Trigonometry and Angles

Trigonometry—the study of triangles—connects angles to ratios that appear throughout mathematics, physics, engineering, and nature. 理解する these relationships unlocks everything from measuring heights to analyzing waves.

Right Triangle Trigonometry

For a right triangle with an acute angle θ:

The Three Main Functions

• Sine (sin): Ratio of opposite side to hypotenuse
• Cosine (cos): Ratio of adjacent side to hypotenuse
• Tangent (tan): Ratio of opposite side to adjacent side

Reciprocal Functions

• Cosecant (csc): 1/sin = Hypotenuse/Opposite
• Secant (sec): 1/cos = Hypotenuse/Adjacent
• Cotangent (cot): 1/tan = Adjacent/Opposite

Common Angle Values

These values come from special triangles: the 30-60-90 triangle (half an equilateral) and the 45-45-90 triangle (half a square).

The Unit Circle

The unit circle extends trigonometry beyond right triangles to all angles.

Definition

• Circle with radius 1, centered at origin
• Angle θ measured counterclockwise from positive x-axis
• Point on circle: (cos θ, sin θ)

Key Relationships

• cos θ = x-coordinate on unit circle
• sin θ = y-coordinate on unit circle
• tan θ = sin θ / cos θ = y/x

Quadrant Signs

• Quadrant I (0°-90°): All positive
• Quadrant II (90°-180°): Sin positive
• Quadrant III (180°-270°): Tan positive
• Quadrant IV (270°-360°): Cos positive

Fundamental Identities

Pythagorean Identity

sin²θ + cos²θ = 1

This comes directly from the Pythagorean theorem applied to the unit circle.

Angle Sum Formulas

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Double Angle Formulas

• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

Inverse Trigonometric Functions

Inverse functions find angles from ratios.

Notation

• arcsin (sin⁻¹): Find angle given sine
• arccos (cos⁻¹): Find angle given cosine
• arctan (tan⁻¹): Find angle given tangent

Ranges

• arcsin: [-90°, 90°] or [-π/2, π/2]
• arccos: [0°, 180°] or [0, π]
• arctan: (-90°, 90°) or (-π/2, π/2)

Example

If sin(θ) = 0.5, what is θ?

θ = arcsin(0.5) = 30° or π/6 radians

Practical Applications

Finding Heights

変換先 find the height of a building:

• Measure angle of elevation (α) from known distance (d)
• Height = d × tan(α)

Navigation

• Compass bearings use angles from north
• Vector components: x = r cos θ, y = r sin θ

Waves and Oscillations

• Sound waves: y = A sin(2πft)
• AC electricity: V = V₀ sin(ωt)
• Light: E = E₀ sin(kx - ωt)

Engineering

• Force components on inclined planes
• Stress analysis in structures
• Signal processing and filtering

Small Angle Approximations

For angles near zero (in radians):

• sin θ ≈ θ
• cos θ ≈ 1
• tan θ ≈ θ

Accuracy

These approximations simplify physics problems (pendulums, optics, etc.).

まとめ

Trigonometry connects angles to the ratios sine, cosine, and tangent—fundamental relationships that appear throughout science and engineering. Starting from right triangles (SOH-CAH-TOA) and extending through the unit circle, these functions describe everything from heights of buildings to electromagnetic waves. Mastering common angle values and key identities provides tools for countless applications.

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