Calculus How To

Double Angle Formulas: Sin, Cos, Tan

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Contents:

  1. Sin double angle formula
  2. Cos double angle formula
  3. Tan double angle formula

Double angle formulas are called “double” angle because they involve trigonometric functions of double angles like sin 2x or cos 2x. In other words, when we use the formula, we’re doubling the angle. These formulas allow you to rewrite a double angle expression like sin 2x and cos 2θ into an expression with a single angle. This means that instead of working with a complicated expression, you can transform it to a simpler one. As well as their ability to help us simplify and solve equations, they can verify identities and are needed for some aspects of calculus involving analytic geometry [1].

List of Double Angle Formulas

Sin double angle formula

The double angle formula for sin is:
sin double angle formula

Cos double angle formula

The cosine has three forms of the double angle formula. Each is derived from the Pythagorean theorem, so you only need to know one (the rest can be derived from the formula and the Pythagorean theorem) [2]. That said, one might suit the format of a particular problem than the others. For example, if you have a lot of sines in an equation, use the first form; Choose whichever one results in fewer calculations for you:
cos double angle formula

Tan double angle formula

tan double angle formula

Example

Example question: Factor and simplify the following expression:
cos4θ – sin4θ.

Step 1: Factor the expression:
The question asks us to factor, so let’s do that first to get it into a “double angle formula”- friendly format:
factor cos sin expression

Step 2: Simplify:

  1. Use the basic trigonometric identity cos2 x + sin2 x = 1 to replace the left-hand side:
    substituting trig identities
  2. Use the first cos double angle formula to replace the right-hand side of the formula:applying the cos double angle formula

The solution is cos 2θ.

References

[1] Double Angle Formulas and Half-Angle Formulas. Retrieved July 11, 2021 from: https://www.math.utah.edu/lectures/math1060/2-5PostNotes.pdf
[2] Joyce, D. (1996). Summary of trigonometric identities. Retrieved July 11, 2021 from: https://www2.clarku.edu/faculty/djoyce/trig/identities.html

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