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§ Trigonometry

Trigonometry (SOH CAH TOA) Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable trigonometry (soh cah toa) worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from evaluate sin/cos/tan of standard angles at the easy level through to find angles using inverse trig at the advanced level.

CCSS.HSG.SRT

What is trigonometry (soh cah toa)?

Trigonometry uses ratios to relate the angles and sides of right triangles through three fundamental functions: sine, cosine, and tangent. The mnemonic SOH CAH TOA helps recall these relationships: SOH means sine equals opposite over hypotenuse, CAH means cosine equals adjacent over hypotenuse, and TOA means tangent equals opposite over adjacent. These ratios form the foundation for solving problems involving right triangles and appear in CCSS.HSG.SRT standards for geometric problem-solving.

Why it matters

Trigonometry appears throughout engineering, physics, and construction where precise angle and distance calculations are essential. Architects use these ratios to determine roof slopes, calculating that a 30° roof angle requires a rise of 6 feet for every 10.4 feet of horizontal distance. Surveyors apply trigonometry to measure inaccessible distances, such as finding a building's height by measuring the angle of elevation and distance from the base. In navigation, pilots use trigonometric functions to calculate flight paths and distances. Video game programmers rely on sine and cosine functions to create realistic motion and rotation effects. Medical imaging technology, including MRI and CT scans, uses advanced trigonometric principles to reconstruct 3D images from 2D data, demonstrating how these basic ratios scale to complex applications.

Common mistakes to watch for

  • ✗Confusing which side is opposite or adjacent, leading to tan(30°) = √3 instead of 1/√3 when the sides are mislabeled
  • ✗Using degrees instead of radians on calculators, producing sin(30) = -0.988 instead of sin(30°) = 0.5
  • ✗Forgetting to use inverse functions when finding angles, writing tan(A) = 0.75 = 36.9° instead of A = tan⁻¹(0.75) = 36.9°

Questions teachers ask

What does SOH CAH TOA stand for?+
SOH CAH TOA is a mnemonic where SOH means Sine = Opposite/Hypotenuse, CAH means Cosine = Adjacent/Hypotenuse, and TOA means Tangent = Opposite/Adjacent. Each letter represents the first letter of the trigonometric function and the sides of a right triangle it relates.
How do you remember which side is opposite and which is adjacent?+
The opposite side is across from the angle in question, while the adjacent side touches the angle. The hypotenuse is always the longest side, opposite the right angle. If working with angle A, the opposite side is across from A, and the adjacent side connects to A.
When should you use inverse trigonometric functions?+
Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle from known side lengths. If you know the ratio and need the angle, apply the inverse. For example, if tan(A) = 0.75, then A = tan⁻¹(0.75) = 36.9°.
What are the exact values for 30°, 45°, and 60° angles?+
For 30°: sin = 1/2, cos = √3/2, tan = 1/√3. For 45°: sin = √2/2, cos = √2/2, tan = 1. For 60°: sin = √3/2, cos = 1/2, tan = √3. These special angles appear frequently in problems and should be memorized.
Why do trigonometric ratios only work for right triangles?+
SOH CAH TOA definitions require a right angle to establish the hypotenuse and create consistent opposite/adjacent relationships. For non-right triangles, mathematicians use the Law of Sines and Law of Cosines, which extend trigonometric principles beyond the basic SOH CAH TOA framework.
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Beginner

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Concepts
Evaluate sin/cos/tan of standard angles
Range
30°, 45°, 60°
Steps
1 step
Example
sin(45°)

Easy

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Concepts
Find sides using SOH CAH TOA
Range
triples scaled 1–2×
Steps
3–4 steps
Example
hyp=10, angle=30°, find opposite

Medium

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Concepts
Find sides, larger triangles
Range
triples scaled 1–3×
Steps
3–4 steps
Example
hyp=25, angle=37°, find opposite

Hard

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Concepts
Find angles using inverse trig
Range
triples scaled 2–5×
Steps
3–4 steps
Example
opposite=20, adjacent=15, find angle

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