All of TRIGONOMETRY in 36 minutes! (top 10 must knows)

Introduction

This tutorial condenses essential trigonometry concepts covered in a 36-minute video by JensenMath. Whether you're preparing for an exam or just looking to refresh your knowledge, this guide will walk you through the top ten must-know topics in trigonometry, providing clear explanations and practical tips.

Step 1: Understand Similar Triangles

• Similar triangles have the same shape but may differ in size.
• Key properties:
  • Corresponding angles are equal.
  • Ratios of corresponding sides are constant.
• Application: Use similar triangles to find unknown lengths in geometry problems.

Step 2: Learn SOHCAHTOA

• SOHCAHTOA is a mnemonic to remember the definitions of sine, cosine, and tangent:
  • Sine (SOH): Opposite side / Hypotenuse
  • Cosine (CAH): Adjacent side / Hypotenuse
  • Tangent (TOA): Opposite side / Adjacent side
• Application: Use these ratios to solve for missing sides or angles in right triangles.

Step 3: Master Sine and Cosine Law

• Sine Law: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )
  • Useful for non-right triangles.
• Cosine Law: ( c^2 = a^2 + b^2 - 2ab \cdot \cos C )
  • Helps find a side or angle when given two sides and the included angle.
• Application: Apply these laws to solve for sides and angles in various triangle configurations.

Step 4: Recognize Special Triangles

• Special triangles include:
  • 30-60-90 triangle: Ratios are ( 1:\sqrt{3}:2 )
  • 45-45-90 triangle: Ratios are ( 1:1:\sqrt{2} )
• Application: Use these ratios to quickly solve problems without a calculator.

Step 5: Utilize the Unit Circle and CAST Rule

• The unit circle is a circle with a radius of 1 centered at the origin.
• The CAST rule indicates the sign of trigonometric functions in each quadrant:
  • Quadrant I: All positive
  • Quadrant II: Sine positive
  • Quadrant III: Tangent positive
  • Quadrant IV: Cosine positive
• Application: Use the unit circle to determine the values of sine, cosine, and tangent for key angles.

Step 6: Analyze Ratios for Angles Greater Than 90 Degrees

• For angles greater than 90 degrees, reference angles (the acute angle formed with the x-axis) can be used.
• The values of sine and cosine will change based on the quadrant.
• Application: Use these concepts to find trigonometric function values for larger angles.

Step 7: Explore Sine and Cosine Functions

• Sine and cosine functions are periodic with a period of ( 2\pi ).
• Their graphs display:
  • Amplitude: Maximum height from the midline.
  • Frequency: Number of cycles in a given interval.
• Application: Use graphs to visualize function behaviors and solve trigonometric equations.

Step 8: Understand Radians

• Radians are another way to measure angles, where ( 2\pi ) radians equals 360 degrees.
• To convert degrees to radians: ( \text{Radians} = \frac{\text{Degrees} \times \pi}{180} )
• Application: Use radians in calculations involving trigonometric functions and the unit circle.

Step 9: Learn Trigonometric Identities

• Key identities include:
  • Pythagorean Identity: ( \sin^2(x) + \cos^2(x) = 1 )
  • Angle Sum and Difference Identities:
    • ( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B )
    • ( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B )
• Application: Use these identities to simplify expressions and solve equations.

Step 10: Solve Trigonometric Equations

• To solve equations, isolate the trigonometric function and use inverse functions.
• Check for all possible solutions within the given range.
• Application: Apply these techniques in problems involving angles and sides in trigonometric contexts.

Conclusion

This guide covers the essential concepts of trigonometry, from foundational principles like similar triangles to advanced topics such as solving trigonometric equations. To deepen your understanding, practice applying these concepts through exercises and real-world applications. Consider using online resources for additional practice and clarification as you continue your trigonometry journey.