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➗ MathFree
De Moivre Theorem Power Calculator for Complex Numbers
Cubing a complex number with modulus 2 and argument 30° gives modulus 8 and argument 90°, which is a real part of 0 and an imaginary part of 8.
B
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➗ MathFree
Degrees minutes seconds arithmetic calculator
18° 42′ 45″ plus 7° 28′ 30″ is 26° 11′ 15″, with the extra seconds and minutes carried over automatically.
F
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➗ MathFree
Complex number rectangular to polar form
Rewrite a complex number by size and direction: 3 + 4i has a modulus of 5 and an argument of about 53.13°, so it reads 5∠53.13°.
A
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➗ MathFree
Angular frequency from period calculator
A wave that repeats every 4 units has an angular frequency of π/2 radians per unit. Enter the length of one full cycle and get ω, ready to use as the B…
F
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➗ MathFree
Triangle area Heron's formula calculator
Sides of 3, 4 and 5 give a semiperimeter of 6 and an area of 6. Enter three side lengths, no angle or height needed, and get the area and perimeter, or a clear…
P
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➗ MathFree
Quotient of complex numbers in polar form calculator
Dividing 8∠150° by 2∠30° gives 4∠120°, or −2 + 3.464101615138i. Enter each complex number as a modulus and an angle, in degrees or radians, and get the quotient…
K
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➗ MathFree
Hyperbolic cotangent calculator
The hyperbolic cotangent of 1 is 1.3130352854993312, the ratio of a cosh of 1.5430806348152437 to a sinh of 1.1752011936438014, and all three values appear…
B
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➗ MathFree
Phase shift of a sinusoidal function
Get the horizontal shift of a sine graph with its direction: for sin(2x + π) the wave moves π/2 to the left.
S
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➗ MathFree
Hyperbolic tangent calculator
tanh(1) is 0.761594155956: enter a real number and get its hyperbolic tangent together with the sinh and cosh that produce it, even for huge values where the…
A
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➗ MathFree
Gravity components on an inclined plane calculator
A 10 kg mass on a 30° slope weighs 98.0665 N: 49.03325 N pull it down the incline and about 84.9280802602 N press it into the surface.
B
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➗ MathFree
Resultant of two vectors calculator
Combine two forces or velocities into one: 3 at 0° plus 4 at 90° gives a resultant of 5 pointing at about 53.13°.
S
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➗ MathFree
Hyperbolic sine double-angle calculator
Evaluate sinh(2x) and see where it comes from: at 0 the result is 0, because sinh(0) is 0 and cosh(0) is 1.
F
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➗ MathFree
Spherical Law of Cosines for Sides Calculator
Get the missing side of a triangle drawn on a sphere: two arcs of 60° and 80° meeting at a 70° angle give the third arc in degrees, where a flat cosine rule…
K
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➗ MathFree
Spherical triangle area from excess
Get a spherical triangle's area straight from its excess, so on a sphere of radius 1 an excess of 0.158407 radians covers an area of 0.158407.
A
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➗ MathFree
Destination point from bearing and distance calculator
Start at latitude 51.4778, longitude −0.0015, head off on a bearing of 45° for 10 km, and you arrive near 51.5413472357, 0.1007456435.
F
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➗ MathFree
Linear speed from angular speed calculator
See how fast a point on a spinning object really travels: 0.5 m from the axis at 12 radians per second, it moves at 6 m per second along its circle.
K
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➗ MathFree
Sign of trig function in quadrant calculator
Sine in quadrant II is positive, the Students of All Students Take Calculus, and you get the same clear plus or minus for all six trig functions in every…
S
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➗ MathFree
Tangent from sine and cosine calculator
Divide sine by cosine and read the tangent at once: a sine of 0.6 with a cosine of 0.8 gives 0.7499999999999999, the plain machine ratio with no hidden…
B
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➗ MathFree
Triangle perimeter calculator
Sides of 3, 4 and 5 give a perimeter of 12. Enter the three lengths and the calculator first confirms they can close into a triangle, then adds them, so an…
A
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➗ MathFree
Distance by triangulation calculator
A baseline of 100 with base angles of 50° and 60° puts the target 92.1604985107 from the first station and 81.5207469096 from the second.
