Choose "value from an angle" or "angle from a value" and enter your numbers. The result comes with a unit circle diagram.
Enter the angle in degrees (°). Negative angles and angles over 360° are fine. For the trig ratio value, you can use decimals or fractions such as 1/2.
Result and figure
Enter an angle (or the value of a trig ratio) in the fields on the left and press "Calculate". The result and a unit circle diagram will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter an angle \(\theta\) in degrees and get \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\) all at once. For special angles such as \(30^\circ\), \(45^\circ\), \(60^\circ\) and \(120^\circ\), the answer is shown both as an exact value like \(\dfrac{\sqrt{3}}{2}\) and as a decimal
- You can also go the other way and find the angle from the value of a trig ratio, as in "which angle gives \(\sin\theta = 0.5\)?" (in the range \(0^\circ\) to \(180^\circ\). If two angles match, both are shown)
- Negative angles and angles over \(360^\circ\) work too. Angles are given in degrees, with the radian measure shown alongside
- The result comes with a unit circle diagram, so you can see at a glance which lengths on the circle \(\sin\theta\) and \(\cos\theta\) are
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
When a value is undefined, as with \(\tan 90^\circ\), the page does not stop with an error. It shows the result with an explanation of why the value is undefined.
What is this calculation used for?
Describing the steepness of roads and ramps (road and accessible design)
A road sign that warns of a "6% grade" means the road rises 6 feet for every 100 feet across. That is exactly \(\tan\theta = 0.06\) (about \(3.4^\circ\)). The ADA Standards limit wheelchair ramps to a slope of 1:12, which is \(\tan\theta = \dfrac{1}{12}\) (about \(4.8^\circ\)).
The tangent value is easier to use on the job than the angle, because it comes straight from the rise and the run. That is why slopes are labeled with this trig ratio as is.
Measuring the height of a tree or building from a distance (surveying)
If you cannot reach the top of something tall with a tape measure, you can find its height from the distance and the angle you look up at it (the angle of elevation): height = distance × \(\tan\theta\). For example, standing 60 feet from a tree with an angle of elevation of \(30^\circ\), the part above your eye level is \(60 \times \tan 30^\circ \approx 34.6\) feet.
Triangulation, used in land surveying and mapmaking, works out distances and positions from measured angles in the same way. Surveying instruments such as total stations also use trig ratios to turn measured angles and distances into coordinates.
Splitting the forces on a slope (physics and mechanics)
Gravity pulls an object on a slope straight down. In physics, this pull is split into two parts: the part that tries to slide the object down the slope (weight × \(\sin\theta\)) and the part that presses it into the slope (weight × \(\cos\theta\)).
On a \(30^\circ\) slope, the force trying to slide the object down is \(\sin 30^\circ = \dfrac{1}{2}\) of its weight, which is half. How well a parking brake holds on a hill, or how fast you slide on a ski run of a given steepness: the physics of slopes always starts from this split.
Moving a character at an angle in games and computer graphics (programming)
When a game moves something "at speed \(v\) in the direction of angle \(\theta\)", the computer moves its position by \(v \times \cos\theta\) across and \(v \times \sin\theta\) up or down. The paths of bullets in a shooting game, the corners of a radar chart, the hands of a clock on the screen: every task that turns an angle into a position uses sin and cos.
The unit circle definition itself (\(x = \cos\theta,\ y = \sin\theta\)) is the program for rotation and circular motion on screen.
Designing the pitch of a roof (construction and carpentry)
In US construction, the steepness of a roof is given as a pitch such as "4/12" (it rises 4 inches for every 12 inches across, so \(\tan\theta = \dfrac{4}{12} = \dfrac{1}{3}\), about \(18.4^\circ\)). Once the pitch is set, the length of a rafter is the run ÷ \(\cos\theta\): a 12-foot run needs about \(12 \div \cos 18.4^\circ \approx 12.65\) feet. Water runoff, snow sliding off and the amount of material are all estimated from this value.
Laying out roof angles with a carpenter's framing square also uses the side ratios of a right triangle, which is the idea of trig ratios itself.
