Overview
This lecture presents the top 10 must-know concepts in trigonometry, including similar triangles, trig ratios, key laws, special triangles, the unit circle, functions, radian measure, identities, and solving equations.
Similar Triangles
- Similar triangles have equal corresponding angles and proportional corresponding sides.
- Proving similarity can be done by Angle-Angle (AA), Side-Side-Side (SSS), or Side-Angle-Side (SAS).
- Proportion equations can solve for unknown sides or angles in similar triangles.
SOHCAHTOA (Primary Trig Ratios)
- SOHCAHTOA is an acronym for sine (sin = opposite/hypotenuse), cosine (cos = adjacent/hypotenuse), and tangent (tan = opposite/adjacent).
- These ratios allow calculation of missing sides or angles in right triangles using a calculator for values.
Sine Law and Cosine Law
- Sine Law: a/sinA = b/sinB = c/sinC, used when knowing two angles and one side, or two sides and a non-included angle.
- Cosine Law: c ² = a² + b² - 2ab cosC, used for non-right triangles when two sides and the included angle, or all three sides, are known.
- Both laws solve for unknown sides or angles in oblique triangles.
Special Triangles
- Isosceles right triangle (45-45-90): side lengths 1-1-√2, exact trig ratios for 45°.
- Equilateral triangle split in half (30-60-90): side lengths 1-√3-2, exact ratios for 30° and 60°.
- Use these triangles for precise values of sin, cos, and tan at special angles.
The Unit Circle and CAST Rule
- The unit circle is a circle of radius 1, centered at the origin, linking coordinates to sine and cosine values.
- CAST Rule identifies which trig ratios are positive in each quadrant (All, Sine, Tan, Cosine).
- Reference angles and their placement on the unit circle determine sign and value for trig ratios.
Trig Ratios for Angles > 90°
- For angles > 90°, use reference angles, CAST rule, and special triangles to find exact trig values.
- E.g., sin(150°) = sin(30°) = 1/2, cos(225°) = -cos(45°) = -√2/2.
Sine and Cosine as Functions
- Sine and cosine are periodic functions with amplitude 1 and period 360°.
- Their graphs oscillate between -1 and 1, repeating every 360°.
Radians
- Radian measure: angle subtended by an arc equal in length to the radius.
- 2π radians = 360°; to convert degrees to radians, multiply by π/180.
Trigonometric Identities
- Reciprocal identities: cosec x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x.
- Quotient identity: tan x = sin x / cos x.
- Pythagorean identity: sin²x + cos²x = 1.
- Identities are true for all x and can be used to prove or simplify other equations.
Solving Trig Equations
- Use reference angles, CAST rule, and knowledge of the unit circle to solve trig equations within given domains.
- For quadratic trig equations, factor and solve for all possible solutions, considering periodicity.
Key Terms & Definitions
- Similar Triangles — Triangles with equal corresponding angles and proportional sides.
- SOHCAHTOA — Mnemonic for primary trig ratios (sin, cos, tan).
- Unit Circle — Circle of radius 1, center at origin, relates angles to (cos, sin).
- CAST Rule — Acronym for which trig ratios are positive in each quadrant.
- Radians — Unit of angle measure based on arc length and radius.
- Trigonometric Identity — Equation true for all angle values, relating trig functions.
Action Items / Next Steps
- Practice solving trig equations and using sine/cosine laws for non-right triangles.
- Memorize special triangle ratios and the CAST rule.
- Convert between degrees and radians as practice.
- Review and apply trigonometric identities in simplifying expressions.