Simple Identities (DP IB Applications & Interpretation (AI): HL): Revision Note

Simple identities

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

• Two trigonometric identities you must know are

  • $\tan \theta =\frac{\sin \theta }{\cos \theta }$

    • This is the identity for tan θ

  • ${\sin }^2\theta +{\cos }^2\theta =1$

    • This is the Pythagorean identity

    • Note that the notation $\sin {}^2\theta$ is the same as ${\left(\sin \theta \right)}^2$

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

• The Pythagorean identity is found using Pythagoras' theorem

  • ${\left(\sin \theta \right)}^2+{\left(\cos \theta \right)}^2=1$

• The tan identity is found using the definition

  • $\tan \theta =\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}=\frac{\sin \theta }{\cos \theta }$

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

  • ${\sin }^2\theta = {1-\cos }^2\theta$

  • ${\cos }^2\theta = {1-\sin }^2\theta$