Trigonometry/Precalculus Study Notes

Trigonometry/Precalculus: Trig #3 Summative (7.1-7.5)

Student Information

  • Name: [Student Name]

  • Date: [Current Date]

Priority Success Criteria

  • Does Not Meet (1): Successfully demonstrates 1-3 of the Priority Success Criteria

  • Approaches (2): Successfully demonstrates 4-6 out of 7 Priority Success Criteria

  • Meets (3): Successfully demonstrates all Priority Success Criteria with 90% accuracy

  • Exceeds (4): Complete all 7 success criteria with 90% accuracy (Standard for Mathematical Practice 6: Attend to Precision)

Success Criteria Overview

  1. Find the exact values of an inverse trigonometric function (SC1)

  2. Find the exact values of a composite function (SC2)

  3. Identify all the possible solutions of trigonometric functions in general form (SC3)

  4. Isolate the trigonometric term using inverse operations (SC4)

  5. Solving trigonometric equations using inverse trigonometric operations (SC5)

  6. Solving trigonometric equations involving multiple angles (SC6)

  7. Use identities to solve trigonometric equations (SC7)

Detailed Success Criteria

SC1: Exact Values of Inverse Trigonometric Functions

  • To find the exact value of an inverse trigonometric function, identify the principal value of the function using its respective range.

Example:

  • Find: $\tan^{-1}(-1)$

    • Solution: The angle whose tangent is -1 is $\frac{3\pi}{4}$ or another equivalent angle depending on the context of the problem.

  • Find: $\csc^{-1}(-2)$

    • Solution: Since $\csc(x) = \frac{1}{\sin(x)}$, the corresponding angle where the sine gives $-\frac{1}{2}$ is at $\frac{7\pi}{6}$ or $\frac{11\pi}{6}$.

SC2: Exact Values of Composite Functions

  • A composite function is evaluated by substituting the inner function into the outer function.

Example:

  • Find: $\sin[\sin(\frac{11\pi}{5})]$

    • Solution: To solve this, first simplify $\frac{11\pi}{5}$ by reducing it within the range of $[0, 2\pi)$ which results in $\frac{11\pi}{5} - 2\pi = \frac{11\pi}{5} - \frac{10\pi}{5} = \frac{\pi}{5}$, hence $\sin(\frac{\pi}{5})$.

  • Find: $\sin[\cos^{-1}(\frac{1}{2})]$

    • Solution: The angle whose cosine is $\frac{1}{2}$ is $\frac{\pi}{3}$ (located in the first quadrant), thus $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.

Additional Considerations

  • Understand the periodicity and symmetry in trigonometric functions to identify all possible solutions in different quadrants.

  • Master the use of trigonometric identities to simplify the process of solving equations and finding exact values.