Formulas and figures
sin (sine) = opposite ÷ hypotenuse
Figure (a right triangle and its trig ratios)
Standard notation
(the usual math form)
\(\sin\theta\)
\(=\)
\(a\)
\(\div\)
\(c\)
In words
(symbols replaced with words)
③
\(\sin\theta\): sine of angle \(\theta\)
\(=\)
①
\(a\): opposite side (across from \(\theta\))
\(\div\)
②
\(c\): hypotenuse
The formula in words
①
In a right triangle, take the
\(a\): opposite side (the side across from angle \(\theta\))
②
divide it by the
\(c\): hypotenuse (the longest side, across from the right angle)
③
and you get the
\(\sin\theta\): sine of angle \(\theta\)
Quick example
In a right triangle with side ratio \(3 : 4 : 5\) (opposite 3, adjacent 4, hypotenuse 5),
\(\sin\theta\)
\(=\)
opposite 3
\(\div\)
hypotenuse 5
\(\sin\theta = \dfrac{3}{5} = 0.6\)
Key idea
Sine is a ratio: if you take the hypotenuse as 1, how high do you rise? Once the angle \(\theta\) is fixed, this ratio has one value no matter how big the triangle is (triangles with the same shape are similar). That is why it can be used as a function of the angle alone.
A popular way to remember the three ratios is SOH-CAH-TOA: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
cos (cosine) = adjacent ÷ hypotenuse
Standard notation
(the usual math form)
\(\cos\theta\)
\(=\)
\(b\)
\(\div\)
\(c\)
In words
(symbols replaced with words)
③
\(\cos\theta\): cosine of angle \(\theta\)
\(=\)
①
\(b\): adjacent side (next to \(\theta\))
\(\div\)
②
\(c\): hypotenuse
The formula in words
①
In a right triangle, take the
\(b\): adjacent side (the side next to angle \(\theta\) that forms the right angle)
②
divide it by the
\(c\): hypotenuse
③
and you get the
\(\cos\theta\): cosine of angle \(\theta\)
Quick example
In a right triangle with side ratio \(3 : 4 : 5\) (opposite 3, adjacent 4, hypotenuse 5),
\(\cos\theta\)
\(=\)
adjacent 4
\(\div\)
hypotenuse 5
\(\cos\theta = \dfrac{4}{5} = 0.8\)
Key idea
Cosine is a ratio: if you take the hypotenuse as 1, how far across do you go? In SOH-CAH-TOA, it is the "CAH" part.
The "co" in cosine comes from the complementary angle (the angle that adds up to \(90^\circ\) with \(\theta\)): \(\cos\theta = \sin(90^\circ - \theta)\). The cosine of an angle equals the sine of its complement. For example, \(\cos 60^\circ = \sin 30^\circ = \dfrac{1}{2}\).
tan (tangent) = opposite ÷ adjacent
Standard notation
(the usual math form)
\(\tan\theta\)
\(=\)
\(a\)
\(\div\)
\(b\)
In words
(symbols replaced with words)
③
\(\tan\theta\): tangent of angle \(\theta\)
\(=\)
①
\(a\): opposite side (across from \(\theta\))
\(\div\)
②
\(b\): adjacent side (next to \(\theta\))
The formula in words
①
In a right triangle, take the
\(a\): opposite side (the side across from angle \(\theta\))
②
divide it by the
\(b\): adjacent side (the side next to angle \(\theta\))
③
and you get the
\(\tan\theta\): tangent of angle \(\theta\)
Quick example
In a right triangle with side ratio \(3 : 4 : 5\) (opposite 3, adjacent 4, hypotenuse 5),
\(\tan\theta\)
\(=\)
opposite 3
\(\div\)
adjacent 4
\(\tan\theta = \dfrac{3}{4} = 0.75\)
Key idea
Tangent tells you how far up you go for every 1 you go across. In other words, it is the slope of the hill itself.
From the definitions, \(\tan\theta = \dfrac{a}{b} = \dfrac{a \div c}{b \div c} = \dfrac{\sin\theta}{\cos\theta}\). The denominator is \(\cos\theta\), so at \(\theta = 90^\circ\), where \(\cos\theta = 0\), \(\tan 90^\circ\) is undefined (you cannot divide by 0).
