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Top 10 Must-Know Trigonometry Concepts

This summary outlines ten essential concepts in trigonometry, covering foundational principles, key formulas, and their applications.

1. Similar Triangles

• Definition: Triangles with the same shape but potentially different sizes.
• Properties:
  • Corresponding angles are equal.
  • Ratios of corresponding sides are equal.
• Applications: Solving for missing sides and angles in triangles.
• Proving Similarity:
  • Angle-Angle (AA): Two pairs of equal angles.
  • Side-Side-Side (SSS): Ratios of all three pairs of sides are equal.
  • Side-Angle-Side (SAS): Two equal ratios of side pairs and one matching pair of angles.

2. SOHCAHTOA

• Purpose: An acronym to remember the three primary trigonometric ratios in a right triangle.
• Definitions (with respect to a reference angle θ):
  • Sine (SOH): sin(θ) = Opposite / Hypotenuse
  • Cosine (CAH): cos(θ) = Adjacent / Hypotenuse
  • Tangent (TOA): tan(θ) = Opposite / Adjacent
• Applications:
  • Finding a missing side length.
  • Finding a missing angle (using inverse trigonometric functions like arcsin, arccos, arctan).

3. Sine Law and Cosine Law

• Purpose: Used to solve for missing sides and angles in non-right (oblique) triangles.
• Cosine Law:
  • Solving for a side: c² = a² + b² - 2ab cos(C) (when two sides and the included angle are known).
  • Solving for an angle: cos(θ) = (b² + c² - a²) / 2bc (when all three sides are known; 'a' is the side opposite angle θ).
• Sine Law: a/sin(A) = b/sin(B) = c/sin(C)
  • Scenarios for Use:
    • Two sides and an angle opposite one of them are known.
    • Two angles and one side are known.

4. Special Triangles

• Purpose: To find exact values of trigonometric ratios for specific angles.
• 45-45-90 Triangle (Isosceles Right Triangle):
  • Angles: 45°, 45°, 90°
  • Side Ratios: 1 : 1 : √2 (legs : hypotenuse)
  • Exact Values:
    • sin(45°) = 1/√2 = √2/2
    • cos(45°) = 1/√2 = √2/2
    • tan(45°) = 1
• 30-60-90 Triangle (Half Equilateral Triangle):
  • Angles: 30°, 60°, 90°
  • Side Ratios: 1 : √3 : 2 (opposite 30° : opposite 60° : hypotenuse)
  • Exact Values:
    • sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3
    • sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3

5. CAST Rule and the Unit Circle

• Unit Circle: A circle with a radius of 1 centered at the origin.
  • Key Concept: For any angle θ, the point where its terminal arm intersects the unit circle is (cos θ, sin θ).
  • Coordinates: Points on the unit circle at 90° intervals are (1,0), (0,1), (-1,0), (0,-1).
• CAST Rule: An acronym to remember which trigonometric ratios are positive in each quadrant of the Cartesian plane.
  • Quadrant I (0°-90°): All positive.
  • Quadrant II (90°-180°): Sine positive.
  • Quadrant III (180°-270°): Tangent positive.
  • Quadrant IV (270°-360°): Cosine positive.

6. Finding Exact Values for Angles > 90°

• Method: Combine the concept of reference angles with the CAST rule.
  1. Determine the quadrant of the principal angle.
  2. Use the CAST rule to determine if the ratio is positive or negative.
  3. Find the reference angle (the acute angle between the terminal arm and the closest x-axis).
  4. Calculate the trigonometric ratio of the reference angle using special triangles.
  5. Apply the sign determined by the CAST rule.

7. Sine and Cosine as Functions

• Concept: Sine and cosine can be viewed as functions where the input is an angle (x) and the output is the trigonometric ratio (y).
• Graphs:
  • Sine Function (y = sin x): A periodic wave oscillating between -1 and 1.
  • Cosine Function (y = cos x): A similar periodic wave, shifted horizontally from the sine wave.
• Key Properties:
  • Amplitude: Half the distance between the maximum and minimum y-values (for basic sine and cosine, amplitude is 1).
  • Period: The horizontal length of one complete cycle (for basic sine and cosine, the period is 360° or 2π radians).

8. Radians

• Definition: An alternative unit for measuring angles, based on the ratio of arc length to radius.
• Calculation: Angle in radians = Arc Length / Radius.
• Key Equivalences:
  • A full circle is 360° or 2π radians.
  • 180° = π radians.
• Conversion:
  • Degrees to Radians: Multiply by π / 180°.
  • Radians to Degrees: Multiply by 180° / π.

9. Trigonometric Identities

• Definition: Equations that are true for all values of the variable.
• Fundamental Identities:
  • Reciprocal Identities:
    • csc(x) = 1 / sin(x)
    • sec(x) = 1 / cos(x)
    • cot(x) = 1 / tan(x)
  • Quotient Identities:
    • tan(x) = sin(x) / cos(x)
    • cot(x) = cos(x) / sin(x)
  • Pythagorean Identity:
    • sin²(x) + cos²(x) = 1
• Application: Proving other trigonometric identities by manipulating both sides of an equation until they are equivalent.

10. Solving Trigonometric Equations

• Process: Finding the angle(s) that satisfy a given trigonometric equation.
• Steps often involve:
  1. Using special triangles and the CAST rule to find reference angles and potential quadrants.
  2. Determining the principal angles within a specified domain (e.g., 0 to 2π).
  3. Expressing the general solution by adding multiples of the period (e.g., + 2πk, where k is an integer) if the domain is unrestricted.
  4. Factoring quadratic trigonometric expressions.