What are sin, cos, and tan?

Sine, cosine, and tangent are the three primary trigonometric functions. In a right triangle, if theta is one of the non-right angles: - sin(theta) = opposite side / hypotenuse - cos(theta) = adjacent side / hypotenuse - tan(theta) = opposite / adjacent = sin(theta) / cos(theta) Memory aid: SOH-CAH-TOA - SOH: Sine = Opposite over Hypotenuse - CAH: Cosine = Adjacent over Hypotenuse - TOA: Tangent = Opposite over Adjacent On the unit circle (radius = 1), for any angle theta measured from the positive x-axis: - cos(theta) = x-coordinate of the point on the circle - sin(theta) = y-coordinate of the point on the circle - tan(theta) = y/x = sin/cos This unit circle definition extends trig to all angles, not just those in a right triangle.

What are the exact values of sin, cos, and tan for common angles?

These exact values appear frequently in maths and should be memorised: 0 degrees: sin=0, cos=1, tan=0 30 degrees (pi/6): sin=1/2, cos=sqrt(3)/2, tan=1/sqrt(3) 45 degrees (pi/4): sin=sqrt(2)/2, cos=sqrt(2)/2, tan=1 60 degrees (pi/3): sin=sqrt(3)/2, cos=1/2, tan=sqrt(3) 90 degrees (pi/2): sin=1, cos=0, tan=undefined Memory trick for sine (0 to 90): sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2 Which simplifies to: 0, 1/2, sqrt(2)/2, sqrt(3)/2, 1 Cosine is just sin read backwards: 1, sqrt(3)/2, sqrt(2)/2, 1/2, 0 For other quadrants, use the reference angle and sign rules: - Quadrant 1 (0-90): all positive - Quadrant 2 (90-180): sin positive, cos and tan negative - Quadrant 3 (180-270): tan positive, sin and cos negative - Quadrant 4 (270-360): cos positive, sin and tan negative

What is the difference between degrees and radians?

Degrees and radians are two units for measuring angles. Degrees: a full circle = 360 degrees. Most intuitive for everyday use. Used in navigation, geography, architecture. Radians: a full circle = 2*pi radians. Defined as the arc length on the unit circle. Used in mathematics, calculus, physics, and programming (most language math libraries use radians). Conversion: - Degrees to radians: multiply by pi/180 (approx 0.01745) - Radians to degrees: multiply by 180/pi (approx 57.296) Common conversions: - 0 degrees = 0 radians - 30 degrees = pi/6 radians - 45 degrees = pi/4 radians - 60 degrees = pi/3 radians - 90 degrees = pi/2 radians - 180 degrees = pi radians - 360 degrees = 2*pi radians Why radians? Calculus formulas (derivatives, Taylor series) are simpler in radians. The derivative of sin(x) is cos(x) only when x is in radians.

What is the Pythagorean identity and how is it used?

The Pythagorean identity is sin^2(x) + cos^2(x) = 1. It follows directly from the Pythagorean theorem applied to the unit circle, where the point (cos x, sin x) is always at distance 1 from the origin. This is the most important trig identity. From it, two more identities follow: - Divide by cos^2(x): tan^2(x) + 1 = sec^2(x) - Divide by sin^2(x): 1 + cot^2(x) = csc^2(x) Uses: - Simplify trig expressions by substituting 1 - sin^2 for cos^2 or vice versa - Solve trig equations by reducing to one function - Verify other identities (start with one side, use the identity to transform it into the other) - In calculus: used in trig substitutions for integrating sqrt(1-x^2) type expressions Verification for 30 degrees: sin^2(30) + cos^2(30) = (1/2)^2 + (sqrt(3)/2)^2 = 1/4 + 3/4 = 1. Confirmed.

What is arcsin, arccos, and arctan?

Arcsin, arccos, and arctan (also written sin^-1, cos^-1, tan^-1) are inverse trig functions. They take a ratio and return the angle that produces it. arcsin(x): the angle whose sine is x. Domain: -1 to 1. Range: -90 to 90 degrees (-pi/2 to pi/2). arccos(x): the angle whose cosine is x. Domain: -1 to 1. Range: 0 to 180 degrees (0 to pi). arctan(x): the angle whose tangent is x. Domain: all real numbers. Range: -90 to 90 degrees (-pi/2 to pi/2). Examples: - arcsin(0.5) = 30 degrees (sin(30) = 0.5) - arccos(0.5) = 60 degrees (cos(60) = 0.5) - arctan(1) = 45 degrees (tan(45) = 1) - arcsin(-1) = -90 degrees Important: each inverse function returns only one value (the principal value), even though multiple angles have the same sine or cosine. sin(30) = sin(150) = 0.5, but arcsin(0.5) returns only 30 degrees. When solving equations, remember there may be additional solutions in other quadrants.

