Trigonometry (SOH CAH TOA)
The SOH CAH TOA mnemonic transforms intimidating trigonometry into manageable steps for finding sides and angles in right triangles. Students master this foundation in CCSS.HSG.SRT before tackling advanced applications like surveying and engineering. Understanding these 3 ratios unlocks problem-solving across construction, navigation, and physics.
Try it right now
Why it matters
Trigonometry appears everywhere in real-world applications. Architects use sine and cosine to calculate roof angles and support beam lengths for buildings up to 50 stories tall. Pilots rely on trigonometric calculations to determine flight paths, with a 1-degree navigation error potentially causing a 92-mile deviation on a 5,280-mile flight. Construction workers use the 3-4-5 triangle principle to ensure foundations are perfectly square, while surveyors measure property boundaries using angle measurements precise to 0.1 degrees. Emergency responders calculate ladder angles for safe rescues, requiring exactly 75.5 degrees for optimal safety. Video game developers use trigonometry to create realistic physics engines, calculating projectile trajectories and character movements 60 times per second.
How to solve trigonometry (soh cah toa)
Trigonometry (SOH CAH TOA)
- sin(A) = Opposite / Hypotenuse (SOH).
- cos(A) = Adjacent / Hypotenuse (CAH).
- tan(A) = Opposite / Adjacent (TOA).
- To find an angle: use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹).
Example: sin(30°) = 12, cos(60°) = 12.
Worked examples
What is cos(45°)?
Answer: √22
- Recall the mnemonic SOH CAH TOA → CAH: cos = adjacent/hypotenuse — SOH = Sine-Opposite-Hypotenuse, CAH = Cosine-Adjacent-Hypotenuse, TOA = Tangent-Opposite-Adjacent.
- Identify what cos means → cos = adjacent/hypotenuse — We need cos(45°), which is the ratio adjacent/hypotenuse.
- Look up the standard value for 45° → cos(45°) = √2/2 — The angles 30°, 45° and 60° have exact values you should memorise.
In a right triangle with opposite = 6 and adjacent = 8, find angle A.
Answer: 36.9°
- Identify the known sides → opposite = 6, adjacent = 8 — We know two sides: the opposite and the adjacent (relative to angle A).
- Choose the right ratio using SOH CAH TOA → We know: opposite + adjacent → use TOA (tan) — We have opposite and adjacent, so we use tan = opposite/adjacent.
- Write the equation → tan(A) = 6 / 8 = 0.75 — Substitute the known side lengths into the tangent ratio.
- Use the inverse function to find the angle → A = tan⁻¹(0.75) = 36.9° — Press tan⁻¹ (or arctan) on your calculator to go from ratio back to angle.
- Sanity check → A = 36.9° (between 0° and 90° ✓) — The answer must be between 0° and 90° for a right triangle. 36.9° is reasonable since opposite < adjacent.
In a right triangle with opposite = 3 and adjacent = 4, find angle A.
Answer: 36.9°
- Identify the known sides → opposite = 3, adjacent = 4 — We know two sides: the opposite and the adjacent (relative to angle A).
- Choose the right ratio using SOH CAH TOA → We know: opposite + adjacent → use TOA (tan) — We have opposite and adjacent, so we use tan = opposite/adjacent.
- Write the equation → tan(A) = 3 / 4 = 0.75 — Substitute the known side lengths into the tangent ratio.
- Use the inverse function to find the angle → A = tan⁻¹(0.75) = 36.9° — Press tan⁻¹ (or arctan) on your calculator to go from ratio back to angle.
- Sanity check → A = 36.9° (between 0° and 90° ✓) — The answer must be between 0° and 90° for a right triangle. 36.9° is reasonable since opposite < adjacent.
Common mistakes
- ✗Students confuse which side is opposite versus adjacent, calculating tan(30°) = √3 instead of 1/√3 when they flip the triangle orientation in their minds.
- ✗Many students forget to switch to inverse functions when finding angles, writing sin(A) = 0.6 instead of A = sin⁻¹(0.6) = 36.9°.
- ✗Students often use degrees when their calculator is in radian mode, getting sin(30) = -0.988 instead of sin(30°) = 0.5.
- ✗Common error involves using the wrong ratio entirely, applying SOH when the problem requires TOA, leading to incorrect setups like sin(A) = 8/6 instead of tan(A) = 8/6.
Practice on your own
Generate unlimited SOH CAH TOA practice problems with step-by-step solutions using MathAnvil's free trigonometry worksheet maker.
Generate free worksheets →