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Use our exact value of trig functions calculator to quickly find the values of the trigonometric functions of some selected angles. Keep reading this short article to learn:

  • How to find the exact values of the trig functions without a calculator;
  • The tricks to calculate the exact value of the trigonometric functions of even more angles;
  • How to use and read our exact value of trig functions calculator.

How do I calculate the exact values of trig functions?

To calculate the exact values of trigonometric functions, you can use a set of rules which relate the various trigonometric functions and their values. As a starter, you can consider the values of the trigonometric functions at the quadrant points.

α\boldsymbol{\alpha}

0°\boldsymbol{0\degree}

90°\boldsymbol{90\degree}

180°\boldsymbol{180\degree}

270°\boldsymbol{270\degree}

sin(α)\boldsymbol{\sin(\alpha)}

00

11

00

1-1

cos(α)\boldsymbol{\cos(\alpha)}

11

1-1

00

11

tan(α)\boldsymbol{\tan(\alpha)}

00

\infty

00

-\infty

🙋 Remember that all trigonometric functions have a periodicity of 2π2\pi: this means that for every angle α\alpha, the exact calculated value of the trig function ff satisfies the rule: f(α)=f(2kπ+α)f(\alpha) = f(2k\pi+\alpha), where kk is any integer number.

Next, you can find the exact value of the trig function calculated in the special right triangle with angles 30°30\degree, 60°60\degree, and 90°90\degree, and 45°45\degree, 45°45\degree, and 90°90\degree. To do so, remember the trigonometric theorems in a right triangle:

sin(α)=oppositehypothenusecos(α)=adjacenthypothenusetan(α)=oppositeadjacent\begin{split} \sin(\alpha) &= \frac{\mathrm{opposite}}{\mathrm{hypothenuse}}\\ \cos(\alpha) &= \frac{\mathrm{adjacent}}{\mathrm{hypothenuse}}\\ \tan(\alpha) &= \frac{\mathrm{opposite}}{\mathrm{adjacent}} \end{split}

By knowing the length of the triangle's sides, we can find that:

α\boldsymbol{\alpha}

30°\boldsymbol{30\degree}

60°\boldsymbol{60\degree}

45°\boldsymbol{45\degree}

sin(α)\boldsymbol{\sin(\alpha)}

1/21/2

3/2\sqrt{3}/2

2/2\sqrt{2}/2

cos(α)\boldsymbol{\cos(\alpha)}

3/2\sqrt{3}/2

1/21/2

2/2\sqrt{2}/2

tan(α)\boldsymbol{\tan(\alpha)}

1/31/\sqrt{3}

3\sqrt{3}

11

We reached a point where we need to introduce a set of slightly more complex rules if you want to learn how to find the exact value of the trig functions without a calculator for even more angles.

You can use the following three sets of rules:

  • Double-angle formulas;
  • Triple-angle formulas; and
  • Half-angle formulas.

The letters are the most important in this context since they allow you to calculate the exact values of the trig functions for angles such as 15°15\degree, 22.5°22.5\degree, and so on. Here they are:

sin(θ2)=1cos(θ)2cos(θ2)=1+cos(θ)2tan(θ2)=1cos(θ)1+cos(θ)\begin{split} \sin\left(\frac{\theta}{2}\right)& = \sqrt{\frac{1-\cos(\theta)}{2}}\\[1em] \cos\left(\frac{\theta}{2}\right)& = \sqrt{\frac{1+\cos(\theta)}{2}}\\[1em] \tan\left(\frac{\theta}{2}\right)& = \sqrt{\frac{1-\cos(\theta)}{1+\cos(\theta)}} \end{split}

Be wary of these formulas, as they can lead easily into multiple nested square roots!

This is not the last trick in our sleeves: you can build as many equations as you want by combining the formulas for the double and triple angles until you isolate a single function. You will probably end in a quadratic equation, so be careful about the sign!