P
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➗ MathFree
Triangle altitude from area and base calculator
A triangle with an area of 30 and a base of 12 has an altitude of 5. Enter the area and the side you treat as the base to get the perpendicular height that goes…
F
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➗ MathFree
Beat frequency of two waves calculator
Waves at 440 Hz and 442 Hz beat at 2 Hz, about two loudness pulses per second. Enter both frequencies in hertz, in any order, and get the beat rate you hear…
K
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➗ MathFree
Negative-angle identity calculator
Get the sign right when an angle flips: sin(30°) is 0.5, so sin(−30°) is −0.5, because sine is odd.
P
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➗ MathFree
Supplementary angle calculator
Get the angle that completes a straight line: 127.5° needs 52.5° to reach 180°, and 65° needs 115°.
B
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➗ MathFree
Gudermannian function calculator
gd(1) is about 0.8657694832396585 radians, or 49.6049374208547°, the angle whose tangent equals sinh(1).
F
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➗ MathFree
Haversine function calculator
The haversine of 90° is 0.5: one minus the cosine, halved, matches the square of the sine of half the angle, 0.707106781187 times itself.
S
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➗ MathFree
Is it a right triangle checker
Find out in one step whether three lengths make a right triangle: 3, 4 and 5 do, because 3² + 4² gives 25, exactly 5².
P
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➗ MathFree
Rafter length from pitch angle and run calculator
Know how long a rafter must be before you cut: a horizontal run of 12 at a 30° roof angle needs a rafter of 13.8564, in whatever unit you measured.
K
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➗ MathFree
Resultant course with current calculator
See where the water really takes your boat: heading 90° at 10 with a current of 3 flowing toward 180°, each hour carries you 10 east and 3 south, so your track…
F
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➗ MathFree
Rotate point about a center calculator
Turn a point around any pivot, not only the origin: rotating (4, 2) by 90° about (1, 1) lands it at (0, 4).
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➗ MathFree
Superposition of two sinusoids calculator
Combine two waves that share one frequency into a single wave: an amplitude of 3 at 0° plus an amplitude of 4 at 90° gives an amplitude of 5.
S
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➗ MathFree
Thirty-sixty-ninety triangle solver
A short leg of 6 gives a long leg of 10.3923048454133 and a hypotenuse of 12. Enter one side, say which one it is, and the fixed 1 : √3 : 2 ratio fills in the…
B
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➗ MathFree
2D cross product magnitude calculator
For vectors (3, 4) and (−2, 5), the 2D cross product is 23: positive, so the second vector lies counterclockwise from the first.
B
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➗ MathFree
Frequency from period calculator
A period of 0.25 gives a frequency of 4, because four full cycles fit into one unit; a period of 4 gives 0.25.
K
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➗ MathFree
Sagitta of a circular arc calculator
A radius of 10 with a 90° central angle gives an arc that rises about 2.9289, worked out from those two values alone.
A
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➗ MathFree
Along-track distance great-circle calculator
Give three coordinate pairs, a route start, its end and a point, and get how far along the great-circle track the closest approach to that point falls, plus its…
S
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➗ MathFree
Angle Between Two Bearings Calculator
Bearings of 350° and 10° are only 20° apart, not 340°. Enter two compass directions in degrees, even negative ones or values above 360, and get the smallest…
K
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➗ MathFree
Evaluate trig expression at an angle
At 30°, the expression 2 sin θ plus 3 cos θ minus tan θ plus 0.5 csc θ comes to 4.020725942164, built from terms of 1, 2.598076211353, −0.57735026919 and 1.
B
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➗ MathFree
NATO mils to degrees calculator
Turn a military bearing into degrees in one step: 1600 NATO mils are exactly 90°, because a full circle holds 6400 mils.
A
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➗ MathFree
Simple harmonic motion position calculator
See the position of an oscillator at one instant: with amplitude 2, angular frequency π/2 and no phase, at time 1 it sits at 0, with a period of 4 and a…
P
Runs in your browser
➗ MathFree
Solar Elevation Angle Calculator
See the Sun's height above the horizon from three angles you already have, with an hour angle that is 0 at local solar noon.
