CCSS.Math.Content.HSG-SRT.D.10
The Standard
(+) Prove the Laws of Sines and Cosines 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 each law from known results, such as altitude-based right triangles, the Pythagorean theorem, or triangle area formulas. They choose the appropriate law from the given side-angle information and solve for missing sides or angles. They also check the ambiguous SSA case and reject impossible answers.
What Mastery Looks Like
- A student can produce a connected algebraic proof, label opposite side-angle pairs correctly, and explain each substitution. Given SSS, SAS, ASA, AAS, or SSA information, the student selects a law, solves accurately, and reports all valid triangles. Answers include units, suitable precision, and a check against the diagram and angle sum.
Common Misconceptions
- Students often mismatch each side with its opposite angle in sine ratios. With SSA data, they may accept only the calculator's acute inverse-sine result and miss a possible supplementary angle. They may also call measurements from one diagram a proof or use the wrong law for SAS and SSS data.
How to Assess It
- Exit ticket: Draw an altitude in a general triangle and use it to derive one form of the Law of Sines. Then find b when A = 40°, B = 65°, and a = 8.
Lesson moves
Ways to Teach It
On graph paper, draw an oblique triangle, drop an altitude, and use labeled right triangles to derive the Law of Cosines algebraically.
Ask: Why does the Law of Cosines become the Pythagorean theorem when the included angle is 90 degrees?
Run a card sort matching SSS, SAS, ASA, AAS, and SSA data sets to a law, two solutions, one solution, or none.
Use two measured bearings and a known baseline to calculate a river width, then compare the result with a scaled map.
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Printable HSG-SRT.D.10 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-SRT.D.10, with an answer key for the teacher on its own page. No account needed.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSG-SRT.C.6
Right-triangle sine and cosine definitions are needed to prove and apply sine and cosine relationships in non-right triangles.
- CCSS.Math.Content.HSG-SRT.C.8
Right triangle sine, cosine, and the Pythagorean Theorem are used to derive and apply the Laws of Sines and Cosines.
- CCSS.Math.Content.HSG-SRT.D.9
The triangle area formula with sine supports a common proof of the Law of Sines and reinforces using included angles.
What This Unlocks
Mastery here sets students up for these next.
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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.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
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