Unit circle definition (for angles of 90° or more)
Figure (the unit circle and trig ratios)
Standard notation
(the usual math form)
\(\cos\theta\)
\(=\)
\(x\)
\(\sin\theta\)
\(=\)
\(y\)
In words
(symbols replaced with words)
②
\(\cos\theta\): cosine of angle \(\theta\)
\(=\)
①
\(x\): \(x\)-coordinate of point \(P\)
④
\(\sin\theta\): sine of angle \(\theta\)
\(=\)
③
\(y\): \(y\)-coordinate of point \(P\)
The formula in words
①
On the circle of radius \(1\) centered at the origin \(O\) (the unit circle), turn a ray (the terminal side) by angle \(\theta\) from the positive \(x\)-axis, and call the point where it meets the circle \(P\). Then the
\(x\)-coordinate of point \(P\)
②
is the
\(\cos\theta\): cosine of angle \(\theta\)
③
and the
\(y\)-coordinate of point \(P\)
④
is the
\(\sin\theta\): sine of angle \(\theta\)
Quick example
When \(\theta = 120^\circ\) (an obtuse angle), the point on the unit circle is \(P\left(-\dfrac{1}{2},\ \dfrac{\sqrt{3}}{2}\right)\), so
\(\cos 120^\circ\)
\(=\)
\(x\)-coordinate \(-\dfrac{1}{2}\)
\(\sin 120^\circ\)
\(=\)
\(y\)-coordinate \(\dfrac{\sqrt{3}}{2}\)
\(\cos 120^\circ = -\dfrac{1}{2}, \quad \sin 120^\circ = \dfrac{\sqrt{3}}{2}, \quad \tan 120^\circ = \dfrac{\sin 120^\circ}{\cos 120^\circ} = -\sqrt{3}\)
Key idea
The right triangle definition only works for angles greater than \(0^\circ\) and less than \(90^\circ\). With the unit circle, trig ratios extend to \(0^\circ\), \(90^\circ\), obtuse angles, \(180^\circ\) and on to any angle up to \(360^\circ\) and beyond. This calculator uses this definition.
Past \(90^\circ\), the point \(P\) moves to the left of the \(y\)-axis and its \(x\)-coordinate becomes negative, so \(\cos\theta\) and \(\tan\theta\) are negative (\(\sin\theta\) stays 0 or more all the way from \(0^\circ\) to \(180^\circ\)).
\(\tan\theta\) is the slope of line \(OP\). When finding the angle from a value, \(\sin\theta = 0.5\) matches two angles, \(30^\circ\) and \(150^\circ\), while \(\cos\theta\) and \(\tan\theta\) give only one answer. The diagram explains why: points at the same height come in mirror-image pairs, but only one point has a given \(x\)-coordinate, and only one terminal side has a given slope. (The exception is when the terminal side lies on the \(x\)-axis: \(\sin\theta = 0\) and \(\tan\theta = 0\) both match \(0^\circ\) and \(180^\circ\).)
In a right triangle, the trig ratios are \(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}},\ \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}},\ \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\). On the unit circle, they are the \(y\)-coordinate of the point \(P\), its \(x\)-coordinate, and the slope of line \(OP\). For special angles such as \(30^\circ\), \(45^\circ\) and \(60^\circ\), the exact values come from the side ratios of the special right triangles (\(1 : \sqrt{3} : 2\) and \(1 : 1 : \sqrt{2}\)).
Symbols and terms
Symbols
| \(\theta\) | theta | A Greek letter often used for angles (there is no settled explanation of why \(\theta\) is used). Besides \(\theta\), \(\alpha\) (alpha) and \(\beta\) (beta) are also used for angles. |
| \(\sin\theta\) | sine theta | The sine of angle \(\theta\): "opposite ÷ hypotenuse" in a right triangle, and the \(y\)-coordinate of the point \(P\) on the unit circle. The name is said to come from the Latin word sinus (a bay or fold). |
| \(\cos\theta\) | cosine theta | The cosine of angle \(\theta\): "adjacent ÷ hypotenuse" in a right triangle, and the \(x\)-coordinate of the point \(P\) on the unit circle. The "co" stands for "complementary": it is the sine of the complementary angle \(90^\circ - \theta\). |
| \(\tan\theta\) | tangent theta | The tangent of angle \(\theta\): "opposite ÷ adjacent" in a right triangle, which is the slope of a line. The name comes from the Latin word for "touching", because it appears as a length on a tangent line to the unit circle. |
| \(a,\ b,\ c\) | a, b, c | On this page, the three sides of the right triangle: \(a\) is the opposite side (across from angle \(\theta\)), \(b\) is the adjacent side (next to angle \(\theta\)), and \(c\) is the hypotenuse. |
| \(P(x,\ y)\) | P of x, y | The point where the terminal side meets the unit circle. \(P\) stands for "point". Its \(x\)-coordinate is \(\cos\theta\) and its \(y\)-coordinate is \(\sin\theta\). |
| \(O\) | O | The origin (the point \((0, 0)\)), from the first letter of "origin". It is the center of the unit circle and the point the terminal side turns around. |
