Smart Trig Calculator
Trigonometric Functions
SINE
Ratio of opposite side to hypotenuse
sin(θ) = opp/hyp
COSINE
Ratio of adjacent side to hypotenuse
cos(θ) = adj/hyp
TANGENT
Ratio of opposite side to adjacent side
tan(θ) = opp/adj
COSECANT
Reciprocal of sine function
csc(θ) = 1/sin(θ)
SECANT
Reciprocal of cosine function
sec(θ) = 1/cos(θ)
COTANGENT
Reciprocal of tangent function
cot(θ) = 1/tan(θ)
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SINE FUNCTION
The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
sin(θ) = opposite / hypotenuse
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COSINE FUNCTION
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
cos(θ) = adjacent / hypotenuse
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TANGENT FUNCTION
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
tan(θ) = opposite / adjacent
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COSECANT FUNCTION
The cosecant is the reciprocal of the sine function, representing the ratio of the hypotenuse to the opposite side.
csc(θ) = 1 / sin(θ) = hypotenuse / opposite
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SECANT FUNCTION
The secant is the reciprocal of the cosine function, representing the ratio of the hypotenuse to the adjacent side.
sec(θ) = 1 / cos(θ) = hypotenuse / adjacent
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COTANGENT FUNCTION
The cotangent is the reciprocal of the tangent function, representing the ratio of the adjacent side to the opposite side.
cot(θ) = 1 / tan(θ) = adjacent / opposite
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Law of Sines
Relates the lengths of sides of a triangle to the sines of its opposite angles.
a / sin(A) = b / sin(B) = c / sin(C) = 2R
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Law of Cosines
Generalizes the Pythagorean theorem for any triangle.
c² = a² + b² - 2ab·cos(C)
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Law of Tangents
Relates the tangent of two angles to the lengths of the opposite sides.
(a - b)/(a + b) = tan[(A - B)/2]/tan[(A + B)/2]
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Law of Cotangents
Relates the cotangent of half an angle to the triangle sides and radius.
cot(A/2) = (s - a)/r
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Pythagorean Theorem
For right triangles, the square of the hypotenuse equals the sum of squares of the other two sides.
a² + b² = c²
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Heron's Formula
Calculates the area of a triangle when the lengths of all three sides are known.
Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2
Laws, Theorems, and Identities
Law of Sines
Relates sides and angles of any triangle
a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines
Generalization of Pythagorean theorem
c² = a² + b² - 2ab·cos(C)
Law of Tangents
Relates tangents of two angles
(a-b)/(a+b) = tan[(A-B)/2]/tan[(A+B)/2]
Law of Cotangents
Relates cotangents to triangle sides
cot(A/2) = (s-a)/r
Pythagorean Theorem
For right triangles only
a² + b² = c²
Heron's Formula
Area from side lengths
Area = √[s(s-a)(s-b)(s-c)]
Trigonometric Terms
Hypotenuse
The longest side of a right triangle, opposite the right angle
c = √(a² + b²)
Opposite Side
The side directly across from the angle of interest
opp = hyp × sin(θ)
Adjacent Side
The side next to the angle (not the hypotenuse)
adj = hyp × cos(θ)
Radians
Angle measurement based on arc length (π rad = 180°)
1 rad ≈ 57.2958°
Unit Circle
Circle with radius 1 used to define trig functions
x² + y² = 1
Reference Angle
Acute angle between terminal side and x-axis
Always between 0 and π/2