Trigonometry Calculator
Calculate sin, cos, tan
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Trigonometry Calculator helps you calculate sin, cos, tan — free, instant, in your browser. You supply Angle (degrees) to Trigonometry; the answer follows at once. In Trigonometry the working sits beside the answer. Trigonometry suits budgeting and planning checks. Trigonometry is privacy-first: Math inputs stay offline. Trigonometry is part of the Math collection on CalcProMaster, alongside sin cos tan, free online trigonometry calculator and more. Keep Trigonometry bookmarked for next time. Try Trigonometry now and keep it handy. Trigonometry is linked from the Math hub page.
What the ratios are at 45°
sin 45° = 0.7071, cos 45° = 0.7071 and tan 45° = 1.0000.
Sine and cosine match because 45° is exactly halfway to 90°: the opposite and adjacent sides of the right triangle are equal, so both ratios are 1/√2. Tangent, being opposite over adjacent, is exactly 1.
Sine and cosine as waves
On a unit circle, sin θ and cos θ trace repeating waves with period 2π ≈ 6.283 and range −1 to 1. The tangent function repeats every π with vertical breaks where cosine is zero.
Plot them with our function grapher: sin(x) or sin(x)*3 shows the amplitude change, and tan(x) shows the asymptotes. Remember the grapher works in radians. Related: law of sines and law of cosines.
Definitions before buttons
In a right triangle with angle θ, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse and tan θ = opposite / adjacent = sin θ / cos θ.
At 45° the triangle is isosceles, so opposite = adjacent. If each leg is 1, the hypotenuse is √2 = 1.4142, giving sin = cos = 1/1.4142 = 0.7071 and tan = 1/1 = 1.
Worked example, in the order the page computes it
Step 1: convert to radians — 45 × π/180 = 0.7854.
Step 2: sin(0.7854) = 0.7071.
Step 3: cos(0.7854) = 0.7071.
Step 4: tan(0.7854) = 1.0000.
The radian step matters because every programming language and spreadsheet evaluates trig functions in radians, not degrees. Passing 45 directly into a radian function returns sin 45 rad = 0.8509, a plausible-looking wrong answer.
Angles worth memorising
- 0° — sin 0, cos 1, tan 0.
- 30° — sin 0.5, cos 0.8660, tan 0.5774.
- 45° — sin 0.7071, cos 0.7071, tan 1.
- 60° — sin 0.8660, cos 0.5, tan 1.7321.
- 90° — sin 1, cos 0, tan undefined.
Why tan 90° is undefined rather than infinite
Tangent is opposite over adjacent. Approaching 90° the adjacent side shrinks toward zero while the opposite side stays fixed, so the ratio grows without bound. Division by zero has no value, which is why the page reports it as undefined rather than printing a huge number that would invite arithmetic on it.
Mistakes worth avoiding
- Feeding degrees into a function that expects radians, or the reverse — the most common trig error in code.
- Using sine for the wrong side. Sine always pairs with the side opposite the angle you named, not the one nearest it.
- Treating the ratios as lengths. They are dimensionless; the actual sides come from multiplying by the hypotenuse.
- Ignoring the quadrant. sin 150° is +0.5, not −0.5, and a triangle diagram alone cannot tell you which.
Frequently Asked Questions
Why are sin 45° and cos 45° the same?
Because a 45° right triangle is isosceles: the two legs are equal, so opposite over hypotenuse and adjacent over hypotenuse are the same fraction, 1/√2 = 0.7071.
Why does the calculator convert to radians first?
Because the underlying sin, cos and tan functions take radians. Degrees are converted with angle × π/180, so 45° becomes 0.7854 rad before any ratio is evaluated.
What is tan 90°?
Undefined. The adjacent side becomes zero at 90°, and opposite/adjacent would require dividing by zero, so the value does not exist rather than being infinite.
How do I find a side from an angle and one known side?
Pick the ratio that pairs your known side with the unknown: opposite = hypotenuse × sin θ, adjacent = hypotenuse × cos θ, opposite = adjacent × tan θ. Then read the ratio from this page for the angle you have.
Is the output in degrees or radians?
The input field takes degrees. The 45° in the steps converts to 0.7854 rad internally, and the ratios are unitless, so no unit is attached to sin, cos or tan.
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