Trigonometry
Grades 9–12 Trigonometry Triangles
Trigonometry turns one angle of a right triangle into ratios between its sides. Learn three names and the mnemonic SOH-CAH-TOA:
SOH: sin θ = opposite ÷ hypotenuse
CAH: cos θ = adjacent ÷ hypotenuse
TOA: tan θ = opposite ÷ adjacent
Using it
30° angle, hypotenuse 10, find opposite:
sin 30° = 0.5 = opposite / 10
opposite = 10 × 0.5 = 5
Backwards — finding the angle
Know the sides, want the angle? The inverse functions: tan θ = 3/4 → θ = arctan(0.75) ≈ 36.9°. Calculators call these sin⁻¹, cos⁻¹, tan⁻¹.
Worth memorizing
sin 30° = 1/2 · cos 30° = √3/2 · sin 45° = cos 45° = √2/2 ≈ 0.707 ·
tan 45° = 1. Those cover most textbook problems cold.
Practice
3-4-5 triangle: sin θ (opposite=3) = ___
Same triangle: cos θ (adjacent=4) = ___
Same triangle: tan θ = ___
sin 30° = 0.5, hypotenuse 10 → opposite = ___
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