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Right Triangle Trig for Machinists

Guide · Updated October 28, 2026

Most shop math comes down to a right triangle: a chamfer, a hole location off an angled edge, a sine bar setup, a taper. If you know any two parts of a right triangle (at least one a side), you can find the rest with three ratios. With angle A, the side opposite A, the side adjacent to A, and the hypotenuse: sin A = opposite ÷ hypotenuse, cos A = adjacent ÷ hypotenuse, tan A = opposite ÷ adjacent. The memory aid is SOH CAH TOA. Pythagoras (a² + b² = c²) gives a third side from two, and the two acute angles always add to 90°.

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Step by step

Sketch the triangle: Draw it and mark the right angle.
Label what you know: Mark the known angle and sides relative to that angle: opposite, adjacent, hypotenuse.
Pick the ratio: Use the one that contains your two knowns and the unknown.
Solve: Rearrange, for example opposite = hypotenuse × sin A.
Use inverse functions for angles: A = atan(opposite ÷ adjacent), and so on; set the calculator to degrees.
Check: Confirm with Pythagoras or the other ratio before cutting.

Values for common angles

5°: sin 0.08716, cos 0.99619, tan 0.08749
10°: sin 0.17365, cos 0.98481, tan 0.17633
15°: sin 0.25882, cos 0.96593, tan 0.26795
20°: sin 0.34202, cos 0.93969, tan 0.36397
25°: sin 0.42262, cos 0.90631, tan 0.46631
29°: sin 0.48481, cos 0.87462, tan 0.55431
30°: sin 0.50000, cos 0.86603, tan 0.57735
35°: sin 0.57358, cos 0.81915, tan 0.70021
40°: sin 0.64279, cos 0.76604, tan 0.83910
45°: sin 0.70711, cos 0.70711, tan 1.00000
50°: sin 0.76604, cos 0.64279, tan 1.19175
55°: sin 0.81915, cos 0.57358, tan 1.42815
60°: sin 0.86603, cos 0.50000, tan 1.73205
70°: sin 0.93969, cos 0.34202, tan 2.74748
75°: sin 0.96593, cos 0.25882, tan 3.73205
80°: sin 0.98481, cos 0.17365, tan 5.67128

Values are calculated to five places. 30°, 45° and 60° are worth remembering: sin 30 = 0.5, tan 45 = 1, cos 60 = 0.5, and sin 60 = cos 30 = 0.86603.

The rearranged formulas

opposite = hypotenuse × sin A
adjacent = hypotenuse × cos A
opposite = adjacent × tan A
hypotenuse = opposite ÷ sin A = adjacent ÷ cos A
A = asin(opp ÷ hyp) = acos(adj ÷ hyp) = atan(opp ÷ adj)
missing side = √(c² − a²) or √(a² + b²)

Worked example: chamfer leg lengths

A print calls for a 0.060 in × 30° chamfer, where 0.060 is the length along the axis and 30° is from the axis. The radial leg is 0.060 × tan 30° = 0.060 × 0.57735 = 0.0346 in. The face width of the chamfer (hypotenuse) is 0.060 ÷ cos 30° = 0.060 ÷ 0.86603 = 0.0693 in.

Worked example: hole on an angled line

A hole sits 2.500 in from a datum corner along a line at 35° from the X axis. X = 2.500 × cos 35° = 2.500 × 0.81915 = 2.0479 in. Y = 2.500 × sin 35° = 2.500 × 0.57358 = 1.4339 in. Check: √(2.0479² + 1.4339²) = √(4.1939 + 2.0561) = √6.2500 = 2.500.

Worked example: angle from two dimensions

A step drops 0.375 in over a horizontal run of 1.250 in. Angle = atan(0.375 ÷ 1.250) = atan(0.300) = 16.70°, or 16° 42'. Set the vise or sine bar to that angle; on a 5 in sine bar the gage block stack is 5 × sin 16.70° = 5 × 0.28734 = 1.4367 in.

Worked example: diagonal of a rectangle

A 3.000 × 4.000 in plate has a diagonal of √(9 + 16) = 5.000 in, the 3-4-5 triangle. That is a quick way to check a layout or a machine square: measure 3 and 4 along two edges and the diagonal must be 5.

Common problems

Calculator in radians instead of degrees (sin 30 gives −0.988), measuring an angle from the wrong reference line, and rounding the angle before using it. Carry four or five decimals through the math and round at the end. For setups, a sine bar stacked with gage blocks turns the trig into a physical angle you can trust.

FAQ

What does SOH CAH TOA mean?

Sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

How do I find an angle from two sides?

Use the inverse function, for example A = atan(opposite ÷ adjacent).

What is tan 30°?

0.57735.

Why is my calculator giving strange answers?

It is probably set to radians; switch to degrees.

How do I find the third side?

Pythagoras: c = √(a² + b²) for the hypotenuse.

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