Trig Ratio from Sides Calculator

Enter Calculation

Opposite side

Adjacent side

Hypotenuse

Enter the three sides of a right triangle relative to angle θ. Leave the hypotenuse blank if you enter both legs — it will be computed for you.

Results

Angle θ

36.8699°

hypotenuse = 5

Ratio	Value
sin θ = opp / hyp	0.6
cos θ = adj / hyp	0.8
tan θ = opp / adj	0.75
csc θ = hyp / opp	1.666667
sec θ = hyp / adj	1.25
cot θ = adj / opp	1.333333

What this calculator does

The Trig Ratio from Sides Calculator computes the six trigonometric ratios — sine, cosine, tangent, cosecant, secant and cotangent — for an acute angle θ in a right triangle, using the lengths of its sides. It also reports the angle θ in degrees and fills in the hypotenuse automatically when you supply only the two legs.

A right triangle labeling the sides relative to angle θ: opposite, adjacent, and hypotenuse.

How to use it

Identify your angle θ. Enter the side opposite θ, the side adjacent to θ, and the hypotenuse (the side across from the right angle). If you know only the two legs, leave the hypotenuse blank and it will be derived from the Pythagorean theorem.

The formulas

The three primary ratios are \(\sin\theta = \frac{O}{H}\), \(\cos\theta = \frac{A}{H}\), and \(\tan\theta = \frac{O}{A}\). The reciprocals are \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), and \(\cot\theta = \frac{1}{\tan\theta}\). When the hypotenuse is unknown it is found with \(c = \sqrt{a^{2} + b^{2}}\).

$$\begin{gathered} \sin\theta = \frac{O}{H}, \quad \cos\theta = \frac{A}{H}, \quad \tan\theta = \frac{O}{A} \\[0.4em] \csc\theta = \frac{H}{O}, \quad \sec\theta = \frac{H}{A}, \quad \cot\theta = \frac{A}{O} \\[1.5em] \text{where}\quad \left\{ \begin{aligned} O &= \text{Opposite} \\ A &= \text{Adjacent} \\ H &= \text{Hypotenuse} = \sqrt{O^{2} + A^{2}} \end{aligned} \right. \end{gathered}$$

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The six ratios (sin, cos, tan and their reciprocals csc, sec, cot) built from opp, adj and hyp.

Worked example

Consider the classic 3-4-5 right triangle with opposite = 3, adjacent = 4, hypotenuse = 5. Then \(\sin\theta = \frac{3}{5} = 0.6\), \(\cos\theta = \frac{4}{5} = 0.8\), and \(\tan\theta = \frac{3}{4} = 0.75\). The reciprocals are \(\csc\theta \approx 1.6667\), \(\sec\theta = 1.25\), and \(\cot\theta \approx 1.3333\). The angle \(\theta = \operatorname{atan}(3/4) \approx 36.87°\).

FAQ

What if I only know the two legs? Leave the hypotenuse field blank or zero — the calculator computes it with the Pythagorean theorem.

Why is my angle close but not exact? Trig values are irrational for most angles; results are rounded for display while the raw value keeps full precision.

Does this work for any triangle? No — these ratios apply only to right triangles. For non-right triangles use the law of sines or cosines.