Here are the double and triple-angle formulas for the three main trigonometric functions:

sin(2θ)=2sin(θ)cos(θ)cos(2θ)=cos2(θ)sin2(θ)tan(2θ)=2tan(θ)1tan2(θ)\begin{split} \sin\left(2\theta\right)& = 2\sin(\theta)\cos(\theta)\\ \cos\left(2\theta\right)& = \cos^2(\theta)-\sin^2(\theta)\\ \tan\left(2\theta\right)& = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \end{split}

And:

sin(3θ)=4sin3(θ)+3sin(θ)cos(3θ)=4cos3(θ)4cos(θ)tan(3θ)=3tan(θ)tan3(θ)13tan2(θ)\begin{split} \sin\left(3\theta\right)& = -4\sin^3(\theta)+3\sin(\theta)\\ \cos\left(3\theta\right)& = 4\cos^3(\theta)-4\cos(\theta)\\ \tan\left(3\theta\right)& = \frac{3\tan(\theta)-\tan^3(\theta)}{1-3\tan^2(\theta)} \end{split}

To complete these calculations, you may have to use some trigonometric equivalences, for example:

  • sin(θ)=±1cos2(θ)\sin(\theta) = \pm \sqrt{1-\cos^2(\theta)}; or
  • cos(θ)=±1/1+tan2(θ)\cos(\theta) = \pm 1/\sqrt{1+\tan^2(\theta)}.

Have fun exploring these possibilities!

Using the periodicty of the trigonometric functions to calculate their exact values in other quadrants

After you calculate the exact values of the trig functions, find the results for the three other quadrants of the trigonometric circle by applying these functions' periodicity and reflective properties.

α\boldsymbol{\alpha}

α\boldsymbol{-\alpha}

(π/2)α\boldsymbol{(\pi/2)-\alpha}

πα\boldsymbol{\pi-\alpha}

(3π/2)α\boldsymbol{(3\pi/2)-\alpha}

sin(α)\sin(\alpha)

sin(α)-\sin(\alpha)

cos(α)\cos(\alpha)

sin(α)\sin(\alpha)

cos(α)-\cos(\alpha)

cos(α)\cos(\alpha)

cos(α)\cos(\alpha)

sin(α)\sin(\alpha)

cos(α)-\cos(\alpha)

sin(α)-\sin(\alpha)

tan(α)\tan(\alpha)

tan(α)-\tan(\alpha)

cot(α)\cot(\alpha)

tan(α)-\tan(\alpha)

cot(α)\cot(\alpha)

And:

α\boldsymbol{\alpha}

α+(π/2)\boldsymbol{\alpha+(\pi/2)}

α+π\boldsymbol{\alpha+\pi}

α+(3π/2)\boldsymbol{\alpha+(3\pi/2)}

sin(α)\sin(\alpha)

cos(α)\cos(\alpha)

sin(α)-\sin(\alpha)

cos(α)-\cos(\alpha)

cos(α)\cos(\alpha)

sin(α)-\sin(\alpha)

cos(α)-\cos(\alpha)

sin(α)\sin(\alpha)

tan(α)\tan(\alpha)

cot(α)-\cot(\alpha)

tan(α)\tan(\alpha)

cot(α)-\cot(\alpha)

How to use our exact value of trig functions calculator

Use our calculator to find the exact value of any trig function by simply inputting the desired angle in degrees or radians. We will print all the values of the trigonometric functions, and if there is a neat expression for these results, print it!

Other specific trigonometric calcualtors

Use our other tools to discover and calculate trigonometry! Try the:

FAQ

How do I calculate the exact value of the trig functions?

To calculate the exact value of the trig functions, you can use one of these methods:

  1. Trigonometric identities;
  2. Periodicity and reflection of the trigonometric functions;
  3. Properties of the right triangles; or
  4. Duplication and halving formulas for the trigonometric functions.

You can also easily find tables of precomputed exact values of the function for the angles, which results in the neatest values.

How do I calculate the exact value of sin(30°) and cos(30°)?

To calculate the values of sin(30°) and cos(30°), remember that those angles appear in the special 30°, -60°, -90° right triangle.

  1. Let's say that the hypotenuse of that triangle is 1 cm long. The shortest side will be then 0.5 cm long.
  2. Use the trigonometry theorems in right triangles to find the value of sin(30°):
    sin(30°) = hypothenuse/opposite = 1/2
  3. To find the value of cos(30°), use the following trigonometric identity:
    cos(30°) = sqrt(1 - sin²(30°)) = sqrt(1 - 1/4) = sqrt(3)/2

Can I find exact values for all the trigonometric functions?

Technically, yes: by using trigonometric identities and other properties of the trigonometric functions, you can compute the exact value of trigonometric functions for any angle. However, the result of such operations often contains nested radicals, and the more exotic the angle, the more complexity of the result. You can find neat results for special angles (30°, 60°, 45°), their immediate multiples and submultiples (halves and doubles), and some other selected angles (like the 18°).

Davide Borchia
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