Right Triangle Trigonometry Guide
A right triangle has one angle of ninety degrees. The other two angles are acute. Their sum is always ninety degrees. Sine, cosine, and tangent describe side relationships. They do not depend on triangle size. They depend on the chosen reference angle.
Why the ratios matter
Sine compares the opposite side with the hypotenuse. Cosine compares the adjacent side with the hypotenuse. Tangent compares the opposite side with the adjacent side. These ratios help you solve height, distance, slope, and direction problems. They also support engineering sketches and classroom work.
Choosing the reference angle
The calculator treats angle A as the reference angle. The opposite side faces that angle. The adjacent side touches that angle. The hypotenuse is always across from the right angle. If you switch to the other acute angle, opposite and adjacent sides trade roles. The hypotenuse stays the same.
Solving from sides
When two sides are known, the missing side comes from the Pythagorean theorem. Two legs create the hypotenuse by squaring both legs, adding them, and taking the square root. A hypotenuse and one leg create the other leg by subtracting squared values, then taking the square root. The angle then comes from inverse trigonometry.
Solving from an angle
When one acute angle and one side are known, every side can be found. A known hypotenuse gives legs through sine and cosine. A known adjacent side gives the opposite side with tangent. A known opposite side gives the adjacent side with tangent. The missing acute angle equals ninety degrees minus angle A.
Accuracy and practical use
Small rounding changes can affect final values. Use more decimal places for technical work. Use fewer decimals for quick homework checks. Always enter positive side lengths. Angle A must be greater than zero and less than ninety degrees. If two entered sides cannot form a right triangle, the calculator returns a warning.
Exports and records
The export buttons help you save the solved values. CSV works well for spreadsheets. PDF works well for sharing a compact report. The example table shows common input patterns. Use it to compare your own triangle with typical results. It also keeps repeated practice organized for later review and sharing.