Trigonometric Equations. Definition Example: Consider:
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Transcript of Trigonometric Equations. Definition Example: Consider:
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 𝑥−2=5sin 𝑥−1
Solve:
5sin 𝑥=3sin 𝑥+√3
Equations Involving Multiple Angles
tan 3𝑥=10≤𝑥<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:
2𝑐𝑜𝑠2𝑥+cos x−1=0 ,0 ≤𝑥<2𝜋
Solve by factoring
2𝑠𝑖𝑛2𝑥−3sin 𝑥+1=0 ,0≤𝑥<360 °
Solve
2𝑠𝑖𝑛2𝑥=sin𝑥+3 ,0 ≤𝑥<2𝜋
Solve
𝑠𝑖𝑛2𝑥+3 𝑠𝑖𝑛𝑥−5=0 ,0≤ 𝑥<360 °
3𝑐𝑜𝑠2 𝑥−4cos 𝑥=−4Solve
Solve
4 𝑠𝑖𝑛2𝑥−1=0 ,0≤ 𝑥<2𝜋
Solve
4𝑐𝑜𝑠2𝑥−3=0 ,0 ≤𝑥<360 °
Separate Two Functions by Factoring
𝑡𝑎𝑛𝑥 𝑠𝑖𝑛2𝑥=3 tan 𝑥 ,0≤ 𝑥<2𝜋
Separate by Factoring
𝑠𝑖𝑛𝑥𝑡𝑎𝑛𝑥=sin𝑥
Solve
𝑡𝑎𝑛2 𝑥𝑐𝑜𝑠𝑥=𝑡𝑎𝑛2𝑥 ,0 ≤𝑥<360 °
Solve
𝑐𝑜𝑡2𝑥𝑠𝑖𝑛𝑥=𝑐𝑜𝑡2 𝑥 ,0≤ 𝑥<2𝜋
Using Identities to Solve Trig Equations (All Solve:
2𝑠𝑖𝑛2𝑥−3cosx=0
𝑐𝑜𝑠2𝑥+3sin 𝑥−2=0
cos 2𝑥+sin 𝑥=0
sin 𝑥cos 𝑥=12
sin 𝑥− cos 𝑥=1
cos 𝑥−sin 𝑥=−1