Trig Triangle Calculator

Enter one acute angle and one known side of a right triangle to find the other two sides and the remaining angle using sine, cosine, and tangent.

Quick Facts

SOH-CAH-TOA
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
The three basic trig ratios relate an acute angle to the triangle's sides.
Angle sum
θ + other acute angle = 90°
The two non-right angles of a right triangle are always complementary.
Pythagorean theorem
a² + b² = c²
Used as a cross-check once all three sides are known.

Your Results

Calculated
Opposite Side
-
Side across from θ
Adjacent Side
-
Side next to θ
Hypotenuse
-
Longest side, opposite the right angle
Other Angle
-
90° − θ

Ready

Enter an angle, choose which side you know, and press Calculate.

Formula and Method for Solving a Right Triangle with Trigonometry

A right triangle has one 90° angle and two acute angles that always add up to 90°. If you know one acute angle (θ) and the length of any one side, trigonometry lets you find the other two sides using the three basic ratios — sine, cosine, and tangent — remembered by the mnemonic SOH-CAH-TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.

How the calculation works

This calculator picks the correct ratio automatically based on which side you say you know. If you enter the opposite side, the hypotenuse is found from hypotenuse = opposite ÷ sin θ, and the adjacent side from adjacent = opposite ÷ tan θ. If you enter the adjacent side, the hypotenuse is adjacent ÷ cos θ, and the opposite side is adjacent × tan θ. If you enter the hypotenuse, the opposite side is hypotenuse × sin θ, and the adjacent side is hypotenuse × cos θ. In every case, the remaining acute angle is simply 90° − θ, since the two non-right angles of a right triangle are complementary.

Common mistakes

  • Mixing up opposite and adjacent: the "opposite" side is the one directly across from the angle θ; the "adjacent" side is the other leg touching θ (not the hypotenuse). Swapping them flips sine and cosine.
  • Degrees vs. radians: this calculator expects the angle in degrees. If you are working from radians, convert first (degrees = radians × 180/π) or the trig ratios will be wrong.
  • Angle out of range: θ must be strictly between 0° and 90° for a right triangle's acute angle — 0° or 90° would collapse the triangle into a line.

Real-world applications

  • Surveying and construction use angle-of-elevation trigonometry to find heights of buildings, towers, or slopes without direct measurement.
  • Navigation and aviation use right-triangle trig to compute bearings, altitude, and distance-to-target from angle and range readings.
  • Carpentry and roofing use it to size rafters, ramps, and staircases from a known rise (opposite), run (adjacent), or slope angle.
  • Physics problems routinely decompose forces or velocities into perpendicular components using the same sine/cosine relationships.

Frequently Asked Questions

What is SOH-CAH-TOA?
SOH-CAH-TOA is a memory aid for the three basic trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. Given one acute angle and one side, you can rearrange the matching ratio to solve for either of the other two sides.
How do I find a missing side if I know an angle and one side?
Pick the trig ratio that relates the side you know to the side you want. For example, if you know the angle and the adjacent side and want the opposite side, use tan(angle) = opposite/adjacent, so opposite = adjacent × tan(angle). This calculator applies the correct ratio automatically based on which side you enter.
How do I find the third angle of a right triangle?
The three interior angles of any triangle sum to 180°. Since a right triangle already has a 90° angle, the two acute angles must add up to 90°. So if one acute angle is θ, the other acute angle is 90° − θ.
How is this different from the Pythagorean theorem?
The Pythagorean theorem (a² + b² = c²) finds a missing side when you already know the other two sides, with no angle needed. Trigonometry (sine, cosine, tangent) is used instead when you know one side and one acute angle and need to find another side or the remaining angle.

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