Trigonometry Calculator
A 30-degree angle in a right triangle gives sin = 0.5, cos = 0.866, and tan = 0.577. This trigonometry calculator computes all six trig ratios, sin, cos, tan, csc, sec, and cot, plus all three side lengths, from a single acute angle and one known side length. Enter any angle between 0 and 90 degrees and choose whether you know the opposite side, the adjacent side, or the hypotenuse. The tool normalizes that known side to a length of 1, applies the sine, cosine, and tangent definitions, and returns the two remaining side lengths along with every trig ratio for that angle. It's built for geometry homework, roof-pitch and ramp-angle checks, quick surveying sketches, and any right-triangle problem where you have one angle and one side and need the rest. Because the outputs are normalized to a known side of 1, multiply every side result by your actual measurement to get real-world lengths in feet, meters, or any unit you're working in.
Quick answer
sin is opposite over hypotenuse, cos is adjacent over hypotenuse, and tan is opposite over adjacent.
What this tells you
- •sin is opposite over hypotenuse, cos is adjacent over hypotenuse, and tan is opposite over adjacent.
- •csc, sec, and cot are the reciprocals of sin, cos, and tan.
- •Pick which side you know, opposite, adjacent, or hypotenuse, and the calculator solves for the other two.
- •The angle must be strictly between 0 and 90 degrees, since a right triangle can't have a 0-degree or 90-degree acute angle.
- •SOH CAH TOA is the standard memory trick: Sin equals Opposite over Hypotenuse, Cos equals Adjacent over Hypotenuse, Tan equals Opposite over Adjacent.
- •Every side result is normalized so your known side equals 1. Multiply all three side lengths by your actual measurement to get real-world sizes.
- •Ratio values round to 4 decimal places, so csc, sec, and cot can look large for small angles since each is 1 divided by a small sine, cosine, or tangent value.
How to Use
- 1Enter an acute angle between 0 and 90 degrees, such as 30, 45, or 62.5. Decimal values are allowed.
- 2Choose which side length you already know: opposite, adjacent, or hypotenuse.
- 3Click Calculate to get all six trig ratios, sin, cos, tan, csc, sec, and cot, plus the three side lengths.
- 4Read the side lengths as multiples of your known side. The calculator sets your known side to 1, so multiply every result by your actual measurement to get real-world lengths.
- 5Use the reciprocal ratios, csc, sec, and cot, if your problem is stated in terms of cosecant, secant, or cotangent instead of sin, cos, or tan.
How It Works
Formula
sin = opp/hyp, cos = adj/hyp, tan = opp/adj. csc = 1/sin, sec = 1/cos, cot = 1/tan.The calculator uses right-triangle trigonometry, the same relationships behind SOH CAH TOA. Given an angle A and one side length, it normalizes the known side to a length of 1 and applies the sine, cosine, and tangent definitions, sin(A) equals opposite over hypotenuse, cos(A) equals adjacent over hypotenuse, and tan(A) equals opposite over adjacent, to solve for the other two sides. The reciprocal functions follow directly: cosecant is 1 divided by sin, secant is 1 divided by cos, and cotangent is 1 divided by tan. Because the triangle is normalized, the real side lengths in your actual triangle are these values multiplied by whatever length your known side truly measures.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
30 degrees, opposite = 1
With the opposite side set to 1, the hypotenuse is 1 / sin(30 degrees) = 2 and the adjacent side is cos(30 degrees) / sin(30 degrees) = 1.7321. This matches the classic 30-60-90 triangle, where the three sides scale in a 1 to square root of 3 to 2 ratio.
45 degrees, hypotenuse = 1
In a 45-45-90 triangle both legs are equal, so sin and cos come out identical and tan lands exactly on 1. With the hypotenuse fixed at 1, each leg equals 1 divided by the square root of 2, which rounds to 0.7071.
60 degrees, adjacent = 1
This is the same 30-60-90 triangle as the first example, viewed from the other acute angle. With the adjacent side fixed at 1, the opposite side is tan(60 degrees) = 1.7321 and the hypotenuse is 1 / cos(60 degrees) = 2. A common real use is finding a ladder's horizontal reach when the base angle is 60 degrees and you know the base distance.
15 degrees, hypotenuse = 1
Shallow angles like 15 degrees come up in low-slope ramp and roof-pitch problems. With the hypotenuse fixed at 1, the opposite side is sin(15 degrees) = 0.2588, which shows how little rise you get from a shallow angle even over a full unit of slope length.
75 degrees, opposite = 1
Steep angles like 75 degrees push tan well above 1, since the opposite side grows much faster than the adjacent side. With the opposite side fixed at 1, the adjacent side is only 0.2679 and the hypotenuse is 1.0353, barely longer than the opposite side itself.
36.87 degrees, adjacent = 1 (3-4-5 triangle)
This angle is the smaller acute angle of a classic 3-4-5 right triangle, scaled down so the adjacent side equals 1. Multiply every side by 4 to get the familiar whole-number triangle: opposite = 3, adjacent = 4, hypotenuse = 5. Builders and carpenters use the 3-4-5 method to square corners without measuring an angle directly.
Common Trig Values
Sine, cosine, and tangent for the most common angles.
| Angle | sin | cos | tan |
|---|---|---|---|
| 15 deg | 0.259 | 0.966 | 0.268 |
| 30 deg | 0.5 | 0.866 | 0.577 |
| 36.87 deg | 0.6 | 0.8 | 0.75 |
| 45 deg | 0.707 | 0.707 | 1 |
| 53.13 deg | 0.8 | 0.6 | 1.333 |
| 60 deg | 0.866 | 0.5 | 1.732 |
| 75 deg | 0.966 | 0.259 | 3.732 |
Common mistakes
- Entering the angle in radians instead of degrees. This calculator expects degrees, so convert radians to degrees by multiplying by 180 and dividing by pi before entering the value.
- Using the wrong side for the ratio. sin uses opposite over hypotenuse, not adjacent over hypotenuse, and mixing up the two sides gives the wrong result every time.
- Confusing sin(30) with sin(0.5). Sin of 30 degrees is 0.5, but that is not the same as the sine of 0.5 radians, which is about 0.4794.
- Entering exactly 0 or 90 degrees. Right-triangle trig breaks down at these limits, tan and sec approach infinity near 90 degrees, so the calculator only accepts values strictly between them.
- Forgetting that side results are normalized. The opposite, adjacent, and hypotenuse values assume your known side equals 1, so multiply all three side results by your actual measurement to get real-world lengths.
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