All the trigonometric functions are calculated for theta in this handy tool: input an angle with any measurement units, and see the values of the trigonometric functions.

Keep reading this article for a quick refresher on:

  • The trigonometric functions;
  • How to calculate for theta any of the trigonometric functions;
  • Other resources to help you with trigonometry.

The trigonometric functions

A trigonometric function is a function that tasks as its argument an angle and returns a dimensionless number corresponding to the length of a segment associated with the angle itself. We identify three fundamental trigonometric functions: the sine, the cosine, and the tangent. We can see all of them and their relationship with angles in the trigonometric circle.

Trigonometric circle

In this picture, you can see how sine and cosine are orthogonal and how they correspond to the vertical and horizontal projection of the radius at a given angle theta. The tangent is slightly more complex to define, but luckily we can use a handy ratio to find its value:

tan(θ)=sin(θ)cos(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

By identifying a right triangle in the trigonometric circle, we can find another fundamental relation of trigonometry, in this case, between sine and cosine.

cos2(θ)+sin2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1

With this relations, you should be able to find the sine, cosine, and tangent for any angle.

Solving for theta: calculate the trigonometric functions for any angle

Calculating the trigonometric functions' values for any angle is no easy task! Only certain angles return neat values; among them, we remember 0°\footnotesize0\degree, 30°\footnotesize30\degree, 45°\footnotesize45\degree, 60°\footnotesize60\degree, and 90°\footnotesize90\degree.
For these angles, we can remember the values of one of the trigonometric functions and find the others with the formula we've seen above. For other angles, it's better to resort to a calculator.

You can find the values of other trigonometric functions, too: the multiplicative inverses of sine, cosine, and tangent are well-defined and useful functions. You don't need to know additional math to calculate them: it suffices to perform a simple division. Here they are:

  • The cosecant: csc(θ)=1/sin(θ)\footnotesize\csc(\theta) = 1/\sin(\theta);
  • The secant: sec(θ)=1/cos(θ)\footnotesize\sec(\theta) = 1/\cos(\theta): and
  • The cotangent: cot(θ)=1/tan(θ)\footnotesize\cot(\theta) = 1/\tan(\theta).

The inverse trigonometric functions

If you are wondering what are the functions that result in an angle when you input the values of the trigonometric functions (hence the segments on the unit circle), here you can discover and learn about the inverse trigonometric functions. These functions take as argument a dimensionless number and return the angle that would produce that argument when fed to the corresponding trigonometric function. Let's understand this concept with one of the inverse trigonometric functions: the inverse sine or arcsin. Given the value 0.5\footnotesize0.5, we compute the inverse sine this way:

arcsin(0.5)=θ\arcsin(0.5) = \theta

Which corresponds to the inverse operation:

sin(θ)=0.5\sin(\theta) = 0.5

We know that the angle that has sine 0.5\footnotesize0.5 is 30°\footnotesize30\degree, hence we can also write:

arcsin(0.5)=30°\arcsin(0.5) = 30\degree

Notice that the arcsine is defined only for values between 1\footnotesize-1 and +1\footnotesize+1, the range of the sine function. Moreover, we notice that arcsin(θ)=arcsin(180°θ)\footnotesize\arcsin(\theta) = \arcsin(180\degree - \theta): the inverse sine is somehow ambiguous!

We identify as many inverse trigonometric functions as the number of trigonometric functions. However, we are usually interested in inverse sine (arcsin), inverse cosine (arccos), and inverse tangent (arctan). You can find these functions in our theta calculator!

Calculating double angle identities

You can use our theta calculator to also find the result of the double angle identities. Thanks to a combination of properties of the trigonometric functions, a bit of ingenuity, and some luck, it may happen that the result of trigonometric functions of double angles is still "neat"!

The three most important trigonometric functions have elegant expressions for their double angle analogs:

  • sin(2θ)=2sin(θ)cos(θ)\footnotesize \sin(2\theta) = 2 \sin(\theta)\cos(\theta);
  • cos(2θ)=cos2(θ)sin2(θ)\footnotesize \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta); and
  • tan(2θ)=2tan(θ)/(1tan2(θ))\footnotesize\tan(2\theta) = 2\tan(\theta)/(1-\tan^2(\theta)).

You can learn everything about double angle identities at our double angle identities calculator!

Other trigonometry calculator

If this tool turned out to be helpful, you might find our other trigonometry calculators useful too. Try our:

FAQ

How do I solve the trigonometric functions for theta?

Let's deal with sine, cosine, and tangent first. One possible approach to solve these functions for theta is:

  1. Find the value of sin(θ). You may have to use a calculator to do so.
  2. Find the value of the cosine by inverting the following formula: cos²(θ) + sin²(θ)=1.
  3. Once you know sine and cosine, calculate the tangent: tan(θ) = sin(θ)/cos(θ).

To find the value of additional trigonometric functions, follow these extra steps:

  • cot(θ) = 1/tan(θ);
  • csc(θ) = 1/sin(θ); and
  • sec(θ) = 1/cos(θ).

How do I solve for θ = 30°?

The angle θ = 30° is one of the few angles resulting in a neat value when used as an argument of the trigonometric functions. To solve the trigonometric functions for theta:

  1. Find the value of sin(30°) = 1/2.
  2. Find the cosine: cos(30°) = sqrt(1 - sin²(30°) = sqrt(3/4) = sqrt(3)/2.
  3. Use the ratio between sine and cosine to find the tangent: `tan(30°) = sin(30°)/cos(30°) = 1/sqrt(3).

What is theta in math?

Theta is a variable usually associated with angles. When you read "theta" in a mathematics or physics book, assume that you are dealing with an angle — any angle — without a specified value. We use theta when we don't expect to meet other angles: if it's not the case, it's the convention to use the Greek letters alpha, beta, gamma, etc.

Davide Borchia
Calculate theta with inverse functions
Output of the function
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