Sine Cosine Tangent Calculator – calculatorpeak.com

About the Sine Cosine Tangent Calculator

Sine, cosine and tangent are the three ratios that connect an angle to the sides of a right triangle. Give them an angle and they tell you the shape; give them a shape and they tell you the angle.

That is the whole of basic trigonometry, and it is more useful than it sounds. Every time someone measures the height of a building from the ground, works out a roof pitch, aims a satellite dish, models a sound wave or animates a bouncing ball, these three ratios are underneath it.

This calculator gives all six ratios for any angle, works backwards from a ratio to the angle, or takes two sides of a right triangle and produces both. Where the answer is exact — and at the special angles it is — the exact value is shown rather than a rounded decimal.

How to Use the Sine Cosine Tangent Calculator

From an angle. Enter it in degrees or radians. All six ratios come back.

From a ratio. Choose sine, cosine or tangent and enter the value. You get the angle, plus the other angles that share that ratio — which arcsin alone will not tell you.

From two sides. Enter the opposite and adjacent sides of a right triangle and the calculator finds the angle and every ratio.

The Three Ratios

For an acute angle θ in a right triangle:

  sin θ = opposite / hypotenuse       SOH
  cos θ = adjacent / hypotenuse       CAH
  tan θ = opposite / adjacent         TOA

And their reciprocals, which are the same information upside down:

  csc θ = 1 / sin θ = hypotenuse / opposite
  sec θ = 1 / cos θ = hypotenuse / adjacent
  cot θ = 1 / tan θ = adjacent / opposite

Note that secant pairs with cosine and cosecant with sine — the opposite way round from what the names suggest. It catches people out constantly.

"Opposite" and "adjacent" depend on which angle

This is the thing to fix in your mind before anything else. The hypotenuse never changes: it is opposite the right angle, always the longest side. But the other two swap roles depending on which acute angle you are working from.

In a 3-4-5 triangle, from the angle facing the 3: opposite is 3, adjacent is 4. From the other acute angle: opposite is 4, adjacent is 3. Same triangle, different ratios.

Beyond the Triangle: the Unit Circle

Right-triangle definitions only work for angles between 0° and 90°. The unit circle extends them to everything.

Draw a circle of radius 1. Take an angle measured anticlockwise from the positive x-axis, and mark where its line crosses the circle. That point's coordinates are (cos θ, sin θ). That is the definition, and the triangle version is the special case where the point is in the first quadrant.

Now everything follows:

Sine is the height, cosine is the width. Above the axis sine is positive;

below it, negative.

At 90° the point is at the top: cos = 0, sin = 1.
At 180° it is at the far left: cos = −1, sin = 0.
Past 360° you have simply gone round again, which is why the ratios repeat.

| Quadrant | Angles | Positive there | |----------|--------|----------------| | 1 | 0–90° | all six | | 2 | 90–180° | sine and cosecant | | 3 | 180–270° | tangent and cotangent | | 4 | 270–360° | cosine and secant |

The reference angle ties it together: the acute angle to the nearest part of the x-axis. The value at 150° is the value at 30° with quadrant 2's signs. Learn the first quadrant and the reference angle, and you know the whole circle.

The Exact Values

At five angles — and their reflections — the ratios are exact rather than endless decimals. These are worth knowing:

| θ | sin | cos | tan | |---|-----|-----|-----| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | undefined |

They come from two triangles you have already met: the 45-45-90 (half a square) and the 30-60-90 (half an equilateral triangle). Nothing to memorise if you can sketch either.

This calculator recognises these angles and returns exact values, which also avoids an ugly problem: Math.tan(Math.PI) in any programming language gives −0.00000000000000012, not 0, because π cannot be stored exactly in binary — the stored value is a hair past π, so the tangent comes out slightly negative. Reporting that as the tangent of 180° would be technically defensible and practically absurd.

Undefined Is Not Infinity

At 90°, the tangent has no value. The adjacent side is zero and the division cannot be done.

