Sine Cosine Tangent Calculator (2024)

With our sine cosine tangent calculator, you will learn everything you need about the most important trigonometric functions: discover how to identify them, their relationship to right triangles, and how to calculate them in the blink of an eye!

Trigonometric functions: how to calculate sine, cosine, and tangent

The trigonometric functions are mathematical functions that associate numerical values to angles according to specific rules. There are many trigonometric functions, but the most important are sine, cosine, and tangent. Since they take an angle as an argument, it's natural to identify them on a circle.

In the image below, you can see a trigonometric circle, a circle with radius 111 where for a specific angle α\alphaα, we highlighted the segments corresponding to sine and cosine.

Sine Cosine Tangent Calculator (1)

Note how the tangent lies outside of the circle, tangent to it, as the name says. While sine and cosine are readily identifiable as the projections of the radius on the vertical and horizontal axis, we need to see the definition of the tangent to understand how to find it and visualize it:

tan(α)=sin(α)cos(α)\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}tan(α)=cos(α)sin(α)

In the picture below, we represented the three trigonometric functions: we calculated sine, cosine, and tangent values for the angles between 0°0\degree and 360°360\degree360°.

Sine Cosine Tangent Calculator (2)

Even from this quick peek into trigonometry, you can identify many properties of these functions:

  • The values calculated for sine, cosine, and tangent are periodic. In particular:
    • Sine and cosine have periodicity equal to 360°360\degree360°; and
    • The tanget has periodicity 180°180\degree180°.
  • Sine and cosine have values comprised between 1-11 and 111. The tangent assumes values between -\infty and \infty.
  • Sine and cosine switch values periodically in the four quadrants of the circle;
  • Sine and cosine are "out of phase" by 90°90\degree90° with each other.

Keep exploring these functions to discover many other, almost endless, properties!

After this introduction to trigonometric functions, it's time to learn how to calculate sine, cosine, and tangent in a triangle. Jump to the next section!

How do I calculate sine, cosine, and tangent in a triangle?

Right triangles and trigonometric functions share a deep connection: you can calculate the values of the latter and the geometric characteristic of the former swiftly without remembering any formulas: thanks to the neat graphic representation of the functions, you can always draw what you need on a piece of paper, and you'll recall the math in no time!

To understand the connection between right triangles and trigonometric functions, look at the detail of the previous image.

Sine Cosine Tangent Calculator (3)

We took an angle and highlighted the values of sine and cosine. As you can see, sine and cosine form a right angle; thus, they correspond to the catheti (legs) of the triangle. The radius of the circle becomes the hypotenuse. Immediately, you can identify the fundamental relationship between sine and cosine:

sin2(α)+cos2(α)=1,\sin^2(\alpha)+\cos^2(\alpha) = 1,sin2(α)+cos2(α)=1,

which we obtain from the Pythagorean theorem. But this alone would not tell us how to calculate sine, cosine, and tangent in a triangle. If you consider the angle α\alphaα, you can find the values of the functions thanks to a set of ratios between the lengths of the triangle's sides:

sin(α)=oppositehypotenusecos(α)=adjacenthypotenusetan(α)=oppositeadjacent\begin{split}\sin(\alpha) & = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\\[1em]\cos(\alpha) & = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\\[1em]\tan(\alpha) & = \frac{\mathrm{opposite}}{\mathrm{adjacent}}\end{split}sin(α)cos(α)tan(α)=hypotenuseopposite=hypotenuseadjacent=adjacentopposite

By rotating the triangle and placing the desired acute angle in the center of the trigonometric circle, you can calculate sine, cosine, and tangent for the second angle, too.

Our sine, cosine, and tangent calculator allows you to see the values of the trigonometric functions directly for a specific angle. You can also insert the known information about a right triangle and see if they are enough to calculate the rest. If you found this tool useful, check out our other calculators:

  • Trigonometry calculator;
  • Cosine triangle calculator;
  • Sine triangle calculator;
  • Trig triangle calculator;
  • Right triangle trigonometry calculator;
  • Trig calculator;
  • Tangent ratio calculator; and
  • Tangent angle calculator.

FAQ

Why can I calculate sine, cosine, and tangent in a triangle?

Right triangles and trigonometric functions are deeply connected. If you look at sine and cosine on a trigonometric circle, you can see them forming a right angle, which allows you to define a right triangle, with catheti corresponding to sine and cosine.

