Find missing angles or sides using SOH CAH TOA with step-by-step working
Trigonometry is the branch of maths that deals with the relationships between the sides and angles of triangles. At GCSE level, trigonometry focuses on right-angled triangles using the three fundamental ratios: sine (sin), cosine (cos), and tangent (tan), commonly remembered through the mnemonic SOH CAH TOA. SOH CAH TOA stands for: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. These ratios allow you to find unknown angles when you know two sides, or find unknown sides when you know an angle and one side. Mastering these relationships is essential for success in GCSE maths Higher tier papers. Our calculator handles both finding a missing angle and finding a missing side. It automatically selects the correct trigonometric ratio based on which values you provide, shows the sin, cos, and tan values for the angle, and provides complete step-by-step working. You can switch between degrees and radians to suit GCSE or A-level requirements. At A-level, trigonometry extends well beyond SOH CAH TOA to include the sine rule, cosine rule, trigonometric identities, and solving equations. Students also work extensively with radians, which become the standard angle measure for calculus. Our calculator supports radians mode for A-level and university students. Trigonometry has practical applications in surveying, navigation, engineering, physics, and architecture. Understanding how to calculate heights, distances, and angles using trigonometric ratios is a fundamental skill that extends far beyond the maths classroom. UK exam boards (AQA, Edexcel, OCR, WJEC) all test trigonometry extensively at GCSE and A-level.
To solve trigonometry problems with this calculator: 1. Select the mode. Choose "Find angle" when you know two sides and want to find the angle. Choose "Find side" when you know an angle and one side and want to find another side. 2. Select the angle unit. Use "Degrees" for GCSE work. Use "Radians" for A-level or university problems where radian measure is required. 3. For "Find angle" mode, enter any two of the three sides: opposite, adjacent, or hypotenuse. The calculator automatically detects which trigonometric ratio to use (sin, cos, or tan) and calculates the angle. 4. For "Find side" mode, enter the known angle and one side. The calculator determines which side can be found from the given information. For example, given a 30-degree angle and a hypotenuse of 10, it calculates the opposite side using sin. 5. Review the results, which include the calculated value, the sin, cos, and tan values of the relevant angle, the formula used, and step-by-step working. 6. Use the step-by-step working as a model for writing out your own exam answers. GCSE and A-level examiners award method marks for showing the correct formula and substitution.
The three basic trigonometric ratios for right-angled triangles are: sin(angle) = opposite / hypotenuse (SOH) cos(angle) = adjacent / hypotenuse (CAH) tan(angle) = opposite / adjacent (TOA) To find a missing angle, use the inverse function: - angle = arcsin(opposite / hypotenuse) - angle = arccos(adjacent / hypotenuse) - angle = arctan(opposite / adjacent) To find a missing side, rearrange the appropriate formula: - opposite = hypotenuse * sin(angle) - adjacent = hypotenuse * cos(angle) - opposite = adjacent * tan(angle) - hypotenuse = opposite / sin(angle) Example: In a right-angled triangle with opposite = 3 and hypotenuse = 5: sin(angle) = 3/5 = 0.6 angle = arcsin(0.6) = 36.87 degrees Converting between degrees and radians: - Radians = degrees * (pi / 180) - Degrees = radians * (180 / pi) Key values to memorise: sin(30) = 0.5, cos(60) = 0.5, sin(45) = cos(45) = sqrt(2)/2 (approximately 0.7071), tan(45) = 1. These exact values are frequently tested in GCSE and A-level maths exams across all UK exam boards.