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Trigonometry Formulas

Definitions
Reciprocal Identities
Even Functions
Pythagorean Identities
Odd Functions
Co-Function Identities

Definitions

Sine Function (sin)



Formula:

sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}
In a right-angled triangle, the sine of an angle θ\theta is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. It's a fundamental trigonometric function used to relate angles to side lengths.
Adjacent Side Opposite Side Hypotenuse θ
Sine of angle θ\theta
Side opposite to angle θ\theta
The longest side opposite the right angle
If the opposite side is 3 units and the hypotenuse is 5 units, then sin⁡θ=35=0.6\sin \theta = \frac{3}{5} = 0.6
Calculating heights, distances, and in wave functions.

Cosine Function (cos)



Formula:

cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}
In a right-angled triangle, the cosine of an angle θ\theta is the ratio of the length of the adjacent side to the angle to the length of the hypotenuse. It's essential for relating side lengths to angles.
Adjacent Side Opposite Side Hypotenuse θ
Cosine of angle θ\theta
Side adjacent to angle θ\theta
The longest side opposite the right angle
If the adjacent side is 4 units and the hypotenuse is 5 units, then cos⁡θ=45=0.8\cos \theta = \frac{4}{5} = 0.8
Determining horizontal distances and in harmonic motion.

Tangent Function (tan)



Formula:

tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}
In a right-angled triangle, the tangent of angle θ\theta is the ratio of the length of the opposite side to the length of the adjacent side. It relates the two sides that form the right angle.
Adjacent Side Opposite Side Hypotenuse θ
Tangent of angle θ\theta
Side opposite to angle θ\theta
Side adjacent to angle θ\theta
If the opposite side is 3 units and the adjacent side is 4 units, then tan⁡θ=34=0.75\tan \theta = \frac{3}{4} = 0.75
Calculating slopes, angles of elevation or depression.

Cosecant Function (csc)



Formula:

csc⁡θ=HypotenuseOpposite Side=1sin⁡θ\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} = \frac{1}{\sin \theta}
The cosecant function is the reciprocal of the sine function. It relates the hypotenuse to the opposite side.
Adjacent Side Opposite Side Hypotenuse θ

θ csc θ y = 1 sin θ 1 -1 1 -1
Cosecant of angle θ\theta
The longest side opposite the right angle
Side opposite to angle θ\theta
If the hypotenuse is 5 units and the opposite side is 3 units, then csc⁡θ=53≈1.6667\csc \theta = \frac{5}{3} \approx 1.6667
Used when the sine value is small and its reciprocal is needed.

Secant Function (sec)



Formula:

sec⁡θ=HypotenuseAdjacent Side=1cos⁡θ\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} = \frac{1}{\cos \theta}
The secant function is the reciprocal of the cosine function. It relates the hypotenuse to the adjacent side.
Adjacent Side Opposite Side Hypotenuse θ

θ sec θ x = 1 cos θ 1 -1 1 -1
Secant of angle θ\theta
The longest side opposite the right angle
Side adjacent to angle θ\theta
If the hypotenuse is 5 units and the adjacent side is 4 units, then sec⁡θ=54=1.25\sec \theta = \frac{5}{4} = 1.25
Useful in scenarios where the cosine value is small.

Cotangent Function (cot)



Formula:

cot⁡θ=Adjacent SideOpposite Side=1tan⁡θ\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} = \frac{1}{\tan \theta}
The cotangent function is the reciprocal of the tangent function. It relates the adjacent side to the opposite side.
Adjacent Side Opposite Side Hypotenuse θ
Cotangent of angle θ\theta
Side adjacent to angle θ\theta
Side opposite to angle θ\theta
If the adjacent side is 4 units and the opposite side is 3 units, then cot⁡θ=43≈1.3333\cot \theta = \frac{4}{3} \approx 1.3333
Applied in trigonometric calculations where the tangent value is large.

