CCSS.Math.Content.HSF-TF.C

CCSS.Math.Content.HSF-TF.C is a cluster heading. These are the individual standards under it.

• CCSS.Math.Content.HSF-TF.C.8

  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.

• CCSS.Math.Content.HSF-TF.C.9

  (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

01

What Students Need to Do

Students show that two trigonometric expressions are equal by transforming one side with known identities and algebra. They use those relationships to simplify expressions and find exact values while tracking quadrants and restrictions.

02

What Mastery Looks Like

Students can transform one side of an equation into the other using known identities and valid algebra. They can find exact values, choose signs by quadrant, and state restrictions.

03

Common Misconceptions

Students often treat an identity as an equation to solve for an angle. They may cancel terms across sums or choose the wrong sign for a square root. Some ignore undefined values or think checking one angle proves equality.

04

How to Assess It

Exit ticket: Simplify (1 − cos²x)/sin x and state where the original expression is defined. Then find cos θ if sin θ = 3/5 and θ is in quadrant II.

1. 01

   Use a paper unit circle and right triangles to label x, y, and radius, then derive x² + y² = 1.

2. 02

   Ask students why dividing by sin²x creates restrictions, then compare the valid domains in two sample derivations.

3. 03

   Run an identity card sort matching 1 − sin²x, cos²x, and other equivalent forms, with students justifying every match.

4. 04

   Have students find the vertical component of a 100 newton force at 75° by expanding sin(45° + 30°).

• CCSS.Math.Content.HSG-CO.C

  Prove geometric theorems

• CCSS.Math.Content.HSG-SRT.D.10

  (+) Prove the Laws of Sines and Cosines and use them to solve problems.

• CCSS.Math.Content.HSG-SRT.D

  Apply trigonometry to general triangles