Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools

Trigonometry Symbols



Notation reference with LaTeX codes for trigonometric functions, identities, and related formulas. Use our Mathematical Keyboard.
Notation confusions·Algebraic traps·Domain & mode errors
basic trigonometric functionsinverse trigonometric functionstrigonometric identitieslaw of sines and cosinesangles and arc lengthsunit circle relationshyperbolic functionsinverse hyperbolic functionscomplex numberssum and difference identities
symbollatex codeexplanation
sin(θ)
\sin(\theta)
Sine function — ratio of the opposite side to the hypotenuse; equals the y-coordinate on the unit circle
cos(θ)
\cos(\theta)
Cosine function — ratio of the adjacent side to the hypotenuse; equals the x-coordinate on the unit circle
tan(θ)
\tan(\theta)
Tangent function — ratio of opposite to adjacent, or equivalently sin(θ)/cos(θ)
cot(θ)
\cot(\theta)
Cotangent function — ratio of adjacent to opposite; reciprocal of tangent
sec(θ)
\sec(\theta)
Secant function — ratio of hypotenuse to adjacent; reciprocal of cosine
csc(θ)
\csc(\theta)
Cosecant function — ratio of hypotenuse to opposite; reciprocal of sine
sin⁻¹(x)
\sin^{-1}(x)
Inverse sine (arcsine) — returns the angle whose sine is x, restricted to [−π/2, π/2]
cos⁻¹(x)
\cos^{-1}(x)
Inverse cosine (arccosine) — returns the angle whose cosine is x, restricted to [0, π]
tan⁻¹(x)
\tan^{-1}(x)
Inverse tangent (arctangent) — returns the angle whose tangent is x, restricted to (−π/2, π/2)
cot⁻¹(x)
\cot^{-1}(x)
Inverse cotangent (arccotangent) — returns the angle whose cotangent is x, restricted to (0, π)
sec⁻¹(x)
\sec^{-1}(x)
Inverse secant (arcsecant) — returns the angle whose secant is x, defined for |x| ≥ 1
csc⁻¹(x)
\csc^{-1}(x)
Inverse cosecant (arccosecant) — returns the angle whose cosecant is x, defined for |x| ≥ 1
sin²(θ) + cos²(θ) = 1
\sin^2(\theta) + \cos^2(\theta) = 1
Pythagorean identity — follows directly from the unit circle equation x² + y² = 1
1 + tan²(θ) = sec²(θ)
1 + \tan^2(\theta) = \sec^2(\theta)
Pythagorean identity for tangent and secant — obtained by dividing sin² + cos² = 1 by cos²
1 + cot²(θ) = csc²(θ)
1 + \cot^2(\theta) = \csc^2(\theta)
Pythagorean identity for cotangent and cosecant — obtained by dividing sin² + cos² = 1 by sin²
sin(2θ) = 2sin(θ)cos(θ)
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
Double angle identity for sine — derived from the angle sum formula with equal angles
cos(2θ) = cos²(θ) − sin²(θ)
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
Double angle identity for cosine — has two alternate forms using the Pythagorean identity
tan(2θ) = 2tan(θ) / (1 − tan²(θ))
\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}
Double angle identity for tangent — derived from the tangent sum formula
a/sin(A) = b/sin(B) = c/sin(C)
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
Law of Sines — relates side lengths to the sines of their opposite angles in any triangle
c² = a² + b² − 2abcos(C)
c^2 = a^2 + b^2 - 2ab\cos(C)
Law of Cosines — generalizes the Pythagorean theorem to any triangle
θ = s / r
\theta = \frac{s}{r}
Angle in radians — defined as the ratio of arc length to radius
s = rθ
s = r\theta
Arc length formula — requires θ in radians; no conversion constant needed
(x, y) = (cos(θ), sin(θ))
(x, y) = (\cos(\theta), \sin(\theta))
Coordinates of a point on the unit circle at angle θ define cosine and sine
tan(θ) = y / x
\tan(\theta) = \frac{y}{x}
Tangent as the ratio of unit circle coordinates; geometrically, the slope of the terminal side
sinh(x)
\sinh(x)
Hyperbolic sine — defined as (eˣ − e⁻ˣ)/2
cosh(x)
\cosh(x)
Hyperbolic cosine — defined as (eˣ + e⁻ˣ)/2
tanh(x)
\tanh(x)
Hyperbolic tangent — defined as sinh(x)/cosh(x)
coth(x)
\coth(x)
Hyperbolic cotangent — defined as cosh(x)/sinh(x), x ≠ 0
sech(x)
\text{sech}(x)
Hyperbolic secant — defined as 1/cosh(x)
csch(x)
\text{csch}(x)
Hyperbolic cosecant — defined as 1/sinh(x), x ≠ 0
sinh⁻¹(x)
\sinh^{-1}(x)
Inverse hyperbolic sine — defined for all real x; equals ln(x + √(x² + 1))
cosh⁻¹(x)
\cosh^{-1}(x)
Inverse hyperbolic cosine — defined for x ≥ 1; equals ln(x + √(x² − 1))
tanh⁻¹(x)
\tanh^{-1}(x)
Inverse hyperbolic tangent — defined for |x| < 1; equals ½ ln((1+x)/(1−x))
coth⁻¹(x)
\coth^{-1}(x)
Inverse hyperbolic cotangent — defined for |x| > 1; equals ½ ln((x+1)/(x−1))
sech⁻¹(x)
\text{sech}^{-1}(x)
Inverse hyperbolic secant — defined for 0 < x ≤ 1
csch⁻¹(x)
\text{csch}^{-1}(x)
Inverse hyperbolic cosecant — defined for x ≠ 0
z = r(cos(θ) + isin(θ))
z = r(\cos(\theta) + i\sin(\theta))
Trigonometric form of a complex number — r is the modulus, θ is the argument
eⁱᶿ = cos(θ) + isin(θ)
e^{i\theta} = \cos(\theta) + i\sin(\theta)
Euler's formula — connects exponential and trigonometric functions through the imaginary unit
zⁿ = rⁿ(cos(nθ) + isin(nθ))
z^n = r^n(\cos(n\theta) + i\sin(n\theta))
De Moivre's theorem — raises a complex number to the n-th power by scaling the modulus and multiplying the argument
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)
Angle sum identity for sine — expands sine of a sum into products of individual functions
sin(α − β) = sin(α)cos(β) − cos(α)sin(β)
\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)
Angle difference identity for sine — same structure as the sum, with signs reversed
cos(α + β) = cos(α)cos(β) − sin(α)sin(β)
\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)
Angle sum identity for cosine — expands cosine of a sum into products of individual functions
cos(α − β) = cos(α)cos(β) + sin(α)sin(β)
\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)
Angle difference identity for cosine — same structure as the sum, with signs reversed
tan(α + β) = (tan(α) + tan(β)) / (1 − tan(α)tan(β))
\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}
Angle sum identity for tangent — undefined when the denominator equals zero
tan(α − β) = (tan(α) − tan(β)) / (1 + tan(α)tan(β))
\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}
Angle difference identity for tangent — undefined when the denominator equals zero