| \(\sqrt{2},\ \sqrt{3}\) | square root of 2, square root of 3 | The positive numbers whose squares are 2 and 3 (\(\sqrt{2} \approx 1.414,\ \sqrt{3} \approx 1.732\)). They appear in the side ratios of the special right triangles, so they show up often in the exact values of trig ratios for special angles. |
| \(^\circ\) (degrees) | degrees | The unit of angle size in degree measure. One full turn is \(360^\circ\). It is the most familiar way to express angles, also used in everyday life. |
| \(\pi\) | pi | The ratio of a circle's circumference to its diameter (\(\approx 3.14159\)). It is used when angles are given in radians: \(180^\circ = \pi\) radians. The letter is said to come from the first letter of the Greek word for "perimeter". |
Terms
| trigonometric ratio | The ratio of two sides of a right triangle. The three basic ones are sin, cos and tan. Once the angle is fixed, each has one value no matter how big the triangle is. Often shortened to "trig ratio". In the US, it is taught in Geometry and again in Algebra 2 or Precalculus. |
| sine | The full name of sin. Historically it goes back to half of a chord (a segment joining two points on a circle). |
| cosine | The full name of cos. It is short for "complement's sine": the sine of the complementary angle (the angle that adds up to 90°), \(\cos\theta = \sin(90^\circ - \theta)\). |
| tangent | The full name of tan. The name comes from the tangent line to a circle: \(\tan\theta\) can be drawn as a length on the line that touches the unit circle at \((1, 0)\). |
| hypotenuse | The longest side of a right triangle, across from the right angle. In the definitions of sin and cos, it is the number you divide by (the denominator). |
| opposite side | The side across from the angle \(\theta\) you are looking at. Note that which side is the opposite side changes depending on which angle you look at. |
| adjacent side | The side next to the angle \(\theta\) you are looking at (the side of the right angle that touches \(\theta\), not the hypotenuse). Note that which side is the adjacent side changes depending on which angle you look at. |
| unit circle | The circle of radius 1 centered at the origin. With the radius set to 1, the coordinates of a point are directly the values of sin and cos, without dividing by the hypotenuse. It is the tool that extends trig ratios to angles of 90° or more. |
| terminal side | The ray from the origin turned by angle \(\theta\) from the positive \(x\)-axis (the angle is in standard position). It shows the size of the angle on the diagram. |
| obtuse angle | An angle greater than 90° and less than 180°. Trig ratios of an obtuse angle cannot be defined with a right triangle, so the unit circle is used. For obtuse angles, cos and tan are negative. |
| right triangle | A triangle with one right angle (90°). It is where trig ratios were first defined. The other two angles add up to 90°. |
| special right triangles | The 45-45-90 triangle (side ratio \(1 : 1 : \sqrt{2}\)) and the 30-60-90 triangle (side ratio \(1 : \sqrt{3} : 2\)). The exact trig values of the special angles come from the side ratios of these two triangles. They are also the shapes of the two classic drafting triangles (set squares). |
| rationalizing the denominator | Rewriting a fraction so there is no square root in the denominator. Multiplying the top and bottom of \(\dfrac{1}{\sqrt{2}}\) by \(\sqrt{2}\) gives \(\dfrac{\sqrt{2}}{2}\). Both are the same value. US textbooks usually give the rationalized form \(\dfrac{\sqrt{2}}{2}\). |
| degree measure | Measuring angles with one full turn set to 360°. It is the main unit for input and output on this page. The number 360 has many factors (a half, a third, a quarter … of it are all whole numbers), and it has been used since ancient times. |
| radian measure (radians) | Measuring an angle by the length of the arc on a circle of radius 1. \(180^\circ = \pi\) radians, and angles are written as fractions of \(\pi\), such as \(60^\circ = \dfrac{\pi}{3}\). Radians are used from Precalculus on and are the main unit in calculus, so this page shows them too. |
| inverse trigonometric function | A function that works backward from the value of a trig ratio to the angle, as in "which θ gives sin θ = 0.5?". Examples are arcsine (\(\sin^{-1}\), arcsin) and friends. Scientific calculators and Excel have them. The "angle from a value" mode on this page is this calculation. |
| special angle | An angle such as 30°, 45°, 60°, 90° or 120° whose trig ratios can be written as exact values (with fractions and square roots). These are the angles of the special right triangles and angles made from them, such as 180° − θ. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to these topics is the fastest way forward.
