Solving Trigonometric Equations
Trigonometry
MATH 103
S. Rook
Overview
• Section 6.1 in the textbook:
– Solving linear trigonometric equations
– Solving quadratic trigonometric equations
2
Basics of Solving Trigonometric
Equations
Basics of Solving Trigonometric
Equations
• To solve a trigonometric equation when the
trigonometric function has been isolated:
– e.g. sin    3
2
– Look for solutions in the interval 0 ≤ θ < period using
the unit circle
• Recall the period is 2π for sine, cosine, secant, &
cosecant and π 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
4
Basics of Solving Trigonometric
Equations (Continued)
– We can use a graphing calculator to help check
(NOT solve for) the solutions
3
sin   
2
3
2
• E.g. For
, enter Y1 = sin
, and look
for the intersection using 2nd → Calc → Intersect
x, Y2 = 
5
Basics of Solving Trigonometric
Equations (Example)
Ex 1: Find all solutions and then check using a
graphing calculator:
tan  3
6
Solving Linear Trigonometric
Equations
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)
8
Solving Linear Trigonometric
Equations
3x  5  5 x  3
• An example of a linear equation:
 2x  5  3
• Solving trigonometric linear
 2x  8
(first degree) equations is very
similar EXCEPT we:
x  4
– 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 x  sin 
• Revert to the trigonometric function after isolating the
variable
– Use the Unit Circle and/or reference angles to
solve
9
Solving Linear Trigonometric
Equations (Example)
Ex 2: Find i) θ, 0° ≤ θ < 360° ii) all degree
solutions
2 cos  3  0
10
Solving Linear Trigonometric
Equations (Example)
Ex 3: Find i) t, 0 ≤ t < 2π ii) all radian solutions
3  3sin t  5 sin t
11
Solving Linear Trigonometric
Equations (Example)
Ex 4: Find i) θ, 0° ≤ θ < 360° ii) all degree
solutions – use a calculator to estimate:
a) 8 cos   1  2 cos   4
b) 33 sin   2  1
c) sin   4  2 sin 
12
Solving Quadratic Trigonometric
Equations
Solving Quadratic Equations
• Recall a Quadratic Equation (second degree)
has the format ax2  bx  c  0
– 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
14
Factoring a Quadratic
• To attempt factoring ax2  bx  c  0 :
– Always look for a GCF (greatest common factor)
• If present, factoring out the GCF simplifies the problem
– Find two numbers that multiply to a·c AND add to b
• Only using the coefficients (numbers)
– If a = 1, we have an easy trinomial
• Can immediately write as two binomials
– If a ≠ 1, we have a hard trinomial
• Expand the trinomial into four terms
• Use grouping
• Alternatively, can also use “Guess and Check”
15
Solving Quadratic Equations Using
the Quadratic Formula
• An equation in the format ax2  bx  c  0 can also be
solved using the Quadratic Formula:
 b  b 2  4ac
x
2a
• To solve a quadratic equation using the Quadratic
Formula:
– Set one side of the quadratic equation to zero
– Plug the values of a, b, and c into the Quadratic
Formula
• a, b, and c are all NUMBERS
– Simplify
16
Solving Quadratic Trigonometric
Equations
• Solving quadratic trigonometric equations is
very similar EXCEPT we:
– Attempt to factor or use the Quadratic Formula on a
trigonometric function instead of a variable
• Can view the trigonometric function as a variable by
making a substitution such as x  cos 
• Revert to the trigonometric function after isolating the
variable
– Use the Unit Circle and/or reference angles to solve
– Be aware of extraneous solutions if fractions OR
functions other than sine or cosine enter into the
equation
17
Solving Quadratic Trigonometric
Equations (Example)
Ex 5: Find i) x, 0 ≤ x < 2π ii) all radian solutions
a) tan x sin x  tan x  0
b) 2 sin 2  7 sin   3
c) cot2 x  cot x  0
18
Solving Quadratic Trigonometric
Equations (Example)
Ex 6: Find i) θ, 0° ≤ θ < 360° ii) all degree
solutions – use a calculator to estimate:
sin   1  sin 
2
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Additional Examples
Ex 7: In a) find all exact degree solutions and in
b) find all exact radian solutions
a)
3
sin  A  50 
2
b)
 
1

cos A    
12 
2

20
Summary
• After studying these slides, you should be able
to:
– Solve Linear Trigonometric Equations
– Solve Quadratic Trigonometric Equations
• Additional Practice
– See the list of suggested problems for 6.1
• Next lesson
– More on Trigonometric Equations (Section 6.2)
21