Trigonometry Lecture: Understanding SOHCAHTOA and Trig Ratios

Introduction to SOHCAHTOA

• SOHCAHTOA: A mnemonic to remember the formulas for sine, cosine, and tangent ratios.
  • Sine (SO): (\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}})
  • Cosine (CA): (\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}})
  • Tangent (TOA): (\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}})

Example with a Right Triangle

• Given: A right triangle with sides 3, 4, 5.
  • (\theta) is the angle with:
    • Opposite: 4
    • Adjacent: 3
    • Hypotenuse: 5
  • Calculate Ratios:
    • (\sin \theta = \frac{4}{5})
    • (\cos \theta = \frac{3}{5})
    • (\tan \theta = \frac{4}{3})

Pythagorean Theorem for Right Triangles

• Formula: (a^2 + b^2 = c^2)
• Example: Sides 5, 12, hypotenuse 13.
  • Calculate missing side using (c^2 = a^2 + b^2).

Special Right Triangles to Remember

• 3-4-5 Triangle
• 5-12-13 Triangle
• 8-15-17 Triangle
• 7-24-25 Triangle
• 9-40-41 Triangle

Evaluating Trig Ratios in Different Quadrants

• Quadrant I: All trig functions positive.
• Quadrant II: Sine positive.
• Quadrant III: Tangent positive.
• Quadrant IV: Cosine positive.

Converting Between Degrees and Radians

• (\pi = 180^\circ)
• (\pi/2 = 90^\circ)

Using Special Angles and Reference Triangles

• 30-60-90 Triangle:
  • Sides: 1, (\sqrt{3}), 2
• 45-45-90 Triangle:
  • Sides: 1, 1, (\sqrt{2})

Reference Angles and Coterminal Angles

• Reference angles help calculate trig functions for any angle.
• Coterminal Angles: Add or subtract (360^\circ) to find equivalent angles.

Calculating Reciprocal Trig Functions

• Cosecant: (\csc \theta = \frac{1}{\sin \theta})
• Secant: (\sec \theta = \frac{1}{\cos \theta})
• Cotangent: (\cot \theta = \frac{1}{\tan \theta})

Examples and Problem Solving

• Problem Solving: How to solve problems involving trig ratios, reference angles, and special triangles.
• Evaluate Trig Functions Without a Calculator: Use memorized values or reference triangles for exact values.
• Expressions and Relationships: Remember "All Students Take Calculus" to determine the sign of trig functions in different quadrants.

Closing Remarks

• Further Study: Explore more examples and problems in trigonometry through suggested playlists and videos.
• Subscription Reminder: Subscribe for more educational content and updates.

This structured breakdown provides a comprehensive guide to understanding the fundamental concepts of trigonometry, specifically focusing on trig ratios, solving problems using special triangles, and navigating through different quadrants. It serves as a condensed study guide for students exploring trigonometry.