Trigonometry Calculator
Find missing sides and angles in right-angled triangles using sin, cos and tan — step by step.
Select Trigonometric Ratio
sin(θ) = O / H
Leave the side you want to calculate blank.
Select Inverse Ratio
θ = sin⁻¹(O / H)
Enter both known side lengths to find angle θ.
Step-by-step solution
Select a mode, pick your ratio, enter the known values and press Calculate
SOH CAH TOA Reference
Trigonometry Calculator Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
When would you actually use this?
Building regulations require wheelchair ramps to have a maximum gradient of about 1 in 20 — that's an angle of just under 3°. A carpenter fitting a roof needs to cut rafters at the exact pitch angle. Trigonometry is the maths that turns angles and distances into precise measurements for real construction.
Worked example
"A ramp rises 0.8 m over a horizontal distance of 6 m. What angle does it make with the ground?"
tan(θ) = opposite / adjacent
tan(θ) = 0.8 / 6 = 0.1333
θ = tan⁻¹(0.1333) = 7.6°
SOH CAH TOA — Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj. Label the triangle first.
Choosing the right ratio
- → Use Sin when you have the opposite and hypotenuse (or need one from the other).
- → Use Cos when you have the adjacent and hypotenuse.
- → Use Tan when you have the opposite and adjacent (no hypotenuse needed).
Inverse functions
To find an angle, use the inverse: sin⁻¹, cos⁻¹, or tan⁻¹ (also written arcsin, arccos, arctan). Make sure your calculator is in degree mode, not radians.
What is Trigonometry?
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. For GCSE Maths, you'll primarily focus on right-angled triangles.
It allows us to find missing side lengths or angles in a right-angled triangle when we have some information already. The key to unlocking these relationships is using the three main trigonometric ratios: Sine (sin), Cosine (cos), and Tangent (tan), often remembered by the acronym SOHCAHTOA.
How to Use the Trigonometry Calculator
Identify Your Triangle
Start by looking at your right-angled triangle. Identify what you already know (angles and side lengths) and what you need to find (a missing side or a missing angle).
Label the Sides
Relative to the angle you are working with (or the angle you want to find), label the three sides of your right-angled triangle: - **Hypotenuse (H):** Always the longest side, directly opposite the right angle. - **Opposite (O):** The side directly opposite the angle you are using. - **Adjacent (A):** The side next to the angle you are using (but not the hypotenuse).
Choose Calculator Mode
Select the appropriate mode on the calculator: - **'Find a missing side'**: If you know one angle (other than the right angle) and one side, and you want to find another side. - **'Find a missing angle'**: If you know the lengths of two sides and you want to find one of the acute angles.
Input Your Values
Enter the known values into the calculator's fields. Make sure your angle is in degrees, as this calculator operates in degrees only.
Select the Ratio & Calculate
Based on the sides and angle you are using, choose the correct SOHCAHTOA ratio (Sin, Cos, or Tan). The calculator will then perform the calculation and display your missing side length or angle.
Common mistakes to avoid
- Incorrectly labelling the sides (Opposite, Adjacent, Hypotenuse):The labels O, A, and H depend entirely on which angle you are focusing on. If you label them incorrectly, you'll use the wrong ratio. Always label the Hypotenuse first (opposite the right angle). Then, for your chosen angle, the Opposite side is directly across from it, and the Adjacent side is next to it (not the Hypotenuse).
- Using the wrong SOHCAHTOA ratio:Each ratio uses a specific pair of sides. Forgetting which pair goes with which ratio will lead to incorrect answers. Remember SOHCAHTOA: Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent. Pick the ratio that involves the two sides you know or need to find.
- Forgetting to use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle:If you're trying to find an angle, you need to 'undo' the sin, cos, or tan operation. Just using sin(ratio) will give you a side length, not an angle. When calculating a missing angle, always use the inverse trigonometric functions: sin⁻¹ (arcsin), cos⁻¹ (arccos), or tan⁻¹ (arctan). These are usually accessed by pressing 'Shift' or '2nd F' on your calculator before the sin/cos/tan button.
Questions students ask
1. What does SOHCAHTOA stand for?
• SOH: Sin(angle) = Opposite / Hypotenuse
• CAH: Cos(angle) = Adjacent / Hypotenuse
• TOA: Tan(angle) = Opposite / Adjacent
2. How do I know which ratio to use?
• If you're using Opposite and Hypotenuse, use SOH (Sine).
• If you're using Adjacent and Hypotenuse, use CAH (Cosine).
• If you're using Opposite and Adjacent, use TOA (Tangent).
3. What is the difference between sin, cos and tan?
• Sine (sin) relates the length of the Opposite side to the length of the Hypotenuse.
• Cosine (cos) relates the length of the Adjacent side to the length of the Hypotenuse.
• Tangent (tan) relates the length of the Opposite side to the length of the Adjacent side.
Each ratio will give a different value for the same angle, as they are comparing different pairs of sides.