Students expand on their knowledge of trigonometry, developing a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between functions.
Unit 7, Trigonometric Identities and Equations, builds on the previous unit on trigonometric functions to expand students’ knowledge of trigonometry. Students develop a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between trigonometric functions. Reasoning flexibly about trigonometric functions and seeing that expressions that look different on the surface can actually act the same on certain domains sets the stage for a study of differentiation and integration, where periodic functions have many useful properties and act as useful tools to study calculus.
Students also apply algebraic techniques to trigonometry. This part of the unit reinforces algebraic skills while also helping students to better understand trigonometric functions graphically and through the unit circle. As students move more flexibly between representations of trigonometric functions, they develop skills in seeing structure in those functions and practice looking at mathematical objects from multiple perspectives and bringing prior knowledge to bear on a new context. This type of relational thinking helps students to see the power of algebraic manipulation and structure in expressions, allowing them to work more flexibly and to see connections more readily.
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Identity | Pythagorean identity |
Cofunction identities | Inverse trig functions (arcsin, arccos, arctan, arcsec, arccsc, arccot) |
Reciprocal identities | $${\mathrm{sin}^{-1}x}$$ notation for inverse trig functions |
Negative angle identities | Linear trigonometric equations |
$$u$$-substitution | Quadratic trigonometric equations |
General solution | Exact solution |
Double angle formula | Sum formula |
Difference formula | Law of Cosines |
Law of Sines |
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.
F.TF.C.8
Derive and verify trigonometric identities using transformations and equivalence of functions.
F.TF.C.8
Derive and use the Pythagorean identity to write equivalent expressions.
F.TF.C.8
Verify trigonometric identities using Pythagorean and reciprocal identities.
F.TF.B.6
F.TF.B.7
Find angle measures using inverse trig functions in right triangles.
F.IF.C.7.E
F.BF.B.4.D
F.TF.B.6
F.TF.B.7
Analyze inverse trigonometric functions graphically.
F.TF.B.7
Solve linear trigonometric equations.
F.TF.B.7
Solve linear trigonometric equations using $$u$$-substitution.
F.TF.B.7
Use inverse trigonometric functions to solve contextual problems.
F.TF.B.6
F.TF.B.7
Solve quadratic trigonometric equations.
F.TF.B.7
F.TF.C.8
Solve trigonometric equations using identities.
F.TF.C.9
Evaluate expressions using sum and difference formulas.
F.TF.C.9
Solve equations and prove identities using sum and difference formulas.
F.TF.C.9
Derive double angle formulas and use them to solve equations and prove identities.
F.TF.C.8
F.TF.C.9
Use trigonometric identities to analyze graphs of functions.
Key: Major Cluster Supporting Cluster Additional Cluster
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