5.3 Solving Trigonometric Equations

5.3Solving Trigonometric Equations JMerrill, 2010

Recall (or Relearn ) • It will be imperative that you know the identities from Section 5.1. Concentrate on the reciprocal, quotient, and Pythagorean identities. • The Pythagorean identities are crucial!

Solve Using the Unit Circle • Solve sin x = ½ • Where on the circle does the sin x = ½ ? Solve for [0,2π] Particular Solutions Find all solutions General solutions

Solving a Trigonometric Equation Using Algebra

Using Algebra Again • Find all solutions to: sin x + = -sin x

You Try • Solve

Solve by Factoring Round to nearest hundredth

You Try • Solve These 2 solutions are true because of the interval specified. If we did not specify and interval, you answer would be based on the period of tan x which is π and your only answer would be the first answer. Verify graphically

Day 2on 5.3 • Quick review of Identities

Fundamental Trigonometric Identities Reciprocal Identities Also true:

Fundamental Trigonometric Identities Quotient Identities

Fundamental Trigonometric Identities These are crucial! You MUSTknow them. Pythagorean Identities

Pythagorean Memory Trick sin2 cos2 tan2cot2 sec2 csc2 (Add the top of the triangle to = the bottom) 1

Sometimes You Must Simplify Before you Can Solve • Strategies • Change all functions to sine and cosine (or at least into the same function) • Substitute using Pythagorean Identities • Combine terms into a single fraction with a common denominator • Split up one term into 2 fractions • Multiply by a trig expression equal to 1 • Factor out a common factor

Recall:Solving an algebraic equation

Solve Hint: Make the words match so use a Pythagorean identity Quadratic: Set = 0 Combine like terms Factor—(same as 2x2-x-1)

Solve

What You CANNOT Do • You cannot divide both sides by a common factor, if the factor cancels out. You will lose a root…

Example Common factor—lost a root No common factor = OK

Squaring and Converting to a Quadratic • Sometimes, you must square both sides of an equation to obtain a quadratic. However, you must check your solutions. This method will sometimes result in extraneous solutions.

Squaring and Converting to a Quadratic • Solve cos x + 1 = sin x in [0, 2π) • There is nothing you can do. So, square both sides • (cos x + 1)2 = sin2x • cos2x + 2cosx + 1 = 1 – cos2x • 2cos2x + 2cosx = 0 • Now what? Remember—you want the words to match so use a Pythagorean substitution!

Squaring and Converting to a Quadratic • 2cos2x + 2cosx = 0 • 2cosx(cosx + 1) = 0 • 2cosx = 0 cosx + 1 = 0 • cosx = 0 cosx = -1

Check Solutions

Functions With Multiple Angles • Solve 2cos3x – 1 = 0 for [0,2π) • 2cos3x = 1 • cos3x = ½ • Hint: pretend the 3 is not there and solve cosx = ½ . • Answer: • But….

Functions With Multiple Angles • In our problem 2cos3x – 1 = 0 • What is the 2? • What is the 3? • This graph is happening 3 times as often as the original graph. Therefore, how many answers should you have? amplitude frequency 6

Functions With Multiple Angles And add the circle once again. Add a whole circle to each of these

Functions With Multiple Angles Final step: Remember we pretended the 3 wasn’t there, but since it is there, x is really 3x:

Practice Problems • Work the problems by yourself. Then compare answers with someone sitting next to you. • Round answers: • 1. csc x = -5 (degrees) • 2. 2 tanx + 3 = 0 (radians) • 3. 2sec2x + tanx = 5 (radians)

Practice – Exact Answers Only (Radians) Compare Answers • 4. 3sinx – 2 = 5sinx – 1 • 5. cos x tan x = cos x • 6. cos2 - 3 sin  = 3

5.3 Solving Trigonometric Equations