How do you calculate sine, cosine and tangent for an angle?

Select the "Trigonometric Functions Values" task, enter the angle and select the unit (degrees, radians or grad). The calculator will immediately provide the values of the six functions. For example: sin(45°) = 0.7071, cos(45°) = 0.7071, tan(45°) = 1.

This is a mnemonic for the three important ratios in a right triangle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. If you label the sides relative to the angle, this mnemonic will help you remember which ratio to use.

What does that notation with a minus sign over sin mean?

This denotes the inverse sine function, also known as arcsine, and returns the angle whose sine is x. It does not represent the reciprocal of the sine. So arcsin(0.5) is the angle in degrees 30, not 1/sin.

What two values are required to solve a right triangle?

It must be either two sides or one side and an acute angle. Two angles alone are not sufficient. Even if two triangles have the same angles they can be enlarged or reduced to any size ratio.

Why is tangent sometimes undefined?

The tangent is defined as the sine divided by the cosine, so it's undefined where the cosine is zero (e.g., at 90 or 270 degrees), and calculators will show a blank field in such cases. For the same reason, secant and cosecant also show a blank field, while cotangent and acotangent are blank where the sine is zero (e.g., at 0 or 180 degrees).

Should you use degrees or radians?

Most right triangles and everyday problems use degrees. Calculations using the unit circle and in calculus use radians, where a full revolution is 2 Pi. Since you can change the units in each input field for angles, you can enter the same angle as either 45 degrees or about 0.7854 radians.

Trigonometry Calculator

Choose a task

What do you want to do?

Pick a task. Give the calculator an angle to evaluate the six trig functions, a ratio to recover the angle, or any two parts of a right triangle to solve for the rest.

Trig function values (sin, cos, tan...)

Inverse trig (find the angle)

Right triangle solver

Your input

Enter two known parts of a right triangle in solver mode; the calculator fills in the rest.

Angle (theta)

Answers

sin(theta)

cos(theta)

tan(theta)

cot(theta)

sec(theta)

csc(theta)

sin(theta)

For theta = 45 deg (°): sine 0.7071, cosine 0.7071, tangent 1. The reciprocals (cosecant, secant, cotangent) are listed on the left.

Steps, table, and graph

Show the calculation steps

Walk through the arithmetic for the answer you are looking at.

Show the special-angle table

List the exact sine, cosine, and tangent of the angles that come up most.

Show the sine and cosine curves

Plot sine and cosine across a full turn and mark where your angle lands.

Trigonometry is the study of the relationships between the angles and sides of a triangle. The word comes from two Greek words, "trigonon" (triangle) and "metron" (measure). This branch of mathematics has three main applications: calculating six trigonometric functions for a given angle, finding an angle using inverse functions, and solving right triangles with any two sides.

Six trigonometric functions

In a right triangle the side opposite to the angle is called the "opposite", the side that is next to the angle is called the "adjacent" and the longest side is called the "hypotenuse". This gives three ratios, which are known as SOH-CAH-TOA.

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The remaining three functions are the reciprocals of the first three respectively. As the calculator displays all six functions simultaneously, no fractions need to be inverted manually.

\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

Here is a simple example: If you enter 45 degrees, the tool will output sin 45 = 0.7071, cos 45 = 0.7071, tan 45 = 1 and their respective reciprocals. At 90 degrees tangent and secant are blank, and at 0 degrees cotangent and cosecant are blank. This is because the denominator is zero at these points and the function is undefined.

Degree, radiant, degree measure

Angles can be measured in different units and this calculator supports the three most commonly used ones. Degrees divide a full circle into 360 equal parts and are suitable for many word problems involving right triangles. Radians, on the other hand, represent an angle as the arc length on a unit circle, where a full circle is 2 Pi. In connection with the unit circle or calculus, radians are often the more natural choice.

180^\circ = \pi \text{ rad} \qquad 1^\circ = \frac{\pi}{180}\text{ rad} \approx 0.01745\text{ rad}

By switching the unit on each input field for angles you can enter and read off angles in degrees, radians or grads. 45 degrees, pi divided by 4 radians and 50 grads are actually the same angle.

Inverse Trigonometric Functions: Find an Angle Given a Ratio

If you know the ratio and want to find the angle that produces this ratio, use an inverse function. The inverse sine is written as arcsin or with a negative exponent of -1 on the sine. This is not the reciprocal of the sine.

\text{if } \sin\theta = x \text{ then } \theta = \arcsin x \qquad \theta = \arccos x \qquad \theta = \arctan x

Each of the inverse functions will return a principal angle. The results from arcsin and arctan are between -90 degrees to 90 degrees, while the results from arccos are between 0 degrees and 180 degrees. Arcsin and arccos only accept values between -1 and 1. This is because there are no real angles whose sine or cosine is outside of this range. In contrast, arctan accepts any value.

For example, arcsin(0.5) is equal to 30 degrees, arccos(0.5) is equal to 60 degrees and arctan(1) is equal to 45 degrees.

Solve a Right Triangle

If you give the solution tool any two parts of a right triangle it can calculate the remaining values. Two independent knowns are sufficient. This could be two sides or one side and an acute angle. Two angles alone is not enough since triangles with equal angles can have different sizes.

