Thursday, March 28, 2013
How do we solve linear trigonometric equations?
Aim: How do we solve linear trigonometric equations?
Solving trigonometric equations are easy as long as you know how to do them.
There are three steps to solving linear equations:
1) Get the trig function alone.
2) Use inverse to solve for angle or use the helpful Unit Circle!
3) Check for other solutions.
Example:
Solve for all values between 0 to 2:
2 sin θ-√3 = 0
1) To get the trig function alone, add √3 to both sides of the equation.
2 sinθ -√3 = 0
+√3 +√3
2 sinθ = √3
2) Divide 2 from both sides to get sinθ completely alone.
2 sinθ = √3
2 2
sinθ = √3
2
3) Now, if you look at the unit circle, you will see that the only places where sin (y-coordinate) is √3 are at π and 2π or in degrees 60°
2 3 3
and 120°. These are the solutions. You can also achieve these solutions by first using the calculator to solve for inverse sine of √3,
2
which will give you 60° and you can subtract it from 180 to get 120° since sine is only positive in the first two quadrants or 180°
Example #2:
tan θ - 2 cos θ tan θ = 0
1) Factor to solve for both tanθ and cosθ. Factor out tanθ since it is
common.
tan θ - 2 cos θ tan θ = 0
tanθ (1- 2 cosθ) = 0
2) Set each equal to 0 and solve.
tanθ = 0 and 1-2 cosθ = 0
-1 -1
-2 cosθ = -1
-2 -2
cosθ = 1/2
3) If you look at the unit circle, you will see that tangent (sin/cos)
equals to 0 at 0° and 180° or 0 and π in radians. The cosine(x-
coordinate) are 60° and 300° or π/3 and 5π/3. These are the
solutions.
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