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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Trigonometric Equations

How would you solve the equation sin θ = 0? You know that θ = 0 is one solution, but this is not the only solution. Any one of the following values of is also a solution.

..., -3π, -2π, -π, 0, π, 2π, 3π,...

You can write this infinite solution set as {nπ: n is an integer}

Example 1

Solving a Trigonometric Equation

Solve the equation

Solution

To solve the equation, you should consider that the sine is negative in Quadrants III and IV and that

Thus, you are seeking values of θ in the third and fourth quadrants that have a reference angle of π/3. In the interval [0, 2π], the two angles fitting these criteria are

By adding integer multiples of to each of these solutions, you obtain the following general solution.

See the figure below.

Solution points of

Example 2

Solving a Trigonometric Equation

Solve cos 2θ = 2 - 3 sin θ, where 0 ≤ θ ≤ 2π.

Solution

Using the double-angle identity cos 2θ = 1 - 2 sin2θ, you can rewrite the equation as follows.

 cos 2θ = 2 - 3 sin θ Given equation 1 - 2 sin2θ = 2 - 3 sin θ Trigonometric identity 0 = 2sin2θ - 3sin θ + 1 Quadratic form 0 = (2 sin θ)(sin θ - 1) Factor.

If 2 sin θ = 0, then sin θ = 1/2 and θ = π/6 or θ = 5π/6. If sin θ - 1, then sin θ = 1 and θ = π/2. Thus, for 0 ≤ θ ≤ 2π, there are three solutions.