Solving Trigonometric Equations

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Solving Trigonometric Equations

• MATH 109 - Precalculus
• S. Rook

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Overview

• Section 5.3 in the textbook
• Basics of solving trigonometric equations
• Solving linear trigonometric equations
• Solving quadratic trigonometric equations
• Solving trigonometric equations with multiple
  angles
• Solving other types of trigonometric equations
• Approximate solutions to trigonometric equations

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Basics of Solving Trigonometric Equations
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Basics of Solving Trigonometric Equations

• To solve a trigonometric equation when the
  trigonometric function has been isolated
• e.g.
• Look for solutions in the interval 0 ? lt period
  using the unit circle
• Recall the period is 2p for sine, cosine, secant,
  cosecant and p for tangent cotangent
• We have seen how to do this when we discussed the
  circular trigonometric functions in section 4.2
• If looking for ALL solutions, add period n to
  each individual solution
• Recall the concept of coterminal angles

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Basics of Solving Trigonometric Equations
(Continued)

• We can use a graphing calculator to help check
  (NOT solve for) the solutions
• E.g. For , enter Y1 sin x,
  Y2 , and look
• for the intersection using 2nd ? Calc ? Intersect

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Basics of Solving Trigonometric Equations
(Example)

• Ex 1 Find all solutions and then check using a
  graphing calculator

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Solving Linear Trigonometric Equations
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Solving Linear Equations

• Recall how to solve linear algebraic equations
• Apply the Addition Property of Equality
• Isolate the variable on one side of the equation
• Add to both sides the opposites of terms not
  associated with the variable
• Apply the Multiplication Property of Equality
• Divide both sides by the constant multiplying the
  variable (multiply by the reciprocal)

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Solving Linear Trigonometric Equations

• An example of a linear equation
• Solving trigonometric linear
  (first degree) equations is very
  similar EXCEPT
  we
• Isolate a trigonometric function of an angle
  instead of a variable
• Can view the trigonometric function as a variable
  by making a substitution such as
  
• Revert to the trigonometric function after
  isolating the variable
• Use the Unit Circle and/or reference angles to
  solve

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Solving Linear Trigonometric Equations (Example)

• Ex 2 Find all solutions

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Solving Quadratic Trigonometric Equations
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Solving Quadratic Trigonometric Equations

• Recall a Quadratic Equation (second degree) has
  the format
• One side MUST be set to zero
• Common methods used to solve a quadratic
  equation
• Factoring
• Remember that the process of factoring converts a
  sum of terms into a product of terms
• Usually into two binomials
• Quadratic Formula

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Solving Quadratic Trigonometric Equations
(Continued)

• The same methods can be used to solve a quadratic
  trigonometric equation
• Substituting a variable for a trigonometric
  function is acceptable so long as there is only
  one trigonometric function present in the
  equation
• e.g. Let y tan x
• Be aware of extraneous solutions if fractions are
  present
• Those solutions which cause the denominator to
  equal 0

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Solving Quadratic Trigonometric Equations
(Example)

• Ex 3 Solve in the interval 0 x lt 2p
• a)
• b)
• c)

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Trigonometric Equations with Two Different
Trigonometric Functions

• Be aware when a quadratic trigonometric equation
  exists with two DIFFERENT trigonometric functions
• Not like Example 3c because after factoring out
  tan x, the equation became two linear
  trigonometric equations
• Recall how we handled two different trigonometric
  functions in section 5.1

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Trigonometric Equations with Two Different
Trigonometric Functions (Continued)

• If we have two different trigonometric functions
  raised to the first power
• Square both sides and apply Pythagorean
  identities to simplify the equation
• E.g.
• Recall that when we square both sides of an
  equation some of the potential solutions will not
  check into the original equation
• MUST check all solutions into the original
  problem
• Discard those solutions that do not check

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Trigonometric Equations with Two Different
Trigonometric Functions (Example)

• Ex 4 Solve in the interval 0 x lt 2p
• a)
• b)
• c)

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Solving Trigonometric Equations with Multiple
Angles
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Solving Trigonometric Equations with Multiple
Angles

• A trigonometric equation with a multiple angle
  has the form kx where k ? 1 (a single-angle
  trigonometric function otherwise)
• To solve a trigonometric equation with
  multiple-angles e.g. 1 cos 3x 3/2
• Isolate the trigonometric function either by
  solving for it or applying a quadratic strategy
• e.g. cos 3x ½

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Solving Trigonometric Equations with Multiple
Angles (Continued)

• Find all solutions in the interval of 0, period)
• e.g.
• Isolate the variable
• e.g.
• If necessary, let n vary to find all solutions in
  the interval 0, 2p)
• e.g.

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Solving Trigonometric Equations with Multiple
Angles (Example)

• Ex 5 Find all solutions in the interval 0,
  2p)

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Other Types of Trigonometric Equations
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Trigonometric Equations and the Sum Difference
Formulas

• Recall the sum and difference formulas
• Valid in both directions
• Given a trigonometric equation involving the
  right-hand side of a sum or difference formula
• Condense into the left-hand side of the formula
• e.g.
• Use previously discussed strategies to solve

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Trigonometric Equations and Multiple-Angle
Formulas

• Recall the double-angle and half-angle formulas
• We can use either the left or right sides of
  these formulas
• Overall goal is to isolate the trigonometric
  function

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Other Types of Trigonometric Equations (Example)

• Ex 6 Solve in the interval 0, 2p)
• a)
• b) sin 6x sin 2x 0
• c) 4 sin x cos x 1
• d)

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Approximate Solutions to Trigonometric Equations
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Approximate Solutions to Trigonometric Equations

• More often than not we run into solutions of
  trigonometric equations that are NOT one of the
  special values on the unit circle
• Solve as normal until the trigonometric function
  is isolated
• Calculate the reference angle
• Use the reference angle AND the sign of the value
  of the trigonometric function to estimate the
  solutions in the interval 0,
  period)

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Approximate Solutions to Trigonometric Equations
(Example)

• Ex 7 Find all solutions in the interval 0, 2p)
  use a calculator to estimate
• a)
• b)

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Summary

• After studying these slides, you should be able
  to
• Solve linear trigonometric equations
• Solve quadratic trigonometric equations
• Solve trigonometric equations with multiple
  angles
• Solve other types of trigonometric equations
  including sum difference formulas, double-angle
  half-angle formulas
• Approximate the solutions to trigonometric
  equations
• Additional Practice
• See the list of suggested problems for 5.3
• Next lesson
• Law of Sines (Section 6.1)