Notation Confusions

Several trigonometric symbols look deceptively similar but carry entirely different meanings. Misreading them leads to wrong calculations even when the underlying method is correct.

The most persistent confusion is between sin⁡−1(x)\sin^{-1}(x) and 1sin⁡(x)\frac{1}{\sin(x)}. The superscript −1-1 on a trigonometric function does not mean "raise to the power −1-1." It denotes the inverse function: sin⁡−1(x)=arcsin⁡(x)\sin^{-1}(x) = \arcsin(x), which returns an angle. The reciprocal of sine is the cosecant: 1sin⁡(x)=csc⁡(x)\frac{1}{\sin(x)} = \csc(x). The notation arcsin⁡\arcsin avoids this trap entirely.

The expression sin⁡2(θ)\sin^2(\theta) means (sin⁡θ)2(\sin\theta)^2 — the square of the output. It does not mean sin⁡(θ2)\sin(\theta^2), which would apply the function to a squared input. These produce different values at almost every angle: sin⁡2(30°)=(12)2=14\sin^2(30°) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}, while sin⁡(30°2)=sin⁡(900°)=sin⁡(180°)=0\sin(30°^2) = \sin(900°) = \sin(180°) = 0.

The abbreviation cis(θ)\text{cis}(\theta) stands for cos⁡θ+isin⁡θ\cos\theta + i\sin\theta and appears in some textbooks for the trigonometric form of complex numbers. It is not a separate function — just shorthand. Not all sources use it.

