How to Calculate an Angle Using Reciprocal Trigonometric Functions
How to Prove Complex Identities by Working Individual Sides of a Trig Proof
How to Combine Reference Angles with Other Techniques to Solve Trigonometric Equations

How to Express Sums or Differences of Trigonometric Functions as Products

It's a good idea to familiarize yourself with a set of formulas that change sums to products. Sum-to-product formulas are useful to help you find the sum of two trig values that aren't on the unit circle. Of course, these formulas work only if the sum or difference of the two angles ends up being an angle from the special triangles:

image0.png

Here are the sum/difference-to-product identities:

image1.png

For example, say you're asked to find

image2.png

without a calculator. You're stuck, right? Well, not exactly. Because you're asked to find the sum of two trig functions whose angles aren't special angles, you can change this to a product by using the sum to product formulas. Follow these steps:

  1. Change the sum to a product.

    Because you're asked to find the sum of two sine functions, use this equation:

    image3.png

    This step gives you

    image4.png
  2. Simplify the result.

    Combining like terms and dividing gives you

    image5.png

    Those angles are represented on the unit circle, so continue to the next step.

    The whole unit circle
    The whole unit circle
  3. Use the unit circle to simplify further.

    image7.png

    Substituting those values in, you get

    image8.png
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