You will often see "tan 90° = ∞". That is misleading. Infinity is not a number you could substitute into an equation, and the tangent here is not a very large quantity — it is no quantity at all. Approach 90° from below and the tangent grows without bound; approach from above and it is large and negative. There is no single value for it to be.

This calculator says undefined, which is the honest answer, and the same applies to secant at 90° and cotangent at 0°.

One Ratio, Many Angles

Ask a calculator for arcsin(0.5) and it answers 30°. That is true but incomplete: 150° also has a sine of 0.5, and so does 390°, and 510°, and infinitely many others.

  sin θ = 0.5   →   θ = 30°, 150°, 390°, 510°, …
  cos θ = 0.5   →   θ = 60°, 300°, 420°, …
  tan θ = 1     →   θ = 45°, 225°, 405°, …

The inverse functions return one principal value because a function may only return one answer. But in a triangle problem the second solution is frequently the one you want — this is exactly what makes the SSA case ambiguous. So this calculator names the companion angle explicitly rather than leaving you with half the picture.

Note that the tangent repeats every 180°, not 360°, because it is a ratio of two things that both flip sign in the opposite quadrant.

Step-by-Step Example

Height of a tree. Standing 20 m away, you measure the angle up to the top as 32°.

  tan 32° = height / 20
  height  = 20 × tan 32° = 20 × 0.6249 = 12.50 m

Add your eye height — say 1.6 m — for about 14.1 m.

A ramp. A ramp rises 0.8 m over a 9.6 m run. What angle?

  tan θ = 0.8 / 9.6 = 0.08333
  θ = arctan(0.08333) = 4.76°

Which is a 1:12 gradient, the usual accessibility standard.

Working backwards. Sin θ = 0.6. Find θ.

  arcsin(0.6) = 36.87°
  And 180 − 36.87 = 143.13° has the same sine.

If this were a triangle problem, both would need checking against the other angles to see which can actually exist.

Understanding Your Result

The three main ratios come with their exact values in brackets where the angle is special.

The reciprocals — cosecant, secant, cotangent — are given for completeness, with "undefined" where the ratio has no value.

The exact values line gives the surd forms, or says plainly that the angle is not one of the special ones.

The angle line shows it in both degrees and radians, with the π form when there is one. In inverse mode it also names the companion angles that share your ratio.

The identity check confirms sin²θ + cos²θ = 1, which holds at every angle without exception. It is the Pythagorean theorem applied to the unit circle, and if it did not come out to 1, something would be badly wrong.

When Should You Use This Calculator?

Measuring heights you cannot reach. Distance plus angle gives height — trees, buildings, masts, cliffs.

Roof pitch and stairs. Rise and run give the angle; the angle and one dimension give the other.

Ramps and gradients. Accessibility standards specify angles or ratios, and this converts between them.

Navigation. Resolving a course into north and east components is exactly cosine and sine.

Physics. Resolving forces, projectile motion, and the components of any vector.

Waves and signals. Sound, light, alternating current and oscillation are all described by sine functions.

Graphics and games. Rotation, circular motion and camera angles.

Surveying and astronomy. Triangulation rests on these three ratios.

Common Mistakes

Calculator in the wrong mode. sin(30°) = 0.5, sin(30 radians) = −0.988. Check this first whenever an answer looks strange.

Swapping opposite and adjacent. Both are relative to the angle you chose. Fix the angle first, then label the sides.

A sine or cosine above 1. Impossible — they are a side over the hypotenuse, and the hypotenuse is longest. If you get 1.4, you divided the wrong way.

Pairing secant with sine. Secant is 1/cos and cosecant is 1/sin. The names mislead.

Treating tan 90° as a number. It is undefined, not infinite.

Taking the inverse answer as the only answer. There is always a companion, and often it matters.

Applying SOHCAHTOA to a non-right triangle. It only works with a 90° angle. For anything else, use the law of sines or cosines.

Rounding mid-calculation. Keep full precision through the working and round once at the end — the ratios are sensitive, and small errors compound quickly.