Sin Cos Tan Calculator — All 6 Trig Functions with Unit Circle

The Sin Cos Tan Calculator computes all six trigonometric functions for any angle in degrees, radians, or gradians. Forward mode: enter an angle and get sin, cos, tan, csc, sec, and cot instantly, with exact values shown for common angles like 30, 45, 60, and 90 degrees. Inverse mode: enter a trig v...

Angle (°)

SIN

0.5

1/2

COS

0.866025

−√3/2

TAN

0.57735

1/√3

CSC (1/sin)

SEC (1/cos)

1.1547

COT (1/tan)

1.73205

Degrees30°

Radians0.52359878

Gradians33.3333

Quadrant 1 — All positive (sin+, cos+, tan+)

Reference angle: 30°

QUICK EXAMPLES

STEP-BY-STEP

Angle: 30 deg = 30° = 0.52359878 rad

Quadrant: 1 (reference angle = 30°)

sin(30°) = 0.5

cos(30°) = 0.8660254

tan(30°) = 0.57735027

csc = 1/sin = 2

sec = 1/cos = 1.15470054

cot = 1/tan = 1.73205081

UNIT CIRCLE

Point on circle = (cos θ, sin θ)

COMMON ANGLES — EXACT VALUES

°	rad	sin	cos	tan
0°	0	0	1	0
30°	π/6	½	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	½	√3
90°	π/2	1	0	Undef
180°	π	0	−1	0
270°	3π/2	−1	0	Undef
360°	2π	0	1	0

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TRIG IDENTITIES & FORMULAS

SIN / COS / TAN REFERENCE

NAME	FORMULA	DESCRIPTION
Sine definition	sin(theta) = opposite / hypotenuse	In a right triangle with angle theta
Cosine definition	cos(theta) = adjacent / hypotenuse	In a right triangle with angle theta
Tangent definition	tan(theta) = opposite / adjacent = sin/cos	Ratio of opposite to adjacent side
Reciprocals	csc=1/sin, sec=1/cos, cot=1/tan	Three reciprocal trig functions
Pythagorean identity	sin^2(x) + cos^2(x) = 1	Fundamental identity from unit circle
Derived identities	tan^2(x)+1=sec^2(x), cot^2(x)+1=csc^2(x)	From the Pythagorean identity
Unit circle	Point at angle theta = (cos theta, sin theta)	x-coordinate is cosine, y-coordinate is sine
Angle to radians	radians = degrees x pi / 180	Convert degrees to radians
Radians to degrees	degrees = radians x 180 / pi	Convert radians to degrees
Double angle (sin)	sin(2x) = 2 sin(x) cos(x)	Double angle formula for sine
Double angle (cos)	cos(2x) = cos^2(x) - sin^2(x)	Double angle formula for cosine
ASTC rule	Q1: all+, Q2: sin+, Q3: tan+, Q4: cos+	Which functions are positive in each quadrant

HOW TO USE

1
Select Angle to sin/cos/tan mode. Choose your angle unit: Degrees (most common), Radians (used in calculus and programming), or Gradians (400 per full circle, used in surveying).
2
Enter your angle. All six trig functions appear instantly: sin, cos, tan (primary) and csc, sec, cot (reciprocals). Exact values like 1/2 and sqrt(3)/2 are shown for common angles.
3
Switch to the inverse mode to find the angle from a trig value. Select arcsin, arccos, or arctan, enter a value, and get the angle in degrees, radians, and gradians.
4
The unit circle diagram shows your angle as a point on the circle, with red dashed lines for the sine projection and blue dashed lines for the cosine projection. The angle arc is drawn from 0 to your angle.
5
The common angles table highlights your current angle and shows exact values for 0, 30, 45, 60, 90, 180, 270, and 360 degrees for quick reference.

WORKED EXAMPLE

30 degrees: sin=0.5 (exact: 1/2), cos=sqrt(3)/2=0.866025, tan=1/sqrt(3)=0.577350. Reciprocals: csc=2, sec=1.154701, cot=sqrt(3)=1.732051. Quadrant 1 (all positive). Reference angle=30. Inverse: arcsin(0.5)=30 degrees=pi/6 rad. arctan(1)=45 degrees. 90 degrees: sin=1, cos=0, tan=undefined.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.