CCSS.Math.Content.HSF-TF.C.8
The standard
Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Common Core State Standards for Mathematics
What this standard means
Students need to connect the unit circle to the identity that the square of sine plus the square of cosine equals 1. They should be able to explain why it is true using x, y coordinates and the Pythagorean Theorem, not just memorize it.
Mastery means students can find a missing trig value from one given value, then choose the correct sign using the quadrant. They also need to move between sine, cosine, and tangent. Common trouble spots are forgetting signs by quadrant, taking both square root answers, and treating tangent like it fits directly into the identity.
Ways to teach it
- Have students label points on a large unit circle and use x squared plus y squared equals 1 to prove the identity.
- Ask students to explain why cosine is negative but sine is positive for an angle in Quadrant II.
- Give four exit ticket problems asking for a missing trig value from one given value and a quadrant.
- Use a ramp diagram with rise, run, and angle to connect sine, cosine, tangent, and sign limits in context.
Plan a lesson for CCSS.Math.Content.HSF-TF.C.8
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Related standards
- CCSS.Math.Content.HSG-SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
- CCSS.Math.Content.HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
- CCSS.Math.Content.HSF-TF.C
Prove and apply trigonometric identities
- CCSS.Math.Content.HSG-SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.