F
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➗ MathFree
Solar azimuth angle calculator from latitude, declination and hour angle
At latitude 40.7128°, with a declination of 23.44° and an hour angle of −30°, the sun stands at 115.289892°, roughly ESE.
F
Runs in your browser
➗ MathFree
Critical angle total internal reflection calculator
Light passing from an index of 1.5 into an index of 1 has a critical angle of 41.810314895779°, or 0.729727656227 radians.
B
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➗ MathFree
Solar declination approximation calculator
On day 172, near the northern summer solstice, the Sun's declination comes out at 23.45°, the highest point of this cosine model; winter days give negative…
A
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➗ MathFree
Parallelogram diagonal calculator from sides and angle
Sides of 5 and 8 meeting at 60° give diagonals of about 11.3578166916005 and exactly 7, in the same unit as the sides.
F
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➗ MathFree
Distance to horizon calculator
With your eyes 1.7 m above the surface, the geometric horizon lies about 4.6541812266 km away, or 2.51305681782 nautical miles.
P
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➗ MathFree
Isosceles triangle from leg and apex angle calculator
Solve a whole isosceles triangle from one equal leg and the angle between the legs: legs of 10 with a 60° apex give a base of 10 and base angles of 60°…
K
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➗ MathFree
Tidal height sinusoid model
Sketch the tide between high and low water from three numbers: with high water at 3.2, low water at 0.4 and a 12.42-hour cycle, the level 3.105 hours after high…
A
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➗ MathFree
Regular polygon apothem calculator from circumradius
Get the distance from the center of a regular polygon to its sides: a hexagon whose corners sit 10 from the center has an apothem of 8.660254.
K
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➗ MathFree
Tangent from line slope calculator
A slope of 1 means the line climbs at 45°: enter any finite slope and get its inclination angle in degrees and radians, plus whether the line rises, falls or…
P
Runs in your browser
➗ MathFree
Triangle height from base and two base angles calculator
A base of 10 with two 45° angles gives a height of 5. Enter the base and the angles at its ends to get the perpendicular height, the angle at the top and the…
S
Runs in your browser
➗ MathFree
Triangle angle from side and altitude calculator
A side of 10 with an altitude of 5 gives a 30° angle. Enter the side next to the angle and the perpendicular height, and get that base angle in degrees and…
S
Runs in your browser
➗ MathFree
Derivative of Logarithmic Function Calculator
The derivative of 6 ln(4x) is 6/x, so at the point 3 it equals exactly 2. Enter a, k and the point to get the simplified derivative and its value, with a check…
F
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➗ MathFree
Power Rule Derivative Calculator for a·x^n
Differentiate one term a·x^n with the power rule: 5x^4 becomes 20x^3, because the coefficient is multiplied by the exponent and the exponent drops by one.
B
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➗ MathFree
Infinite limit from the right calculator for rational functions
As x approaches 1 from the right, (x + 1)/(x² − 2x + 1) goes to positive infinity: the denominator has a double zero there and the numerator none.
K
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➗ MathFree
Limit Product and Quotient Rule Calculator
If f(x) tends to 6 and g(x) to 3 at the same point, the product law gives 18 and the quotient law gives 2.
F
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➗ MathFree
Equation of tangent line calculator
At 2, the curve 3x^2 - 2x + 1 touches its tangent at (2, 9), with slope 10 and a y-intercept of -11.
K
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➗ MathFree
Implicit Derivative of an Ellipse Calculator
Get the tangent slope of an ellipse at a point on it: with semi-axes 5 and 3 and the point (4, 1.8), the implicit derivative gives a slope of −0.8 and the…
B
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➗ MathFree
Concave up intervals calculator
For x³ − 3x the second derivative is 6x, so the curve is concave up exactly on (0, +∞). Enter the coefficients, highest power first, to get both derivatives and…
S
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➗ MathFree
Critical Points of a Polynomial Calculator
The critical points of x³ − 3x + 2 are −1 and 1, where its derivative 3x² − 3 equals zero. Enter the coefficients, highest power first, and get the derivative…
B
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