By calculating the ratios between hypotenuse and catheti, you can find a measure of the "slope" of the segment in analysis, which is nothing but the result of a trigonometric function.

How do I calculate sine, cosine, and tangent in a 3-4-5 triangle?

To calculate sine, cosine, and tangent in a 3-4-5 triangle, follow these easy steps:

  1. Place the triangle in a trigonometric circle with an acute angle in the center.

  2. Identify the adjacent and opposite catheti to the angle.

  3. Compute the results of the trigonometric functions for that angle using the following formulas:

    • sin(α) = opposite / hypotenuse;

    • cos(α) = adjacent / hypotenuse; and

    • tan(α) = opposite / adjacent.

  4. Rotate and flip the triangle to find the values of sine, cosine, and tangent for the other acute angle, following the same steps.

How do I calculate the sine in a right triangle?

To calculate the sine in a right triangle, follow these steps:

  1. Identify the angle α in the analysis.

  2. Assign the label opposite to the cathetus not touching the angle.

  3. Assign the label adjacent to the cathetus touching the angle.

  4. Calculate the sine with the ratio of the opposite cathetus and the hypotenuse:
    sin(α) = opposite/hypotenuse.

  5. To calculate the sine of the other acute angle ß, repeat Steps 2-4 or note that, in this case, sin(ß) = cos(α) thanks to the property sin(90° - α) = cos(α).

How do I calculate the tangent if I know sine and cosine?

To calculate the tangent if you know sine and cosine, simply compute the ratio between these two quantities:

tan(α) = sin(α)/cos(α)

Since both sine and cosine go to zero in some instances (but never at the same time), the function has singularities where it is not defined: we find them at every argument of the form 90° + k × π, where k is an integer.

Sine Cosine Tangent Calculator (2024)

FAQs

How to calculate sin cos tan on calculator? ›

Simply enter the value of the angle in degrees and push the "sin," "cos," or "tan" button. Convert the sine of an angle into the measure of the angle. Input the value of the sine, then hit the button that says "arcsin," or "sin-1." Convert the cosine or tangent of an angle into the measure of the angle.

How do you memorize sine, cosine tangent? ›

The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English: Sine = Opposite ÷ Hypotenuse. Cosine = Adjacent ÷ Hypotenuse. Tangent = Opposite ÷ Adjacent.

How to calculate sin, cos, and tan? ›

Sin Cos Tan Formula

Now as per sine, cosine and tangent formulas, we have here: Sine θ = Opposite side/Hypotenuse = BC/AC. Cos θ = Adjacent side/Hypotenuse = AB/AC. Tan θ = Opposite side/Adjacent side = BC/AB.

Why is my calculator not doing trigonometry? ›

Before using your calculator to find the values of the trigonometric ratios of angles measured in degrees, you need to ensure that it is set to use the correct units. Always check that your calculator is using the correct system of angle measurement before using trigonometric ratios.

Can you calculate sin and cos without calculator? ›

Another way to solve, knowing the formulas of addition and subtraction identities such as sin(x + y) = sin(x)cos(y) + cos(x)sin(y) and cos(x + y) = cos(x) cos (y) – sin(x) sin(y).

Why is sin opposite over hypotenuse? ›

The sine is always the measure of the opposite side divided by the measure of the hypotenuse. Because the hypotenuse is always the longest side, the number on the bottom of the ratio will always be larger than that on the top.

What sin is 0? ›

The value of sin 0° is 0.

How to solve sin cos tan by hand? ›

To calculate sine , divide the length of the side opposite the angle by the length of the hypotenuse . For cosine , divide the length of the adjacent side by the length of the hypotenuse . And for tangent , divide the length of the opposite side by the length of the adjacent side .

What is the math behind sin? ›

The formula for sine is opposite over hypotenuse. In other words sin (x) is equal to the length of the side opposite x and the length of the hypotenuse.

What is the actual formula for sine? ›

The sine of an angle of a right-angled triangle is the ratio of its perpendicular (that is opposite to the angle) to the hypotenuse. The sin formula is given as: sin θ = Perpendicular / Hypotenuse. sin(θ + 2nπ) = sin θ for every θ

How to solve degree in calculator? ›

Degree is the initial default angle mode for a standard scientific calculator and a programmable scientific calculator. Press [MODE] to cycle through screens on the display until the angle mode selection screen is appears. Input the number that corresponds to the angle mode you want to select.

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