| Ratios and rates (Grade 6) |
|
| The coordinate plane (Grades 5–6) |
|
| Circles (Grade 7) |
|
| Similar triangles (Grade 8 and Geometry) |
|
| The Pythagorean theorem (Grade 8) |
|
| Square roots (Grade 8 and Algebra 1) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find sin θ (opposite ÷ hypotenuse)
| Opposite a | 3 |
| Hypotenuse c | 5 |
| sin θ = a ÷ c | =B1/B2 |
Table to find cos θ (adjacent ÷ hypotenuse)
| Adjacent b | 4 |
| Hypotenuse c | 5 |
| cos θ = b ÷ c | =B1/B2 |
Table to find tan θ (opposite ÷ adjacent)
| Opposite a | 3 |
| Adjacent b | 4 |
| tan θ = a ÷ b | =B1/B2 |
Table to find sin, cos and tan from an angle
| Angle θ (degrees) | 60 |
| sin θ | =SIN(RADIANS(B1)) |
| cos θ | =COS(RADIANS(B1)) |
| tan θ | =TAN(RADIANS(B1)) |
Table to find the angle from sin θ (0° to 180°)
| Value of sin θ | 0.5 |
| Angle 1 (degrees) | =DEGREES(ASIN(B1)) |
| Angle 2 (degrees) | =180-B2 |
Excel's SIN, COS and TAN functions take the angle in radians, so to work in degrees, convert with the RADIANS function first. The other way around, answers from ASIN (arcsine) and similar functions come back in radians, so convert them to degrees with the DEGREES function.
The first three tables use a right triangle with side ratio 3 : 4 : 5. The answers are 0.6, 0.8 and 0.75.
The fourth table uses an angle of 60°: sin θ = 0.8660…, cos θ = 0.5 and tan θ = 1.7320….
The fifth table finds the angles where sin θ = 0.5: Angle 1 is 30 and Angle 2 is 150 (between 0° and 180°, two mirror-image angles have the same sine). For cos, use just one formula, "=DEGREES(ACOS(B1))". For tan, use "=DEGREES(ATAN(B1))" and add 180 if the answer is negative.
The first three tables use a right triangle with side ratio 3 : 4 : 5. The answers are 0.6, 0.8 and 0.75.
The fourth table uses an angle of 60°: sin θ = 0.8660…, cos θ = 0.5 and tan θ = 1.7320….
The fifth table finds the angles where sin θ = 0.5: Angle 1 is 30 and Angle 2 is 150 (between 0° and 180°, two mirror-image angles have the same sine). For cos, use just one formula, "=DEGREES(ACOS(B1))". For tan, use "=DEGREES(ATAN(B1))" and add 180 if the answer is negative.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find sin θ (opposite ÷ hypotenuse)
| Opposite a | 3 |
| Hypotenuse c | 5 |
| sin θ = a ÷ c | =B1/B2 |
Table to find cos θ (adjacent ÷ hypotenuse)
| Adjacent b | 4 |
| Hypotenuse c | 5 |
| cos θ = b ÷ c | =B1/B2 |
Table to find tan θ (opposite ÷ adjacent)
| Opposite a | 3 |
| Adjacent b | 4 |
| tan θ = a ÷ b | =B1/B2 |
Table to find sin, cos and tan from an angle
| Angle θ (degrees) | 60 |
| sin θ | =SIN(RADIANS(B1)) |
| cos θ | =COS(RADIANS(B1)) |
| tan θ | =TAN(RADIANS(B1)) |
Table to find the angle from sin θ (0° to 180°)
| Value of sin θ | 0.5 |
| Angle 1 (degrees) | =DEGREES(ASIN(B1)) |
| Angle 2 (degrees) | =180-B2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the angle or side lengths with your own numbers.
