Trigonometry

Trigonometry

Learn trigonometric ratios, standard angles, quadrants, the unit circle, and law of sines/cosines.

Lessons included in this chapter:

• Angles — The Foundation: A full circle is 360 degrees. Acute is < 90°, right is 90°, obtuse is > 90°, straight is 180°.
• Radians — The Natural Unit: 180° equals exactly π radians. Radians match the radius length to circle arcs.
• Right-Angled Triangles: Hypotenuse is always the longest side. Opposite is across from the angle; adjacent is next to it.
• The Six Trigonometric Ratios: sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj. The secondary ratios are just their flips.
• Standard Trigonometric Values: A simple root progression model helps memorise sin 0° to 90° from √0/2 to √4/2.
• The Unit Circle: For any angle θ, x = cos θ and y = sin θ. Quadrant flags determine the signs.
• Trigonometric Graphs: Sine and cosine are smooth waves bounded between -1 and 1. Tangent shoots to infinity.
• The Pythagorean Identity: It is just a² + b² = c² mapped to the coordinates of a unit circle.
• Complementary & Even-Odd Identities: sin(90-θ) = cos θ; cos is an even function (eats negative signs) while sine is odd.
• Angle Sum Formulas: sin(A+B) and cos(A+B) mix and match terms according to compound geometric projections.
• Double-Angle Formulas: A special compound result when A = B. It forms the base for resolving wave frequencies.
• Heights and Distances: elevation matches the gaze going up; depression matches the gaze looking down.
• Laws for Any Triangle: a/sin A = b/sin B = c/sin C for proportions; c² = a² + b² - 2ab cos C when SAS or SSS applies.
• Inverse Trigonometric Functions: sin⁻¹(x) takes a ratio and returns the primary principal angle. Pay attention to range constraints.