Lesson Overview:

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• Use sum and difference identities to evaluate trigonometric functions.

• Use sum and difference identities to solve trigonometric equations.

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Part A: Using sum and difference identities to evaluate trigonometric functions

For example, you are given this problem

cos(-)= -cos

To prove that both sides are equal, we first look at the left-hand side (LHS). Cos (-) equals cos(π)cos(θ) + sin(π)sin(θ) when put into the formula above. Cos π we know equals -1, and when multiplied by cos(θ), you get -cos(θ). sin(π) we know equals 0, so when we multiply that by sin(θ), we get 0. -cos(θ)+0= cos(θ), which is the right-hand side (RHS).

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Part B: Use sum and difference identities to solve trigonometric equations. 

When it comes to solving for exact trig equations, you’ll probably see something like this:

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Sin(75) degrees can be split into 30 and 45 degrees, which are on the unit circle. We then plug this into our formula, then change the degrees to their radian counterparts, and solve/simplify. This would equal 84.

Example 2:

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Here’s a good video to help you understand Sum and Difference Identities

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Next, we will use the Power Pattern technique to solve trigonometric equations:

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The keyword when using the Power Pattern technique is “Isolate”, which means isolate the trig function and x. The difference between the 2 algebraic techniques is when the trig function is squared, we use the Power Pattern. Therefore, we know that after we isolate the trig function, the answer will be both positive and negative. This gives us 4 solutions for x. After finding the reference angle and plugging in the n-value for all quadrants, we have all 4 answers for x, as seen in the bottom half of the image below.

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Sum, Difference and Double Angle Identities on the AP Exam:

Here is an example of a problem you may get on this subject on an AP or IB exam:​

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As shown in the picture, we’re essentially doing the opposite of what we just did.

sin(α)cos(β) + cos(α)sin(β) equals sin(α + β), so we plug in the 65 and 25 to the smaller equations. That adds to sin90, which equals 1.

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Sum, Difference and Double Angle Identities in the Real World

You are designing a chair for your grandmother. While drawing up the blue prints you come across a problem, the angles of the chair are not adding up correctly. You need an exact measurement. You must find the cosine of this angle.

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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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