Simple Identities (DP IB Analysis & Approaches (AA)): Revision Note

Simple identities

What are the trigonometric identities relating sin, cos and tan?

  • Two trigonometric identities you must know are

    • tan space theta space equals space fraction numerator sin space theta over denominator cos space theta end fraction

      • This is the identity for tan θ

    • sin squared theta space plus space cos squared theta space equals space 1

      • This is the Pythagorean identity

      • Note that the notation sin space squared theta is the same as left parenthesis sin space theta right parenthesis to the power of space 2 end exponent

Examiner Tips and Tricks

Both identities can be found in the formula booklet.

How do I derive these identities?

  • Draw a right-angled triangle with a hypotenuse equal to 1

  • Label one of the other angles θ

  • Use right-angled trigonometry to find the lengths of the other sides

    • The side adjacent to this angle is cosθ

    • The side opposite to this angle is sinθ

Right-angled triangle labelled with hypotenuse 1, angle theta, adjacent side as cos(theta) in blue, and opposite side as sin(theta) in red.
  • The Pythagorean identity is found using Pythagoras' theorem

    • open parentheses sin theta close parentheses squared plus open parentheses cos theta close parentheses squared equals 1

  • The tan identity is found using the definition

    • tan theta equals Opposite over Adjacent equals fraction numerator sin theta over denominator cos theta end fraction

Examiner Tips and Tricks

You do not need to learn these derivations, but they might help you with your internal assessment.

How are the trigonometric identities used?

  • Trigonometric identities are used to change an equation into a form that allows it to be solved

  • Rearranging the Pythagorean identity often makes it easier to work with

    • sin squared theta equals blank 1 minus space cos to the power of 2 space end exponent theta

    • cos squared theta equals blank 1 minus space sin squared theta

Examiner Tips and Tricks

If you are asked to show that one thing is identical to another, look at what parts are missing –  for example, if tan x has gone it must have been substituted

Worked Example

Show that the equation 2 sin squared space x minus cos space x equals 0 can be written in the form a cos squared space x plus b cos space x plus c equals 0, where a, b and c are integers to be found.

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