Calculate sine, cosine, and tangent for any angle in degrees or radians.
Sine, cosine, and tangent are ratios of the sides of a right triangle. For an angle θ in a right triangle: sin(θ) = opposite/hypotenuse (SOH), cos(θ) = adjacent/hypotenuse (CAH), tan(θ) = opposite/adjacent (TOA). The mnemonic SOH CAH TOA is how millions of students worldwide remember these definitions.
These ratios are defined with respect to a specific angle — the "opposite" and "adjacent" sides depend on which angle you're working with. The hypotenuse is always the longest side and always opposite the right angle — it never changes regardless of which angle you're analyzing. The opposite and adjacent sides swap roles depending on which acute angle is being referenced.
The unit circle (radius = 1, centered at the origin) extends trigonometric functions beyond the 0°-90° range of right triangles. For any angle θ measured counterclockwise from the positive x-axis, a point on the unit circle has coordinates (cos θ, sin θ). This definition allows trig functions to handle angles in all four quadrants and beyond 360°.
Key unit circle values worth memorizing: sin(0°)=0, cos(0°)=1; sin(30°)=½, cos(30°)=√3/2; sin(45°)=√2/2, cos(45°)=√2/2; sin(60°)=√3/2, cos(60°)=½; sin(90°)=1, cos(90°)=0. The pattern for sine: 0, ½, √2/2, √3/2, 1 — values corresponding to √0/2, √1/2, √2/2, √3/2, √4/2, revealing the elegant structure underlying these seemingly arbitrary values.
Degrees divide a full rotation into 360 parts — a historically convenient number with many divisors. Radians measure angles as the ratio of arc length to radius — a full circle is 2π radians because the circumference of a unit circle is 2π. The conversion: radians = degrees × π/180; degrees = radians × 180/π.
Radians are the natural unit for trigonometry in mathematics and physics. When you differentiate sin(x) in calculus, the simple result d/dx[sin(x)] = cos(x) holds only when x is in radians. In degrees, the derivative carries an additional π/180 factor that complicates every calculation. This is why all scientific and mathematical software defaults to radians.
Architecture and construction: calculating roof pitch angles, ramp gradients, and structural load distribution. Navigation: triangulating position using known landmarks, GPS satellite positioning uses trigonometry to solve for receiver location from multiple satellite distances. Physics: projectile motion, wave equations (sin and cos describe oscillations), alternating current analysis in electrical engineering. Computer graphics: all rotation, scaling, and perspective transformation in 3D rendering uses matrices built from sine and cosine values. Signal processing: Fourier analysis decomposes any periodic signal into sums of sine and cosine waves of different frequencies.