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Introduction to Trigonometry

Trigonometry studies the relationship between angles and side lengths in triangles. It appears in physics, engineering, architecture, music, and computer graphics — anywhere waves, rotations, or spatial relationships appear.

In this lesson

  1. The three trig ratios
  2. SOH-CAH-TOA
  3. The unit circle
  4. Key identities
  5. Applications

1 The Three Trig Ratios

In a right triangle with an angle θ, the three sides are: the hypotenuse (longest side, opposite the right angle), the opposite side (across from angle θ), and the adjacent side (next to angle θ).

The three primary trig ratios define relationships between these sides:

The three primary trig ratios
sinsine(θ) = opposite / hypotenuse
coscosine(θ) = adjacent / hypotenuse
tantangent(θ) = opposite / adjacent = sin(θ)/cos(θ)

2 SOH-CAH-TOA

The mnemonic SOH-CAH-TOA helps remember: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Using Trig Ratios
In a right triangle, angle θ = 30°, hypotenuse = 10. Find the opposite and adjacent sides.
1sin(30°) = opposite/hypotenuse → opposite = 10 × sin(30°) = 10 × 0.5 = 5
2cos(30°) = adjacent/hypotenuse → adjacent = 10 × cos(30°) = 10 × (√3/2) ≈ 8.66
Answer: Opposite = 5, adjacent ≈ 8.66

Key angles to memorize: sin(30°)=0.5, cos(30°)=√3/2, tan(30°)=1/√3. sin(45°)=√2/2, cos(45°)=√2/2, tan(45°)=1. sin(60°)=√3/2, cos(60°)=0.5, tan(60°)=√3.

3 The Unit Circle

The unit circle is a circle of radius 1 centered at the origin. For any angle θ, the point where the terminal side of θ intersects the unit circle has coordinates (cos θ, sin θ). This extends trig beyond 0°-90° to all angles.

This is why sin and cos range from −1 to +1: they are coordinates on a unit circle. At 0°: (cos 0°, sin 0°) = (1, 0). At 90°: (0, 1). At 180°: (−1, 0). At 270°: (0, −1). These coordinates are exact values, not approximations.

Radians vs Degrees

Radians are the natural unit for angles in mathematics and physics. 2π radians = 360°. π radians = 180°. π/2 radians = 90°. To convert: degrees × π/180 = radians. Radians appear in calculus because d/dx[sin(x)] = cos(x) only when x is in radians.

4 Key Identities

Pythagorean identity: sin²θ + cos²θ = 1. This follows directly from the Pythagorean theorem applied to the unit circle.

Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Even/odd: cos(−θ) = cos θ (even — symmetric about y-axis). sin(−θ) = −sin θ (odd — antisymmetric).

sin²θ means (sin θ)², not sin(θ²)

The notation sin²θ is shorthand for (sin θ)² — square the result of sine, not the input. This matters for the Pythagorean identity: sin²θ + cos²θ = 1 means (sin θ)² + (cos θ)² = 1.

5 Applications

Navigation: bearings and headings use angles. Finding distances between GPS coordinates uses trig. The haversine formula for great-circle distance is entirely trigonometric.

Physics: wave motion (sound, light, water waves) is described by sinusoidal functions y = A·sin(ωt + φ). Simple harmonic motion (springs, pendulums) involves sine and cosine. Alternating current in electrical engineering is sinusoidal.

Computer graphics: rotations are implemented using trig. Rotating a point (x, y) by angle θ gives (x cos θ − y sin θ, x sin θ + y cos θ). Every 3D game uses this hundreds of times per frame.

Practice Problems

In a right triangle, the opposite side is 7 and hypotenuse is 25. Find sin θ, cos θ, and tan θ.
Adjacent = √(25²−7²) = √(625−49) = √576 = 24. sin θ = 7/25 = 0.28, cos θ = 24/25 = 0.96, tan θ = 7/24 ≈ 0.292
Convert 135° to radians.
135 × π/180 = 3π/4 ≈ 2.356 radians
If sin θ = 3/5, find cos θ using the Pythagorean identity.
sin²θ + cos²θ = 1. (3/5)² + cos²θ = 1. 9/25 + cos²θ = 1. cos²θ = 16/25. cos θ = 4/5