This solving tool internally calculates based on the Pythagorean theorem and the fact that the sum of two acute angles in a right triangle is always 90 degrees.

a^2 + b^2 = c^2 \qquad \alpha + \beta = 90^\circ \qquad \text{area} = \tfrac{1}{2}ab \qquad h = \frac{ab}{c}

Here a and b are the two legs, c is the hypotenuse, α is the angle opposite to a and β is the angle opposite to b. The height h is an altitude drawn from the right corner onto the hypotenuse, which divides the triangle into two smaller triangles similar to each other and to the original triangle.

A complete example:

Assume the two legs of a right triangle have lengths a = 3 and b = 4. By the Pythagorean theorem, the hypotenuse has length 5, which is a well-known Pythagorean triple.

c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \qquad \alpha = \arctan\frac{3}{4} \approx 36.87^\circ \qquad \beta \approx 53.13^\circ

The area is half of the product of 3 and 4, so it's 6. The perimeter is 12. And then the altitude to the hypotenuse is going to be the product of 3 and 4 divided by 5, which is 2.4. So if you put in a = 3 and b = 4 into the solution mode, you will get exactly the same result.

The solution tool can also be used in reverse. If the angle beta = 30 degrees, the opposite side b = 10 and the hypotenuse c = 20 are known, then the tool immediately shows alpha = 60 degrees and the remaining cathetus a = 17.32.

Special angles to remember:

Some angles have simple, exact values that come up again and again. The following table shows these angles. The calculator can show the same table if needed.

Angle	sin	cos	tan
0 deg	0	1	0
30 deg	1/2	sqrt 3 / 2	1 / sqrt 3
45 deg	sqrt 2 / 2	sqrt 2 / 2	1
60 deg	sqrt 3 / 2	1/2	sqrt 3
90 deg	1	0	undefined

The ratio of the sides in a 30-60-90 triangle is 1 to square root of 3 to 2 while the ratio of the sides in a 45-45-90 triangle is 1 to 1 to square root of 2. In both triangles, if you know one side length, then you can immediately figure out the other two by using these ratios.

Applications of trigonometry.

The use of trigonometry is not limited to what you learn in the classroom. Surveyors and construction workers use trigonometry to determine heights and distances that cannot be measured directly, such as the pitch of a roof or the steepness of a slope.

Physicists use sine and cosine to break forces and velocities into components and represent different oscillations and waves with the same functions. Everything from pendulum motion to sound and light is affected by them.

Navigation, astronomy, computer graphics and medical imaging all rely on these relationships to convert angles into positions and vice versa.

Used symbols:

The symbols used in the above equations are shown in the table below.

Symbol	Meaning	Example
theta	The angle being evaluated	45 deg
a, b	The two legs of the right triangle	3 and 4
c	The hypotenuse, opposite the right angle	5
alpha, beta	The acute angles, opposite a and b	36.87 and 53.13 deg
h	The altitude to the hypotenuse	2.4
x	A trig ratio fed to an inverse function	0.5

This calculator is for general information and educational purposes only. The solution tool assumes that in a right triangle the right angle is C. For non-right triangles use the law of sines or the law of cosines.

Frequently asked questions

How do you calculate sine, cosine and tangent for an angle?: Select the "Trigonometric Functions Values" task, enter the angle and select the unit (degrees, radians or grad). The calculator will immediately provide the values of the six functions. For example: sin(45°) = 0.7071, cos(45°) = 0.7071, tan(45°) = 1.
What is Soh-Cah-Toa?: This is a mnemonic for the three important ratios in a right triangle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. If you label the sides relative to the angle, this mnemonic will help you remember which ratio to use.
What does that notation with a minus sign over sin mean?: This denotes the inverse sine function, also known as arcsine, and returns the angle whose sine is x. It does not represent the reciprocal of the sine. So arcsin(0.5) is the angle in degrees 30, not 1/sin.
What two values are required to solve a right triangle?: It must be either two sides or one side and an acute angle. Two angles alone are not sufficient. Even if two triangles have the same angles they can be enlarged or reduced to any size ratio.
Why is tangent sometimes undefined?: The tangent is defined as the sine divided by the cosine, so it's undefined where the cosine is zero (e.g., at 90 or 270 degrees), and calculators will show a blank field in such cases. For the same reason, secant and cosecant also show a blank field, while cotangent and acotangent are blank where the sine is zero (e.g., at 0 or 180 degrees).
Should you use degrees or radians?: Most right triangles and everyday problems use degrees. Calculations using the unit circle and in calculus use radians, where a full revolution is 2 Pi. Since you can change the units in each input field for angles, you can enter the same angle as either 45 degrees or about 0.7854 radians.

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Arcsin CalculatorFind the inverse sine (arcsin) of any value from -1 to 1. Read the angle in degrees, radians, gradians, or turns, with the cosine, the complementary angle, common values, and a graph.

Arccos CalculatorFind the inverse cosine (arccos) of any value from -1 to 1. Read the angle in degrees, radians, gradians, or turns, with the sine, the complementary angle, common values, and a graph.

Arctan CalculatorFind the inverse tangent (arctan) of any real number. Read the angle in degrees, radians, gradians, or turns, with the sine, cosine, complementary angle, a four-quadrant atan2 mode, common values, and a graph.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Trigonometric functions
Definitions of the six trig functions, their reciprocals, and inverse functions.
Wikipedia: Solution of triangles
How the known parts of a triangle determine the rest, including the right-triangle case.