trigonometric equations. definition example: consider:
TRANSCRIPT
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Trigonometric Equations
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Definition
A trigonometric equation is an equation that contains a trig expression with a variable, such as .
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Example:
A solution to this equation is because
Is the only solution to this equation?
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Consider:
This graph shows 5 different solutions to the equation So how can we represent all of the solutions?
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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.
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Find 4 more solutions to
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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
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Solve:
3sin π₯β2=5sin π₯β1
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Solve:
5sin π₯=3sin π₯+β3
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Equations Involving Multiple Angles
tan 3π₯=10β€π₯<2π
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Solve:
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Solve:
,
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Trig Equations in Quadratic FormForm: where is a trig function Solve by usual Quadratic
Methodsa. Factorb. Quadratic Formulac. Square Root Method
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Solve by factoring:
2πππ 2π₯+cos xβ1=0 ,0 β€π₯<2π
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Solve by factoring
2π ππ2π₯β3sin π₯+1=0 ,0β€π₯<360 Β°
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Solve
2π ππ2π₯=sinπ₯+3 ,0 β€π₯<2π
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Solve
π ππ2π₯+3 π πππ₯β5=0 ,0β€ π₯<360 Β°
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3πππ 2 π₯β4cos π₯=β4Solve
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Solve
4 π ππ2π₯β1=0 ,0β€ π₯<2π
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Solve
4πππ 2π₯β3=0 ,0 β€π₯<360 Β°
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Separate Two Functions by Factoring
π‘πππ₯ π ππ2π₯=3 tan π₯ ,0β€ π₯<2π
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Separate by Factoring
π πππ₯π‘πππ₯=sinπ₯
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Solve
π‘ππ2 π₯πππ π₯=π‘ππ2π₯ ,0 β€π₯<360 Β°
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Solve
πππ‘2π₯π πππ₯=πππ‘2 π₯ ,0β€ π₯<2π
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Using Identities to Solve Trig Equations (All Solve:
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2π ππ2π₯β3cosx=0
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πππ 2π₯+3sin π₯β2=0
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cos 2π₯+sin π₯=0
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sin π₯cos π₯=12
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sin π₯β cos π₯=1
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cos π₯βsin π₯=β1