Lesson Overview:

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• Solve trigonometric equations using algebraic techniques.

• Solve trigonometric equations using basic identities.

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Steps to solving trigonometric equations:

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Part A: Solve trigonometric equations using algebraic techniques

There are 2 types of algebraic techniques, Linear and Power Pattern. First, we will focus on the Linear Pattern technique

Example:

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The key word for the Linear Pattern technique is “Isolate” which means, isolate the x. After we isolate we then identify the reference angle, then find solutions between 0 and 2π.

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We then proceed to finding Period which is found using 2π over the B value, which in this case is one, as shown in the image above. Finally, we finish solving our equation by finding general form, N is all integers.

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Part B: Solve trigonometric equations using basic identities 

When it comes to solving trigonometric equations using basic identities, you’ll probably see something like this:

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sin^2(x)+cos^2(x)=1

2sin(x)−1=0

tan^2(2x)−1=0

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Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved. You can use trigonometric identities along with algebraic methods to solve the trigonometric equations.

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An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was exclude from the domain of the original equation.

When you solve trigonometric equations, sometimes you can obtain an equation in one trigonometric function by squaring each side, but this technique may produce extraneous solutions.

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Find all the solutions of the equation in the interval 

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The equation contains both sine and cosine functions.

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We rewrite the equation so that it contains only cosine functions using the Pythagorean Identity 

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Factor

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Use zero product property

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Solve for the solutions of the given equation in the interval

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This Video is very helpful to help you understand how to solve equations using trigonometric identities.

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Solving Trig Identities on the AP Exam:

There are some handy equations that are vital to helping you solve questions or reach the answer in a different way. 

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Sum, Difference and Double Angle Identities in the Real World

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Conclusion:

Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most practical use is to find exact values of an angle that can be written as a sum or difference using the familiar values for the sine, cosine and tangent of the 30°, 45°, 60° and 90° angles and their multiples.

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