Chapter 5-trig

1. BIG PICTURE

  • Solving trig equations means finding the angles that satisfy trigonometric functions being equal to a specific value.

  • Connecting to algebra: just like solving x² = 1 gives two solutions, sin x = 1/2 can give multiple angles.

  • Trig equations generally have multiple answers because trigonometric functions are periodic, meaning they repeat values over intervals.

2. TYPES OF TRIG EQUATIONS

A. Basic equations

  • Recognize: equations of the form sin x = a, cos x = a, tan x = a.

  • Best strategy: Isolate the trig function and find reference angles.

  • Why it works: Basic ratio definitions of trig functions apply.

B. Quadratic trig equations

  • Recognize: equations like sin²x − sin x = 0.

  • Best strategy: Factor the equation.

  • Why it works: Similar to solving quadratic equations, factoring helps find solutions.

C. Equations requiring identities

  • Recognize: equations that need simplification with trig identities.

  • Best strategy: Use known identities to simplify the equation.

  • Why it works: Identities help transform and solve the function more easily.

D. Equations with multiple angles

  • Recognize: equations featuring terms like sin(2x) or cos(3x).

  • Best strategy: Use angle identities to reduce complexity.

  • Why it works: Reducing to single angles allows for easier solving.

3. STEP-BY-STEP SOLVING TEMPLATES

  1. Isolate the trig function (e.g., sin x = a).

  2. Find the reference angle (use inverse trig functions).

  3. Use the unit circle to find all solutions (consider all quadrants).

  4. Write the general solution (include periodic terms).

  5. Restrict to the interval if needed (e.g., [0, 2π]).

  • Each step helps clarify the path to finding all valid solutions.

4. UNIT CIRCLE CONNECTION

  • Solutions correspond to angles from the unit circle, where the x-coordinate represents cos and y-coordinate represents sin.

  • Quadrant rules (ASTC) help determine the sign of trig functions in various quadrants.

  • Multiple answers visually come from the periodic nature of the unit circle.

5. GENERAL SOLUTION FORMULAS

  • For sin x = a → x = θ + 2πk and x = π − θ + 2πk.

  • For cos x = a → x = ±θ + 2πk.

  • For tan x = a → x = θ + πk.

  • k represents the number of full circles (complete cycles) that can be added to the solution, indicating periodicity.

6. INTERVAL RESTRICTIONS

  • To find solutions in intervals like [0, 2π], list all solutions and filter out those that fall outside this interval.

  • Common mistakes include missing solutions or failing to correctly limit solutions to the specified range.

7. COMMON MISTAKES

  • Forgetting the second solution (especially for sin and cos).

  • Mixing radians and degrees when solving equations.

  • forgetting to include periodic terms (+2πk or +πk).

  • Algebra mistakes, such as errors in factoring or sign mistakes from the unit circle.

8. FULLY WORKED EXAMPLES

  • Provide detailed examples for each type of equation, showing every step and reasoning:

    • For basic equations: solve sin x = 1/2.

    • For quadratic: solve sin²x − sin x = 0.

    • For identities: problem showing simplified steps.

9. QUICK RECOGNITION GUIDE

  • If you see quadratic → factor first.

  • If multiple angles → solve then divide by the coefficient of x.

  • If trig functions on both sides → move everything to one side.

10. PRACTICE PROBLEMS

  1. Solve sin x = 1/2 (0 to 2π)

  2. Solve cos x = -1/2 (0 to 2π)

  3. Solve tan x = √3 (0 to 2π)

  4. Solve sin²x - sin x = 0 for [0, 2π]

  5. Solve sin(2x) = 1 (0 to 2π)

Advanced:

  1. Solve 2sin²x - sin x - 1 = 0 for [0, 2π]

  2. Solve 3cos(3x) - 1 = 0 for [0, 2π]

  3. Solve sin²x + cos²x = 1 for [0, 2π]