Figure

Math and relationships

Sine and cosine

Relate a right triangle’s angle and sloping length to its horizontal and vertical components.

In a right triangle, sine relates an angle to the opposite side and hypotenuse; cosine relates it to the adjacent side and hypotenuse. Measure the angle from the stated reference before choosing the relationship.

Find rise and run from a slant

Split a sloping length into its horizontal run and vertical rise.1

Which function?

For an angle measured up from horizontal, cos × slant gives the run. sin × slant gives the rise. Enter the angle in degrees without a length unit.

Right triangle: a 10 foot slant makes a 30 degree angle above horizontal. The horizontal run is about 8.66 feet and the vertical rise is 5 feet. Cosine finds the run; sine finds the rise.10 ftrunrise30°
Schematic, not to scale.

Try it · a 10 ft slant at 30°

Use 2nd if Feet is above its key.

  1. Start fresh

    Press Clear entry, then Clear entry, then Select length units.

  2. Cosine of the angle

    Press 30, then Cosine, then Multiply.

  3. Multiply by the slant

    Press 10, then feet, then Equals.
  1. Horizontal run · rounded to 1/16 in.8 feet 7 and 15/16 inches

About 8.66 ft. Your fraction setting controls rounding.

Find the rise of that same slant

  1. Start the other calculation

    Press Clear entry, then Clear entry.

  2. Sine of the same angle

    Press 30, then Sine, then Multiply.

  3. Multiply by the same slant

    Press 10, then feet, then Equals.
  1. Vertical rise5 feet 0 inches

Also useful for

Diagonal framing braces, sloping pipe or duct, ramp layouts and projecting an angled support onto a wall or floor.

Remember

These relationships use a right triangle. If the angle is measured from vertical, rise and run swap roles.

Find sine, cosine and tangent

Enter the angle in degrees without a length unit.

Start a new calculation

  1. Press Clear entry, then Clear entry.

Calculate

  1. Angle in degrees

    Press 12, then Sine.
  1. Example result0.2079117

Keep in mind

For cosine or tangent, enter the angle, then press cos or tan.

Cosine of 33°

Enter the angle in degrees without a length unit.

Start a new calculation

  1. Press Clear entry, then Clear entry.

Calculate

  1. Angle in degrees

    Press 33, then Cosine.
  1. Example result0.8386706

Display assumptions

  1. Fraction precision: 1/16″; Fraction format: Simplified. ↩

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 →