Trigonometric Identities and Equations

• Derive the Pythagorean identity of $${{\mathrm{sin}^{2}x+\mathrm{cos}^2x=1}}$$ by connecting sines and cosines on the unit circle with right triangles.
• Use appropriate notation to write powers of trigonometric functions.
• Derive the remaining Pythagorean identities through manipulation of $${{\mathrm{sin}^{2}x+\mathrm{cos}^2x=1}}$$.
• Identify any domain restrictions that follow from the manipulation of Pythagorean identities. Describe why these domain restrictions exist. 
• Use Pythagorean identities to evaluate trigonometric functions.

• Students will likely struggle with the conventions of how to write the square of trigonometric ratios, and there are a bunch of common errors that can result from this misunderstanding. Require that students use precise mathematical notation so as not to get confused. Here is a great Dr. Math article from The Math Forum on the topic: “Interpreting Expressions Involving Cosine Squared.”
• Quick Questions on Proving Trig Identities by Sam Shah is an article on proving trig identities that is useful to read before going through this with students.

April claims that $${1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}={1\over{\mathrm{sin}^2({{\theta}})}}}$$ is an identity for all real numbers $${{\theta}}$$ that follows from the Pythagorean identity.

1. For which values of $${{\theta}}$$ are the two functions $$f({{\theta}})=1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}$$ and $$g({{\theta}})={1\over{\mathrm{sin}^2({{\theta}})}}$$ defined?
2. Show that the equation $${1+{{\mathrm{cos}^2({{\theta}})}\over{\mathrm{sin}^2({{\theta}})}}={1\over{\mathrm{sin}^2({{\theta}})}}}$$ follows from the Pythagorean identity.