Figure

Math and relationships

Inverse cosine

Find an angle from a cosine ratio or from horizontal reach and sloping length.

Inverse cosine finds an angle from an adjacent-side-to-hypotenuse ratio. A horizontal reach and a sloping brace supply one practical example; the same relationship also works with a known cosine ratio.

Find a brace’s angle

Find the angle between a brace and the horizontal in a right triangle.

Why cosine?

Horizontal reach ÷ brace length gives the cosine of this angle. Inverse cosine, cos⁻¹, turns that ratio into an angle.

Right triangle: a 10 foot brace slopes up from the left over an 8 foot horizontal reach. Theta is the angle between the brace and the horizontal at the left end.8 ft10 ftθ
Schematic, not to scale.

Try it · 8 ft reach, 10 ft brace

Both measurements are in feet, so divide their numbers.

  1. Start fresh

    Press Clear entry, then Clear entry.

  2. Horizontal reach ÷ brace length

    Press 8, then Divide, then 10, then Equals.

  3. Find the angle

    Press Second, then Inverse cosine.
  1. Angle above horizontalabout 36.87°

Also useful for

Framing braces, diagonal supports, sloping pipe or duct, and checking a ramp angle when you know its horizontal reach and slant length.

Remember

cos⁻¹ finds an angle. It does not mean 1 ÷ cos.

Find an angle from a cosine ratio

Enter the cosine ratio to find its angle in degrees.1

Start a new calculation

  1. Press Clear entry, then Clear entry.

Calculate

  1. Cosine ratio

    Press .06753, then Second, then Inverse cosine.
  1. Example result86.12787°

Display assumptions

  1. Angle decimal places: Variable. ↩

Keep the guide beside your calculation.

These examples are built into Figure. Open Help while you calculate, then return to the same guide and reading position.

See Help in Figure →