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Trigonometric Equations
Definition
A trigonometric equation is an equation that contains a trig expression with a variable, such as .
Example:
A solution to this equation is because
Is the only solution to this equation?
Consider:
This graph shows 5 different solutions to the equation So how can we represent all of the solutions?
Since the period of the sine function is first find all solutions in Those solutions are and .
Any multiple of can be added to these values and the sine is still So, all solutions can be given by with n any integer.
Find 4 more solutions to
Equations involving a Single Trig FunctionTo solve an equation involving a single trig function:1. Isolate the function on one side
of the equation.2. Solve for the variable
Solve:
3sin π₯β2=5sin π₯β1
Solve:
5sin π₯=3sin π₯+β3
Equations Involving Multiple Angles
tan 3π₯=10β€π₯<2π
Solve:
Solve:
,
Trig Equations in Quadratic FormForm: where is a trig function Solve by usual Quadratic
Methodsa. Factorb. Quadratic Formulac. Square Root Method
Solve by factoring:
2πππ 2π₯+cos xβ1=0 ,0 β€π₯<2π
Solve by factoring
2π ππ2π₯β3sin π₯+1=0 ,0β€π₯<360 Β°
Solve
2π ππ2π₯=sinπ₯+3 ,0 β€π₯<2π
Solve
π ππ2π₯+3 π πππ₯β5=0 ,0β€ π₯<360 Β°
3πππ 2 π₯β4cos π₯=β4Solve
Solve
4 π ππ2π₯β1=0 ,0β€ π₯<2π
Solve
4πππ 2π₯β3=0 ,0 β€π₯<360 Β°
Separate Two Functions by Factoring
π‘πππ₯ π ππ2π₯=3 tan π₯ ,0β€ π₯<2π
Separate by Factoring
π πππ₯π‘πππ₯=sinπ₯
Solve
π‘ππ2 π₯πππ π₯=π‘ππ2π₯ ,0 β€π₯<360 Β°
Solve
πππ‘2π₯π πππ₯=πππ‘2 π₯ ,0β€ π₯<2π
Using Identities to Solve Trig Equations (All Solve:
2π ππ2π₯β3cosx=0
πππ 2π₯+3sin π₯β2=0
cos 2π₯+sin π₯=0
sin π₯cos π₯=12
sin π₯β cos π₯=1
cos π₯βsin π₯=β1
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