Enter any angle to get all six trig functions, the reference angle, quadrant, and exact values for standard angles, plus a Pythagorean/sum-difference/double-angle identity reference.
The unit circle is the single picture that ties all six trigonometric functions together: sine and cosine are literally the y- and x-coordinates of a point at angle θ on a circle of radius 1, and the other four functions (tangent, cosecant, secant, cotangent) are all built from ratios of those two. This tool evaluates all six at any angle and pairs the result with the identities most useful for simplifying trig expressions in engineering calculations.
For a point on a circle of radius 1 at angle θ (measured counterclockwise from the positive x-axis), cos θ is the x-coordinate and sin θ is the y-coordinate. The other four functions follow directly: tan θ = sin θ / cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ = cos θ/sin θ. Because sin θ and cos θ are just coordinates on a circle, their values repeat every 360° — trig functions are inherently periodic.
The reference angle is the acute angle (0°-90°) between the terminal side of θ and the x-axis — it tells you the magnitude of the trig ratios, while the quadrant tells you their sign. Quadrant I: all six functions positive. Quadrant II: only sine and cosecant positive. Quadrant III: only tangent and cotangent positive. Quadrant IV: only cosine and secant positive — the classic "All Students Take Calculus" mnemonic for which functions are positive in each quadrant.
The most-used identity in all of trigonometry follows directly from the Pythagorean theorem applied to the unit circle: sin²θ + cos²θ = 1. Dividing through by cos²θ gives 1 + tan²θ = sec²θ; dividing through by sin²θ gives 1 + cot²θ = csc²θ. These three identities let you convert between any two of sine, cosine, tangent, secant, and cosecant without needing a calculator.
Sum/difference: sin(A±B) = sinA·cosB ± cosA·sinB, and cos(A±B) = cosA·cosB ∓ sinA·sinB. Double-angle (set A = B above): sin(2θ) = 2·sinθ·cosθ, and cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. These identities are how you evaluate non-standard angles exactly (e.g., 75° = 45° + 30°) and how oscillation and wave problems in engineering simplify products of sinusoids into sums.
tan θ and sec θ are undefined wherever cos θ = 0 (at 90° and 270°), since both involve dividing by cos θ. Similarly, cot θ and csc θ are undefined wherever sin θ = 0 (at 0° and 180°). These correspond to vertical asymptotes in the graphs of tangent, cotangent, secant, and cosecant.
Multiples of 30° and 45° (0°, 30°, 45°, 60°, 90°, and their reflections around the circle) have exact values expressible with simple square roots — these are the angles that appear constantly in textbook problems and the FE exam because they don't require a calculator to evaluate precisely.
Any oscillating or rotating engineering quantity — AC voltage, mechanical vibration, wave motion, rotating machinery phase angles — is naturally described with sine and cosine functions of time or angle, and the identities above are exactly what let you combine, simplify, and analyze sums of such signals (like combining two AC voltage phases).
The reference angle is what you actually look up in a table of standard values — you find the magnitude from the reference angle, then apply the correct sign based on which quadrant the original angle falls in. This two-step process (reference angle for magnitude, quadrant for sign) is faster and less error-prone than memorizing separate values for every angle around the full circle.