How to calculate it in Python
import math
# Find sin, cos and tan from an angle
angle_degree = 60
angle_radian = math.radians(angle_degree) # the math module's trig functions take radians
print(f"sin {angle_degree}° = {math.sin(angle_radian)}")
print(f"cos {angle_degree}° = {math.cos(angle_radian)}")
print(f"tan {angle_degree}° = {math.tan(angle_radian)}")
# Find the angle from the value of a trig ratio (0° to 180°)
sin_value = 0.5
angle_first = math.degrees(math.asin(sin_value)) # arcsine; the answer is in radians, so convert to degrees
angle_second = 180 - angle_first # sin(180° − θ) = sin θ gives the other angle
print(f"Angles where sin θ = {sin_value}: {angle_first}° and {angle_second}°")
Runs with just the math module from the standard library. The trig functions (sin, cos, tan) take the angle in radians, so the key is to convert degrees to radians with math.radians() first. Running this example prints sin 60° = 0.8660…, cos 60° = 0.5000… and tan 60° = 1.7320…, and the angles 30.0° and 150.0° where sin θ = 0.5. To find an angle from cos, use math.acos() (one answer). For tan, use math.atan() and add 180 if the answer is negative.
How to write it in LaTeX and other math languages (copy and paste)
sin (sine) = opposite ÷ hypotenuse
sin θ = a/c
\sin\theta = \dfrac{a}{c}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>sin</mi><mo>⁡</mo><mi>θ</mi>
<mo>=</mo>
<mfrac><mi>a</mi><mi>c</mi></mfrac>
</mrow>
</math>
sin theta = a/c
Sin[30 Degree]
sin(30*Pi/180);
sind(30)
sin θ = a/c
cos (cosine) = adjacent ÷ hypotenuse
cos θ = b/c
\cos\theta = \dfrac{b}{c}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>cos</mi><mo>⁡</mo><mi>θ</mi>
<mo>=</mo>
<mfrac><mi>b</mi><mi>c</mi></mfrac>
</mrow>
</math>
cos theta = b/c
Cos[30 Degree]
cos(30*Pi/180);
cosd(30)
cos θ = b/c
tan (tangent) = opposite ÷ adjacent
tan θ = a/b
\tan\theta = \dfrac{a}{b}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>tan</mi><mo>⁡</mo><mi>θ</mi>
<mo>=</mo>
<mfrac><mi>a</mi><mi>b</mi></mfrac>
</mrow>
</math>
tan theta = a/b
Tan[30 Degree]
tan(30*Pi/180);
tand(30)
tan θ = a/b
Unit circle definition (for angles of 90° or more)
cos θ = x, sin θ = y (point P(x, y) on the unit circle)
\cos\theta = x,\quad \sin\theta = y
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mi>x</mi>
<mo>,</mo>
<mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mi>y</mi>
</mrow>
</math>
cos theta = x, sin theta = y
{Cos[120 Degree], Sin[120 Degree]}
evalf([cos(120*Pi/180), sin(120*Pi/180)]);
[cosd(120), sind(120)]
cos θ = x, sin θ = y
How to have ChatGPT do the calculation
You are a math calculation assistant for trigonometric ratios. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 1. Find sin 135°, cos 135° and tan 135°. Give each both as an exact value (with fractions and square roots) and as a decimal. 2. For 0° ≤ θ ≤ 180°, find every angle θ where sin θ = 0.5. In Python, use the math module and convert degrees to radians before calculating. Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
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Solves a right triangle from any two known values, finding the other sides, angles, area and height with the Pythagorean theorem and trig ratios
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Finds the third side of a right triangle from the other two with a² + b² = c², step by step
Triangle Calculator (Solve for Sides, Angles, Area and More)
Solves any triangle from three values with the law of cosines and the law of sines, including the area and the inscribed and circumscribed circles
Harmonic Addition Calculator (a sin θ + b cos θ as R sin(θ + α), with Max and Min)
Combines a sin θ + b cos θ into a single sine wave R sin(θ + α), with the exact amplitude R = √(a² + b²), α as a fraction of π, and the maximum and minimum
Complex Number to Polar Form Calculator (Modulus, Argument and Quadrant Steps with a Complex Plane Graph)
Converts a + bi to polar form r(cos θ + i sin θ), with the exact modulus and the argument as a fraction of π. Also converts polar form back to rectangular form
De Moivre's Theorem Calculator (Powers and nth Roots of Complex Numbers with Steps and a Polygon Graph)
Finds powers and nth roots of a complex number with De Moivre's theorem, with steps, and draws the roots as a regular polygon on a circle
