trigonometric, identities, and graphing graphing graphing grap

Trigonometric Identities and Graphing

1. Basic Trigonometric Functions

  • Addition of Angles:
    • Alpha over two added to ascent alpha gives:
      4623\frac{4\sqrt{6}}{23}

2. Cosecant and Secant Identities

  • For cosecant or secant of the double angle:
    • Write it as a reciprocal.
    • Example Answer:
      −169120\frac{-169}{120}

3. Identities Involving Cosines

  • Identity:
    • ±1+cos⁡(α)/2\pm 1 + \cos(\alpha) / 2
    • Example Answer:
      5−2610\frac{\sqrt{5} - 2\sqrt{6}}{10}

4. Expanded Tangent Identity

  • Expanded Identity for Tangent:
    • tan⁡(β)−tan⁡(α)1+tan⁡(β)tan⁡(α)\frac{\tan(\beta) - \tan(\alpha)}{1 + \tan(\beta)\tan(\alpha)}
    • Simply write this identity without substitutions.

5. Cosine Double Angle Identity

  • Cosine Double Angle Identity:
    • cos⁡(2α)=cos⁡2(α)−sin⁡2(α)\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)
    • Misunderstanding mentioned:
    • Incorrect: cos⁡2(α)+sin⁡2(α)\cos^2(\alpha) + \sin^2(\alpha)
    • Correct: 11 (Pythagorean Identity)

6. Sine Double Angle Identity

  • Identity for Sine:
    • sin⁡(β/2)=±(1−cos⁡(β))2\sin(\beta / 2) = \frac{\pm(1 - \cos(\beta))}{2}
    • Example Answer:
      2626\frac{\sqrt{26}}{26}

7. Alternative Tangent Form

  • Using sine version for tangent:
    • Symptoms listed:
    • sin⁡(α)1+cos⁡(α)\frac{\sin(\alpha)}{1 + \cos(\alpha)} or 1−cos⁡(α)sin⁡(α)\frac{1 - \cos(\alpha)}{\sin(\alpha)}.
    • Example Answer:
      5+265 + 2\sqrt{6}

8. Expanded Cosine Identity

  • Expanded Cosine Identity:
    • cos⁡(β)cos⁡(α)−sin⁡(β)sin⁡(α)\cos(\beta)\cos(\alpha) - \sin(\beta)\sin(\alpha)
    • Example Answer:
      −246+565\frac{-24\sqrt{6} + 5}{65}

9. Cotangent Identity

  • Cotangent is the reciprocal of tangent:
    • Example Answer:
      −54\frac{-5}{4}

10. Sine Addition Formula

  • Sine Addition Formula:
    • sin⁡(α+β)=sin⁡(α)cos⁡(β)+cos⁡(α)sin⁡(β)\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)
    • Example Answer:
      12+10665\frac{12 + 10\sqrt{6}}{65}

11-15: Various Trigonometric Identities

  • 11: Cosine Double Angle Identity:
    • Identity with answer presented.
  • 12: Sine Difference Identity:
    • Identity with answer presented.
  • 13: Double Angle Times Factor:
    • Example:
      Four times the double angle\text{Four times the double angle}
  • 14: Half the double angle.
  • 15: Tangent Sum Identity:
    • Example Answer: Negative value presented

16-19: Specific Values and Calculations

  • Example Answers include calculations for specific values such as:
    • Negative Value:
      2−32 - \sqrt{3}
  • Range of Answers for question 17 with correct values.

20-22: Co-functions and Homework Assignment

  • Co-functions include values for angles such as:
    • sin⁡(79),cot⁡(3π28),sin⁡(29π66),cos⁡(76∘)\sin(79), \cot(\frac{3\pi}{28}), \sin(\frac{29\pi}{66}), \cos(76^{\circ})
  • Homework layout identical to the test in format and number of problems.

23-36: Structuring for Test Preparation

  • A strict necessity is highlighted for correct memorization of identities for tests. Specificity for questions approaching is essential.
  • Amplitude and Period must be calculated for various assignments:
    • Graphing involves determining shifts.
  • All graphical representations require checking original points to avoid inaccuracies.

Graphing Transformations and Periodicity

  • Basic Steps:
    1. Identify amplitude and period.
    2. Determine phase shift and vertical shift.
    3. Tick marks and labelling.

Summary for Test Prep

  • Focus on reviewing trigonometric identities and functions.
  • Graphs should reflect transformations cleanly.
  • Mastery of these will be critical for performance on upcoming examinations:
    • Memorize identities thoroughly; Failing to understand them effectively can impact grade negatively.