Reciprocal Identities

Secant Reciprocal Identity



Formula:

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
The secant function is the reciprocal of the cosine function. Geometrically, on a unit circle, it represents the distance from the center to where a line from the origin at angle θ intersects the secant line (vertical line at x = 1). This relationship is fundamental in calculus and trigonometric substitutions.
sec⁡θ=1cos⁡θ=hypotenuseadjacent\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}
Used in calculus, especially in integration involving trigonometric functions
The Pythagorean identity sec⁡2θ=1+tan⁡2θ\sec^2 \theta = 1 + \tan^2 \theta can be derived using this

Cosecant Reciprocal Identity



Formula:

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}
The cosecant function is the reciprocal of the sine function. On a unit circle, it represents the distance from the origin to where a line at angle θ intersects the cosecant line (horizontal line at y = 1). This relationship mirrors the secant identity but works with the perpendicular component.
csc⁡θ=1sin⁡θ=hypotenuseopposite\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}
Essential in integration techniques and solving differential equations
Used to derive the Pythagorean identity csc⁡2θ=1+cot⁡2θ\csc^2 \theta = 1 + \cot^2 \theta

Cotangent Reciprocal Identity



Formula:

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}
The cotangent function is both the reciprocal of tangent and the ratio of cosine to sine. This dual nature makes it particularly useful in simplifying expressions. Unlike secant and cosecant which reference the hypotenuse, cotangent relates the two legs of the right triangle directly.
cot⁡ heta=\ rac1 an heta=\ rac1\ racsin⁡ hetacos⁡ heta=\ raccos⁡ hetasin⁡ heta\cot \ heta = \ rac{1}{\ an \ heta} = \ rac{1}{\ rac{\sin \ heta}{\cos \ heta}} = \ rac{\cos \ heta}{\sin \ heta}
Useful in solving trigonometric equations and in calculus integration
Can be used to convert between tangent and cotangent expressions

Pythagorean Identities

Primary Pythagorean Identity



Formula:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
This fundamental identity comes directly from the Pythagorean theorem applied to the unit circle. When you place a point P on a unit circle, its coordinates (cos⁡ heta,sin⁡ heta)(\cos \ heta, \sin \ heta) form a right triangle. Since the radius (hypotenuse) is 1, the squares of sine and cosine must sum to 1. This becomes the foundation for deriving most other trigonometric identities.

{[-100, -50, 50, 100].map(coord => ( ))} sin θ cos θ 1 θ 1 -1 1 -1

From Pythagorean theorem: x2+y2=r2x^2 + y^2 = r^2 on unit circle where r=1r=1
Fundamental in calculus, physics, and simplifying complex trig expressions
All other trig identities can be derived from this one

Tangent Pythagorean Identity



Formula:

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta
This identity emerges when you divide the primary identity by cos⁡2 heta\cos^2 \ heta. Since  an heta=\ racsin⁡ hetacos⁡ heta\ an \ heta = \ rac{\sin \ heta}{\cos \ heta} and sec⁡ heta=\ rac1cos⁡ heta\sec \ heta = \ rac{1}{\cos \ heta}, dividing sin⁡2 heta+cos⁡2 heta=1\sin^2 \ heta + \cos^2 \ heta = 1 by cos⁡2 heta\cos^2 \ heta gives us this relationship.
sin⁡2θ+cos⁡2θcos⁡2θ=1cos⁡2θ\frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}
sin⁡2θcos⁡2θ+1=sec⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta} + 1 = \sec^2 \theta
tan⁡2θ+1=sec⁡2θ\tan^2 \theta + 1 = \sec^2 \theta
Useful in calculus, especially when working with tangent derivatives
Can be used to simplify expressions involving tangent and secant

Cotangent Pythagorean Identity



Formula:

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta
This identity is obtained by dividing the primary identity by sin⁡2 heta\sin^2 \ heta. Since cot⁡ heta=\ raccos⁡ hetasin⁡ heta\cot \ heta = \ rac{\cos \ heta}{\sin \ heta} and csc⁡ heta=\ rac1sin⁡ heta\csc \ heta = \ rac{1}{\sin \ heta}, we get this complementary relationship to the tangent identity.
sin⁡2θ+cos⁡2θsin⁡2θ=1sin⁡2θ\frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}
1+cos⁡2θsin⁡2θ=csc⁡2θ1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \csc^2 \theta
1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta
Particularly useful when working with inverse trigonometric functions
Mirrors the tangent identity but uses reciprocal functions

