Overview
This lecture is a condensed introduction to trigonometry, covering key concepts like right triangles, the Pythagorean theorem, SOHCAHTOA, the unit circle, trigonometric laws, and fundamental identities.
Right Angle Triangles & Pythagorean Theorem
- A right triangle has one 90° angle.
- The Pythagorean theorem states: a² + b² = c², where c is the hypotenuse.
- Hypotenuse is always the longest side, opposite the right angle.
- Only positive values for lengths are used in geometry.
SOHCAHTOA & Trig Functions
- SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Use inverse trig functions to find angles given side lengths.
- Formula triangles help visualize which ratios to use.
The Unit Circle & Radians
- The unit circle has radius 1; used to define sine and cosine as coordinates.
- 1 radian subtends an arc equal to radius; full circle = 2π radians = 360°.
- Degrees to radians: degrees × (π/180); radians to degrees: radians × (180/π).
- Coordinates of points on the unit circle: (cos θ, sin θ).
Special Triangles & Gradients
- 30-60-90 and 45-45-90 triangles yield common sine/cosine values.
- Tangent of an angle (tan θ) is the slope (gradient) of the line from the origin to (cos θ, sin θ).
Laws of Sines and Cosines
- Law of Sines: sin A/a = sin B/b = sin C/c, used when two angles and a side or two sides and a non-enclosed angle are known.
- Law of Cosines: c² = a² + b² − 2ab cos C, used when two sides and the enclosed angle or all three sides are known.
Trigonometric Identities
- Pythagorean Identity: sin²θ + cos²θ = 1.
- Reciprocal Identities: 1/sinθ = cosecθ; 1/cosθ = secθ; 1/tanθ = cotθ.
- Angle Addition/Subtraction:
- sin(A±B) = sinA cosB ± cosA sinB
- cos(A±B) = cosA cosB ∓ sinA sinB
- tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB)
- Double Angle Identities:
- sin2θ = 2 sinθ cosθ
- cos2θ = cos²θ − sin²θ
- tan2θ = 2 tanθ / (1 − tan²θ)
Key Terms & Definitions
- Hypotenuse — the longest side of a right triangle, opposite the right angle.
- Radians — angular unit where one radian is the angle subtended by an arc equal to the circle’s radius.
- Unit Circle — a circle with radius 1, centered at the origin, used for defining trig functions.
- Pythagorean Identity — sin²θ + cos²θ = 1; relates the sine and cosine of an angle.
- SOHCAHTOA — mnemonic for remembering trig ratios.
Action Items / Next Steps
- Practice solving right triangles using SOHCAHTOA and the Pythagorean theorem.
- Convert angles between degrees and radians.
- Solve problems using the laws of sines and cosines.
- Memorize key trigonometric identities.