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:
    \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:
    \frac{-169}{120}

3. Identities Involving Cosines

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

4. Expanded Tangent Identity

• Expanded Identity for Tangent:
  • \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(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)
  • Misunderstanding mentioned:
  • Incorrect: \cos^2(\alpha) + \sin^2(\alpha)
  • Correct: 1 (Pythagorean Identity)

6. Sine Double Angle Identity

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

7. Alternative Tangent Form

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

8. Expanded Cosine Identity

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

9. Cotangent Identity

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

10. Sine Addition Formula

• Sine Addition Formula:
  • \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)
  • Example Answer:
    \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:
    \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 - \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(\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.