Trigonometry Word Problem Solver
Choose a real-world scenario, enter your measurements, and get a complete step-by-step SOHCAHTOA solution with a visual diagram.
How to Solve Trigonometry Word Problems
Trigonometry word problems bridge the gap between abstract math and the physical world. Every problem follows the same fundamental pattern: identify the right triangle hidden in the scenario, label the sides as opposite, adjacent, and hypotenuse relative to the known or target angle, then select the appropriate trig ratio (sine, cosine, or tangent) to solve for the unknown. Our solver automates this process while showing you every step so you learn the method.
The key to success is the SOHCAHTOA mnemonic, which encodes the three primary trig ratios used in right triangle problems. Once you identify which sides you know and which you need, the correct ratio follows directly.
The SOHCAHTOA Framework
SOH: sin(θ) = Opposite / Hypotenuse | CAH: cos(θ) = Adjacent / Hypotenuse | TOA: tan(θ) = Opposite / Adjacent
Types of Problems This Solver Handles
Our solver covers six common real-world scenarios that students encounter in homework and exams:
- Ladder Problems: Given the ladder length (hypotenuse) and the angle with the ground, find how high the ladder reaches (opposite) and how far the base is from the wall (adjacent). Uses both sin and cos.
- Shadow and Height: Given the distance from a building (adjacent) and the angle of elevation, find the building height (opposite). Uses tan directly.
- Ramp / Incline: Given the desired rise (opposite) and ramp angle, find the ramp length (hypotenuse) and horizontal run (adjacent). Uses sin and tan.
- Airplane Climb: Given the flight distance along the climb path (hypotenuse) and climb angle, find the altitude gained (opposite) and horizontal distance covered (adjacent).
- Tree Height: Given the distance from the tree (adjacent), angle of elevation, and your eye height, find the total tree height. Combines tan with an offset for eye level.
- Kite Flying: Given the string length (hypotenuse) and angle with the ground, find the kite's altitude (opposite) and horizontal distance (adjacent).
Worked Example: The Ladder Problem
Suppose a 20-foot ladder leans against a wall at a 65-degree angle with the ground. To find how high the ladder reaches, identify the triangle: the ladder is the hypotenuse (20 ft), the wall is the opposite side (unknown height), and the ground is the adjacent side. Since we have the hypotenuse and want the opposite, we use sin:
sin(65°) = height / 20 → height = 20 × sin(65°) = 20 × 0.9063 = 18.13 ft
For the base distance, use cos: base = 20 × cos(65°) = 20 × 0.4226 = 8.45 ft. Every scenario in our solver follows this same identify-select-compute pattern.
Tips for Solving Word Problems on Your Own
Always start by drawing the triangle. Label every piece of information the problem gives you: side lengths, angles, and which side you need to find. Then ask: relative to the angle I am working with, is my unknown side the opposite, adjacent, or hypotenuse? The answer determines which ratio to use. If the problem gives you the opposite and adjacent, you want tan. If it gives you the hypotenuse and you need the opposite, use sin. Practice this identification step and the actual computation becomes straightforward.
Angle of Elevation vs. Angle of Depression
Many word problems involve looking up (angle of elevation) or looking down (angle of depression). The angle of elevation is measured upward from horizontal to the line of sight. The angle of depression is measured downward from horizontal. By alternate interior angles, the angle of depression from the top of a building equals the angle of elevation from the ground observer. This means you can use either angle interchangeably depending on which triangle you set up.
Frequently Asked Questions
What is SOHCAHTOA and when do I use it?
SOHCAHTOA is a mnemonic for the three primary trigonometric ratios in right triangles. SOH means sin = Opposite / Hypotenuse, CAH means cos = Adjacent / Hypotenuse, and TOA means tan = Opposite / Adjacent. You use it whenever you have a right triangle and know at least one side length and one acute angle (or two sides) and need to find another measurement. It only works for right triangles — for non-right triangles, you need the Law of Sines or Law of Cosines.
How do I know which trig ratio to use?
Look at what you know and what you need relative to the angle in question. If you have the hypotenuse and need the opposite side (or vice versa), use sine. If you have the hypotenuse and need the adjacent side, use cosine. If you have one leg and need the other leg (opposite and adjacent), use tangent. A quick way to remember: circle the two sides involved (the one you know and the one you want), then pick the ratio that uses exactly those two sides.
Can this solver handle problems with two unknowns?
Each scenario is designed to solve for all unknown sides given the minimum required inputs. For example, the ladder problem takes the ladder length and angle, then calculates both the height reached and the base distance. Internally, it applies sin and cos separately to find each unknown. If your problem has truly two independent unknowns (two missing pieces of information), you would need two equations — which typically means two given measurements beyond the right angle.
Why does the tree height problem ask for eye height?
When you stand at ground level and measure the angle of elevation to the top of a tree, you are actually measuring from your eye level, not from the ground. The trig calculation gives you the height of the tree above your eyes. To get the total tree height, you must add your eye height (typically about 1.5 to 1.8 meters) to the calculated value. This is a common source of small errors in real-world measurements that our solver accounts for.
Do the diagrams update when I enter values?
The diagrams show a fixed illustrative representation for each scenario type to help you visualize the triangle setup. They are designed to clarify which side is which (hypotenuse, opposite, adjacent) and where the angle is located. The solution section provides the precise numerical answers. Future updates may include dynamically scaled diagrams based on your inputs.
What units does the solver use?
The solver is unit-agnostic — it works with whatever units you enter. If you input measurements in feet, the output is in feet. If you use meters, the output is in meters. Just make sure all your inputs use the same unit system. Angles are always entered in degrees, as that is the most common format in word problems and practical applications.