Right-Triangle Trigonometry (SOH-CAH-TOA)
From Vectors to Triangles
In 1D and 2D motion we often break a vector (displacement, velocity, or force) into components along perpendicular axes (x: east–west, y: north–south). Whenever the vector and its components form a right triangle, the trigonometric functions sine, cosine, and tangent relate the sides of the triangle to the angle θ measured from the +x-axis:
- SOH — Sine θ = Opposite / Hypotenuse
- CAH — Cosine θ = Adjacent / Hypotenuse
- TOA — Tangent θ = Opposite / Adjacent
For a vector →v of magnitude |v| at angle θ: vx = |v| cosθ, vy = |v| sinθ. Likewise for displacement: Δx = |Δr| cosθ, Δy = |Δr| sinθ.
Drag the Vector: Read Off Components with SOH-CAH-TOA
Drag the arrowhead. Watch Δx, Δy, and angle θ update. Toggle the labels to see Opposite, Adjacent, and Hypotenuse.
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Real-World Connection
A drone flies at a speed of 10 m/s at an elevation angle of 30°. Its ground speed is vx = 10 cos 30° ≈ 8.66 m/s while its climb rate is vy = 10 sin 30° = 5.0 m/s. Trigonometry lets engineers plan climb paths, estimate travel time, and check if motors can supply enough vertical thrust.