 1 = 0 Sin x = ½ Ref. Angle: 6 Quad: I, II 5 x= , 6 6 Sin x - 1 = 0 Sin x = 1 x= 2 SOLVING TRIG EQUATIONS Solve the following on the interval [0, 2π) 2Sin²x + 3Cos x – 3 = 0 Convert all expressions to one trig function 2 (1 – Cos²x) + 3Cos x – 3 = 0 2 – 2Cos²x + 3Cos x – 3 = 0 0 = 2Cos²x – 3Cos x + 1 SOLVING TRIG EQUATIONS 0 = 2Cos²x – 3Cos x + 1 0 = (2Cos x – 1) (Cos x – 1) 2Cos x - 1 = 0 Cos x = ½ Ref. Angle: 3 Quad: I, IV 5 x= , 3 3 Cos x - 1 = 0 Cos x = 1 x= 0 SOLVING TRIG EQUATIONS The last type of quadratic equation would be a problem such as: ( Sec x + 1 )² = Tan²x What do these two trig functions have in common? When you have two trig functions that are related through a Pythagorean Identity, you can square both sides. SOLVING TRIG EQUATIONS (Sec x + 1)² = Tan²x Sec²x + 2Sec x + 1 = Sec²x - 1 2 Sec x + 1 = -1 Sec x = -1 Cos x = -1 x= When you have a problem that requires you to square both sides, you must check your answer when you are done! SOLVING TRIG EQUATIONS Sec x + 1 = Tan x 1 1 0 x= SOLVING TRIG EQUATIONS (Cos Cosx x+ +1)² 1 = Sin Sin²xx Cos²x + 2Cos x + 1 = 1 – Cos² x 2Cos² x + 2 Cos x = 0 Cos x (2 Cos x + 2) = 0 Cos x = 0 Cos x = - 1 3 x= , x= 2 2 SOLVING TRIG EQUATIONS Cos x + 1 = Sin x 3 x= , , 2 2 Cos 1 Sin 2 0 11 2 3 3 Cos 1 Sin 2 2 0 1 -1 Cos 1 Sin -1 1 0 5.3 SOLVING TRIG EQUATIONS SOLVING TRIG EQUATIONS Equations involving multiply angles Solve the equation for the angle as your normally would Then divide by the leading coefficient SOLVING TRIG EQUATIONS Solve the following trig equation for all values of x. 2Sin 2x + 1 = 0 2Sin 2x = -1 Sin 2x = - ½ 7 2n 2x = 6 7 n x= 12 11 2n 2x = 6 11 n x= 12 SOLVING TRIG EQUATIONS x x 3 T an 3 0 T an - 1 2 2 x 3 n 2 4 x 7 n 2 4 x 3 2n 2 2 x 7 2n 2 2 Redundant Answer SOLVING TRIG EQUATIONS Solve the following equations for all values of x. a) 2Cos 3x – 1 = 0 b) Cot (x/2) + 1 = 0 SOLVING TRIG EQUATIONS 2Cos 3x - 1 = 0 2Cos 3x = 1 Cos 3x = ½ 3x = 3 2n 2n x= 9 3 5 2n 3x = 3 5 2n x= 9 3 SOLVING TRIG EQUATIONS x Cot 1 0 2 x 3 n 2 4 x 3 2n 2 2 x Cot - 1 2 SOLVING TRIG EQUATIONS Topics covered in this section: Solving basic trig equations Finding solutions in [0, 2π) Find all solutions Solving quadratic equations Squaring both sides and solving Solving multiple angle equations Using inverse functions to generate answers SOLVING TRIG EQUATIONS Find all solutions to the following equation: Sec²x – 3Sec x – 10 = 0 (Sec x + 2) (Sec x – 5) = 0 Sec x + 2 = 0 Sec x = -2 Cos x = - ½ 2 2n x= 3 4 2n 3 Sec x – 5 = 0 Sec x = 5 1 Cos x = 5 1 x = Cos 5 -1 SOLVING TRIG EQUATIONS a) b) c) One of the following equations has solutions and the other two do not. Which equations do not have solutions. Sin²x – 5Sin x + 6 = 0 Sin²x – 4Sin x + 6 = 0 Sin²x – 5Sin x – 6 = 0 Find conditions involving constants b and c that will guarantee the equation Sin²x + bSin x + c = 0 has at least one solution. SOLVING TRIG FUNCTIONS Find all solutions of the following equation in the interval [0, 2π) Sec²x – 2 Tan x = 4 1 + Tan²x – 2Tan x – 4 = 0 Tan²x – 2Tan x – 3 = 0 (Tan x + 1) (Tan x – 3) = 0 Tan x = -1 Tan x = 3 SOLVING TRIG FUNCTIONS Tan x = -1 3 7 X , 4 4 Tan x = 3 x = ArcTan 3 ref. angle: 71.6º Quad: I, III x = 71.6º, 251.6º