Trigonometry Formulas for Electrical Engineers

Sine, cosine and tangent formulas for engineers: the six trig functions, degrees and radians, quadrant signs and power factor angle examples.

The formulas

On the unit circle, cos θ is the x coordinate and sin θ is the y coordinate. In the first quadrant (0° to 90°) all functions are positive; in the second only sin is positive, in the third only tan, and in the fourth only cos. tan θ and sec θ are not defined at 90° and 270°, where cos θ = 0. cot θ and csc θ are not defined at 0° and 180°, where sin θ = 0. Check that your calculator is in the right angle mode: sin 30 is 0.5 in degrees but −0.988 in radians.

Worked example: reactive power from a power factor of 0.8

A load draws P = 100 kW at a lagging power factor of 0.8.

1. φ = arccos(0.8) = 36.87°
2. sin φ = 0.6 and tan φ = 0.75
3. Q = P tan φ = 100 × 0.75 = 75 kVAR
4. S = P / cos φ = 100 / 0.8 = 125 kVA

Worked example: 150° in degrees and radians

1. 150° × π / 180 = 2.618 rad
2. 150° is in the second quadrant, so sin is positive and cos is negative: sin 150° = 0.5 and cos 150° = -0.866
3. tan 150° = 0.5 / -0.866 = -0.5774

Frequently asked questions

Should I use degrees or radians?

Use whichever your angle is written in, and make sure the calculator mode matches. Electrical phase angles are normally in degrees; angular frequency in rad/s is in radians.

Why does tan 90° have no value?

tan θ = sin θ / cos θ and cos 90° = 0, so the division has no value. The tangent grows without limit as the angle approaches 90°.

Why does cos 90° show 0 and not 6e-17?

The computer holds π only approximately, so a plain cos of 90° in floating point gives a tiny leftover. The calculator treats an angle within a billionth of a degree of a multiple of 90° as exactly that multiple.

How do I find the angle from a power factor?

Take the inverse cosine of the power factor: a power factor of 0.8 gives 36.87°. Use this calculator in the other direction to check that cos of that angle returns 0.8.

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