CCSS.Math.Content.HSF-TF.C.9
The Standard
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students derive angle formulas using coordinates, rotations, or earlier identities. They use the formulas to find exact values, verify identities, and solve equations or applied problems.
What Mastery Looks Like
- Students can build a valid proof from unit-circle coordinates, rotations, or known identities. They can select the correct formula and calculate exact values. They can also state restrictions when tangent is undefined.
Common Misconceptions
- Students often switch the signs in the sine and cosine formulas. They may expand tangent without using sine divided by cosine. They also overlook values that make a tangent denominator zero.
How to Assess It
- Exit ticket: derive the formula for cos(α − β), then use an angle formula to find the exact value of sin 75°. Require each algebraic step.
Lesson moves
Ways to Teach It
Place tracing paper over a unit circle, rotate a marked vector by a second angle, and record how its coordinates change.
Ask students to explain in writing why cosine of a difference has a plus sign while sine of a difference has a minus sign.
Run a card match pairing angle expressions such as sin 75° with formula expansions and exact values.
Model two successive heading changes for a drone, then use angle sums to calculate the final horizontal and vertical components.
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Printable HSF-TF.C.9 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSF-TF.C.9, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSF-TF.A.2
Addition formulas need sine and cosine as unit-circle coordinates for arbitrary angle measures, not just triangle ratios.
- CCSS.Math.Content.HSF-TF.C.8
The Pythagorean identity and quadrant reasoning support manipulating sine, cosine, and tangent in the more advanced angle formulas.
Keep exploring
Related Standards
- CCSS.Math.Content.HSF-TF.B.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms...
- 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
Prove and apply trigonometric identities
- CCSS.Math.Content.HSG-SRT.D.11
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, result...
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