Trigonometry: Ratios, Functions, and Triangle Solving
Trigonometry relates the angles of a triangle to the ratios of its sides. The three primary functions — sine, cosine, and tangent — are the foundation of physics, engineering, navigation, and signal processing.
SOH-CAH-TOA (Right Triangles)
sin(θ) = Opposite ÷ Hypotenuse
cos(θ) = Adjacent ÷ Hypotenuse
tan(θ) = Opposite ÷ Adjacent = sin(θ) ÷ cos(θ)Key Angle Values
- sin(0°)=0, sin(30°)=0.5, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1
- cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=0.5, cos(90°)=0
- tan(45°)=1, tan(60°)=√3≈1.732, tan(30°)=1/√3≈0.577
Solving Any Triangle
Law of Sines (any triangle):
a/sin(A) = b/sin(B) = c/sin(C)Law of Cosines (SAS or SSS):
c² = a² + b² − 2ab×cos(C)Inverse Functions
arcsin (sin⁻¹): gives angle from sin ratio
arccos (cos⁻¹): gives angle from cos ratio
arctan (tan⁻¹): gives angle from tan ratioCalculate trig functions: Free Trigonometry Calculator
Key Angle Values to Know
- 0°: sin=0, cos=1, tan=0
- 30°: sin=0.5, cos=√3/2≈0.866, tan=1/√3≈0.577
- 45°: sin=cos=√2/2≈0.707, tan=1
- 60°: sin=√3/2≈0.866, cos=0.5, tan=√3≈1.732
- 90°: sin=1, cos=0, tan=undefined
Real-World Applications
Trigonometry is the mathematics of angles and is indispensable in engineering, architecture, navigation, and physics. Surveyors calculate distances across rivers and valleys using the tangent function. Architects design ramps (tan θ = rise/run), roof pitches, and staircase angles. GPS receivers use trigonometric calculations to determine position from satellite distances. Sound engineers model wave interference with sine and cosine functions. Bridge and skyscraper designs rely on trigonometric analysis of forces. Even animation and game development use sine and cosine for smooth oscillatory motion.
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees divide a full circle into 360 equal parts (historical, from Babylonian astronomy). Radians measure angle by the arc length subtended on a unit circle: one full circle = 2π radians ≈ 6.283 rad. Radians are the natural unit in calculus because the derivative of sin(x) is cos(x) only when x is in radians. To convert: radians = degrees × π/180; degrees = radians × 180/π.
When should I use sin, cos, or tan?
In a right triangle: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. The mnemonic SOH-CAH-TOA helps. Use sin or cos when the hypotenuse is known; use tan when you know two legs and want the angle or the third side without using the hypotenuse.
What is the law of cosines and when is it used?
The law of cosines extends the Pythagorean theorem to non-right triangles: c² = a² + b² - 2ab·cos(C). It is used when you know three sides (to find an angle) or two sides and the included angle (to find the third side). For right triangles (C=90°), cos(90°)=0, and the formula reduces to the Pythagorean theorem.