Co-Function Identities

Sine-Cosine Co-Function Identity



Formula:

sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta
This identity shows that sine of a complementary angle (π/2 - θ) equals the cosine of the original angle θ. Geometrically, this represents the relationship between vertical and horizontal components when an angle is rotated to its complement. When you take π/2 (90°) and subtract θ, you get the complementary angle, and its sine equals the original angle's cosine.
Used in solving trigonometric equations and simplifying expressions involving complementary angles
This forms the basis for other co-function identities

Cosine-Sine Co-Function Identity



Formula:

cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta
The complement of the cosine relationship: cosine of a complementary angle equals the sine of the original angle. This is essentially the same relationship as sine-cosine co-function but from the other perspective. It demonstrates the symmetric nature of complementary angles.
Helpful in problems involving right triangles and complementary angles
Direct complement to the sine co-function identity

Tangent-Cotangent Co-Function Identity



Formula:

tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta
This identity relates the tangent of a complementary angle to the cotangent of the original angle. Since tangent is sine/cosine and cotangent is cosine/sine, this relationship follows naturally from how sine and cosine swap roles in complementary angles.
Useful in calculus and advanced trigonometric manipulations
Connected to the reciprocal relationship between tangent and cotangent

Cotangent-Tangent Co-Function Identity



Formula:

cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta
The complement of the tangent-cotangent relationship. Shows that the cotangent of a complementary angle equals the tangent of the original angle. This follows from the reciprocal nature of these functions and their behavior with complementary angles.
Used in solving trigonometric equations involving complementary angles
Mirror of the tangent co-function identity

Secant-Cosecant Co-Function Identity



Formula:

sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta
The secant of a complementary angle equals the cosecant of the original angle. Since secant and cosecant are reciprocals of cosine and sine respectively, this relationship mirrors the fundamental sine-cosine co-function identity.
Important in advanced trigonometric calculations and proofs
Follows from the basic sine-cosine relationship combined with reciprocal identities

Cosecant-Secant Co-Function Identity



Formula:

csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta
Completes the set of co-function identities by showing that the cosecant of a complementary angle equals the secant of the original angle. This relationship is the natural consequence of how reciprocal functions behave with complementary angles.
θ π/2 - θ 90° In any triangle, angles sum to 180° We have: • One right angle (90°) • One angle θ • Third is π/2 - θ (complementary)
Used in complex trigonometric proofs and calculations
Final piece of the co-function identity set

Even Functions

Cosine Even Identity



Formula:

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta
Cosine is an even function - reflecting angle around y-axis gives same value.
θ -θ cos θ Even Function: cos(-θ) = cos(θ) When we reflect angle θ across x-axis: • x-coordinate stays the same • Both angles have same cosine value • Only y-coordinate changes sign
Simplifying expressions with negative angles
Even functions have no sign change with negative input

Secant Even Identity



Formula:

sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \theta
Secant is even because it's reciprocal of cosine
Working with negative angles in secant expressions
Inherits evenness from cosine function

Odd Functions

Sine Odd Identity



Formula:

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta
Sine is an odd function - reflecting angle around origin gives opposite value
Wave functions, oscillations with negative angles
Odd functions change sign with negative input

Tangent Odd Identity



Formula:

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \theta
Tangent is odd as it's sine (odd) divided by cosine (even)
Simplifying tangent expressions with negative angles
Sign changes when angle becomes negative

Cosecant Odd Identity



Formula:

csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \theta
Cosecant is odd because it's reciprocal of sine
Working with negative angles in cosecant expressions
Inherits oddness from sine function

Cotangent Odd Identity



Formula:

cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \theta
Cotangent is odd as it's cosine (even) divided by sine (odd)
Working with negative angles in cotangent expressions
Sign changes when angle becomes negative