The notation sinh⁡\sinh denotes the hyperbolic sine function, defined as ex−e−x2\frac{e^x - e^{-x}}{2}. It is not sine evaluated at a variable called hh. Hyperbolic functions are a separate family with their own identities and properties.

The letter ee in Euler's formula eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta is Euler's number, the mathematical constant approximately equal to 2.718282.71828. It is not a variable and cannot be treated as one in algebraic manipulation.

The notation ∣z∣|z| applied to a complex number means the modulus — the distance from zz to the origin in the complex plane: ∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}. This extends the real-number absolute value but is not the same as "removing the sign."

Algebraic Traps

Trigonometric functions do not obey the algebraic rules that apply to multiplication and addition. Treating them as if they do produces errors that no amount of careful arithmetic can fix.

The most common mistake: sin⁡(α+β)≠sin⁡α+sin⁡β\sin(\alpha + \beta) \neq \sin\alpha + \sin\beta. Sine does not distribute over addition. The correct expansion is the angle sum identity: sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta. The same applies to cosine and tangent — none of them distribute.

The expression sin⁡(2θ)\sin(2\theta) is not the same as 2sin⁡(θ)2\sin(\theta). The first applies sine to a doubled argument; the second doubles the output of sine. They coincide only at isolated angles. The correct relationship is the double angle identity: sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta.

The expressions sin⁡2θ+cos⁡2θ\sin^2\theta + \cos^2\theta and sin⁡(2θ)+cos⁡(2θ)\sin(2\theta) + \cos(2\theta) look similar at a glance but are unrelated. The first is the Pythagorean identity and always equals 11. The second is a sum of trigonometric functions at a doubled angle and varies with θ\theta.

The ±\pm sign in half-angle formulas is not a free choice. It is determined by the quadrant of θ2\frac{\theta}{2}. For sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}, use ++ when θ2\frac{\theta}{2} falls in a quadrant where sine is positive, and −- when it falls where sine is negative. The formula provides the magnitude; the quadrant provides the sign.

Domain and Mode Errors

Input and output conventions in trigonometry carry strict requirements that are easy to overlook — and the consequences are silently wrong answers, not error messages.

The degree symbol matters. sin⁡(30)\sin(30) with no degree symbol means sine of 3030 radians — approximately −0.988-0.988. sin⁡(30°)\sin(30°) means sine of 3030 degrees — exactly 12\frac{1}{2}. When no unit symbol is present, radians are assumed. This mismatch is the most common source of calculator errors: the calculator is in the wrong mode, the answer looks plausible enough to go unquestioned, and the error propagates through subsequent work.

Radians carry no unit symbol precisely because they are dimensionless — the ratio of two lengths. The absence of a symbol is itself the indicator. When an expression like sin⁡(π/6)\sin(\pi/6) appears, the π/6\pi/6 is in radians. When an expression like sin⁡(30°)\sin(30°) appears, the degree symbol is explicit. Mixing the two in a single computation is a reliable path to error.

The inverse trigonometric functions take numbers as inputs and return angles as outputs. The domain of arcsin⁡\arcsin is [−1,1][-1, 1] — only numbers in this interval have a corresponding angle. The output of arcsin⁡\arcsin is an angle in [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]. Students frequently reverse these, attempting to compute arcsin⁡(2)\arcsin(2) (undefined) or expecting arcsin⁡\arcsin to return a number outside its range.

The composition cos⁡−1(cos⁡(x))=x\cos^{-1}(\cos(x)) = x holds only when x∈[0,π]x \in [0, \pi]. Outside this interval, the inverse "folds" the result back into its restricted range. For example, cos⁡−1(cos⁡(5π3))=cos⁡−1(12)=π3\cos^{-1}(\cos(\frac{5\pi}{3})) = \cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}, not 5π3\frac{5\pi}{3}. The same caution applies to arcsin⁡(sin⁡(x))\arcsin(\sin(x)) outside [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

Parentheses in trigonometric notation are optional for single variables — sin⁡θ\sin\theta and sin⁡(θ)\sin(\theta) are identical. But for compound expressions, parentheses are mandatory. sin⁡α+β\sin\alpha + \beta means (sin⁡α)+β(\sin\alpha) + \beta, while sin⁡(α+β)\sin(\alpha + \beta) applies sine to the entire sum. Omitting parentheses does not merely look informal — it changes the mathematical meaning.