Ch. 5 – Analytic Trigonometry 5.3 – Solving Trig Equations

Solving Trig Equations When solving trig equations, get the trig function by itself first, then think unit circle or use a calculator to finish! Ex: Find all solutions of 2sinx – 1 = 0 over the interval [0, 2π). Solve for sin! 2sinx = 1 sinx = ½ Now we can use inverse trig functions (think unit circle!) Since sinx = ½ at both and , those are our answers! Remember: x is an angle here!

Solving Trig Equations Ex: Find all solutions of cosx + 1 = -cosx . Solve for cos! 1 = -2cosx - ½ = cosx Find cos-1 (- ½) to get… We also want the other solution where cos(x) = -½ , so think unit circle to get … However, we want ALL SOLUTIONS of x, not just on [0, 2π)! Since the period of sine and cosine is 2π, if we add/subtract multiples of 2π, we also get solutions for x! To indicate this in our answer, we add n times the period n signifies some integer value Final answer: and

Ex: Find all solutions of 3tan2x – 1 = 0 . Solve for tan! 3tan2x = 1 tan2x = 1/3 tanx = Now we can use inverse trig functions! Find the tan-1 of both answers to get … However, we want ALL SOLUTIONS of x, not just on [0, 2π)! Since the period of tangent is π, if we add/subtract multiples of π, we also get solutions for x, so… Final answer:

Find all solutions to over the interval [0, 2π).

Find all solutions to over the interval [0, 2π).

Find all solutions to .

Find all solutions to .

Ex: Find all solutions of sinx – sinx cos2x = 1/8 on the interval [0, 2π). We can use trig identities to simplify the left side of the equation to one trig function!

Ex: Find all solutions of 2sin2x – sinx – 1 = 0 on the interval [0, 2π). Factor first… (2sinx + 1)(sinx – 1) = 0 Now we have two equations to solve… Think unit circle to find all values on [0, 2π) that satisfy one of those equations… All three of these are possible answers!

Ex: Find all solutions of 4cos2(3x) – 1 = 0 . Solve for cos… 4cos2(3x) = 1 cos2(3x) = 1/4 cos(3x) = ± ½ Now we have two equations to solve. However, we won’t divide by 3 until the trig function has left the equation… Final answers:

Ex: Find all solutions of csc (4x) – 2 = 0 on the interval [0, π). Solve for csc, then flip it to solve for sin… csc(4x) = 2 sin(4x) = ½ Using unit circle knowledge, we know… Now find all solutions to these equations on the interval [0, π)…

Find all solutions to .

Find all solutions to over the interval [0, 2π).