About

Cofunction identities relate a trigonometric function of an angle to the corresponding function of its complement. The complement of an angle θ is 90° − θ (or π2 − θ in radians). Misidentifying cofunctions leads to incorrect phase shifts in signal processing, wrong triangle solutions in surveying, and flawed structural load decompositions. This calculator computes all six trigonometric functions for any input angle and simultaneously evaluates their cofunctions at the complementary angle, verifying the identities sin(θ) = cos(90° − θ) and the analogous relations for tangent - cotangent and secant - cosecant pairs.

The tool handles edge cases where functions are undefined (e.g., tan(90°) or csc(0°)). Results assume standard Euclidean geometry. For angles outside 0° - 90°, the complement may be negative; the identities still hold algebraically but lose geometric meaning as complementary angles of a right triangle. Precision is limited to IEEE 754 double-precision floating-point arithmetic (~15 significant digits).

Formulas

The six cofunction identities state that each trigonometric function of an angle equals the cofunction of its complement:

sin(θ) = cos(90° − θ)

cos(θ) = sin(90° − θ)

tan(θ) = cot(90° − θ)

cot(θ) = tan(90° − θ)

sec(θ) = csc(90° − θ)

csc(θ) = sec(90° − θ)

In radian form, the complement of θ is π2 − θ. The reciprocal functions are defined as:

cot(θ) = 1tan(θ)

sec(θ) = 1cos(θ)

csc(θ) = 1sin(θ)

Where θ = the input angle, and 90° − θ = the complementary angle. Degree-to-radian conversion uses: θ_rad = θ_deg × π180.

Reference Data

Angle θ	sin(θ)	cos(θ)	tan(θ)	cot(θ)	sec(θ)	csc(θ)
0°	0	1	0	undef	1	undef
15°	0.2588	0.9659	0.2679	3.7321	1.0353	3.8637
30°	0.5	0.8660	0.5774	1.7321	1.1547	2
36°	0.5878	0.8090	0.7265	1.3764	1.2361	1.7013
45°	0.7071	0.7071	1	1	1.4142	1.4142
54°	0.8090	0.5878	1.3764	0.7265	1.7013	1.2361
60°	0.8660	0.5	1.7321	0.5774	2	1.1547
72°	0.9511	0.3090	3.0777	0.3249	3.2361	1.0515
75°	0.9659	0.2588	3.7321	0.2679	3.8637	1.0353
90°	1	0	undef	0	undef	1
120°	0.8660	−0.5	−1.7321	−0.5774	−2	1.1547
135°	0.7071	−0.7071	−1	−1	−1.4142	1.4142
150°	0.5	−0.8660	−0.5774	−1.7321	−1.1547	2
180°	0	−1	0	undef	−1	undef
270°	−1	0	undef	0	undef	−1
360°	0	1	0	undef	1	undef

Frequently Asked Questions

The tangent function equals sin(θ) ÷ cos(θ). At θ = 90°, cos(90°) = 0, producing a division by zero. The function approaches ±∞ from either side. Similarly, cot(0°), sec(90°), and csc(0°) are undefined for analogous reasons. This calculator marks such cases explicitly rather than displaying unreliable large numbers.

Yes, algebraically. The identities sin(θ) = cos(90° − θ) and their counterparts are valid for all real values of θ. However, the geometric interpretation as complementary angles in a right triangle only applies when 0° < θ < 90°. For angles outside this range, the complement (90° − θ) becomes negative or exceeds 90°, which is still mathematically valid but no longer represents a physical triangle.

IEEE 754 double-precision provides approximately 15-17 significant decimal digits. Near critical angles like 90° or 0°, functions like tan or csc approach infinity, and small rounding errors are magnified. For example, cos(90°) may compute as ~6.12 × 10⁻¹⁷ rather than exactly 0 due to the irrational nature of π. This calculator rounds display output to the selected decimal places but uses full double-precision internally.

The prefix "co-" in cosine, cotangent, and cosecant literally means "complement of." Cosine is the sine of the complement, cotangent is the tangent of the complement, and cosecant is the secant of the complement. This naming convention dates to Edmund Gunter's work in the early 1600s. The cofunction identity is therefore a tautology embedded in the naming itself.

This calculator accepts decimal radian input. To enter π/6, compute the decimal approximation: π/6 ≈ 0.5236. The preset buttons provide exact standard angles (30°, 45°, 60° equivalents). For maximum precision at standard angles, use degree mode, which avoids the compounding error of first approximating π and then computing the trig function.

Secant and cosecant are simply the reciprocals of cosine and sine respectively. Calculator manufacturers omit them because sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ) require no dedicated hardware logic. This tool computes them directly, displays them alongside their cofunctions, and handles the undefined cases (division by zero) that manual reciprocal calculation often misses.