Common Core Standards: Math
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given find sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Students should know that the squared sine and squared cosine functions complete each other. Seriously. The equation sin2(θ) + cos2(θ) = 1 will hold true for any angle θ. You might want to avoid students' confusion by first telling them that sin2(θ) is the same as (sinθ)2.
Given a reference triangle and the SOHCAHTOA definitions of sine and cosine, students should be able to prove the Pythagorean identity. Or at least get a vague sense of understanding as to where this identity comes from. If students are really struggling with the proof, walk students through it step by step using a lot of pretty pictures of triangles.
Once they've proved the identity, students can use it to find sinθ, cosθ, or tanθ when given a value for sinθ, cosθ, or tanθ and the quadrant of the angle. If they ever slip up, remind them of ASTC. Several examples also help.
Students should know that we can use Pythagorean identities to find missing trigonometric values. That should be obvious, since that's what they'll most likely be doing anyway. But they should also know that these identities help simplify trigonometric expressions. This may seem like an unnecessarily complicated way of simplifying, but trust us. They'll thank you later.
In calculus (prepare for screams of horror from your students), they'll need these identities when integrating functions. They're just necessary stepping stones along the river of math that leads to calculus—their dream destination.
See? All students really do take calculus.