Trigonometry studies relationships between angles and sides of triangles, especially right triangles, and extends to periodic functions on the unit circle.
Key concepts
- Angles: measured in degrees or radians.
- Right-triangle ratios: sine (sin), cosine (cos), tangent (tan) defined as opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent.
- Reciprocals: cosecant (csc = 1/sin), secant (sec = 1/cos), cotangent (cot = 1/tan).
- Unit circle: defines trig functions for all real angles; coordinates (cos θ, sin θ) on circle radius 1.
- Identities: Pythagorean (sin²θ + cos²θ = 1), angle-sum/difference (sin(a±b), cos(a±b)), double-angle, half-angle.
- Inverse functions: arcsin, arccos, arctan for recovering angles from ratios.
- Graphs: sine and cosine are periodic with period 2π; tangent has period π with vertical asymptotes.
- Applications: waves and oscillations, signal processing, navigation, surveying, engineering, computer graphics.
Basic example
- For a right triangle with hypotenuse 10 and angle θ where sin θ = 0.6, opposite = 6, adjacent = 8 (since 6²+8²=10²), so cos θ = 0.8 and tan θ = 0.75.
If you want, I can explain unit circle visuals, derive identities, solve example problems, or show how to compute values with a portable calculator.
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