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
• To solve a trigonometric equation when the trigonometric function has been isolated:– e.g.– 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
2
3sin
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
5
2
3sin
2
3
Basics of Solving Trigonometric Equations (Example)
Ex 1: Find all solutions and then check using a graphing calculator:
6
3tan
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
• 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
9
4
82
352
3553
x
x
x
xx
sinx
Solving Linear Trigonometric Equations (Example)
Ex 2: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions
10
03cos2
Solving Linear Trigonometric Equations (Example)
Ex 3: Find i) t, 0 ≤ t < 2π ii) all radian solutions
11
tt sin5sin33
Solving Linear Trigonometric Equations (Example)
Ex 4: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate:
a)
b)
c)12
4cos21cos8
sin24sin
12sin33
Solving Quadratic 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 Formula14
02 cbxax
Factoring a Quadratic
• To attempt factoring :– 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
02 cbxax
Solving Quadratic Equations Using the Quadratic Formula
• An equation in the format can also be solved using the Quadratic Formula:
• 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
02 cbxax
a
acbbx
2
42
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 • 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
cosx
Solving Quadratic Trigonometric Equations (Example)
Ex 5: Find i) x, 0 ≤ x < 2π ii) all radian solutions
a)
b)
c)
18
0cotcot2 xx
3sin7sin2 2
0tansintan xxx
Solving Quadratic Trigonometric Equations (Example)
Ex 6: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate:
19
2sin1sin
Additional Examples
Ex 7: In a) find all exact degree solutions and in b) find all exact radian solutions
a)
b)
20
2
350sin A
2
1
12cos
A