Basic trigonometry

Direct-current circuits often reduce to straight addition: 12 V here plus 12 V there. AC circuits add quantities that point in different directions—like two tugboats pulling a barge at angles. Trigonometry is the tool that lets you find the resultant (the single equivalent pull) without guessing. Before you touch phasors or power triangles, you must be comfortable with a right triangle and three buttons on your calculator.

A right triangle has one 90° corner. The side across from the right angle is the hypotenuse—always the longest side. Pick any acute angle (less than 90°) in the triangle and label it θ (theta). Relative to θ:

• Opposite — the side across from θ, not touching it.
• Adjacent — the side next to θ that is not the hypotenuse.
• Hypotenuse — always opposite the 90° corner.

If you change which angle you call θ, opposite and adjacent swap. Hypotenuse never changes.

For angle θ in a right triangle:

Each ratio is a pure number (no units) for a given angle. A 30° angle always has the same sine whether the triangle is drawn on paper or scaled to a motor diagram.

If you know one acute angle and one side, trigonometry finds the others.

Example. Hypotenuse = 10 units, θ = 30°. Find the opposite side.

sin 30° = opposite / 10 opposite = 10 × sin 30° = 10 × 0.5 = 5 units

Same pattern for cos (adjacent) and tan (when you know adjacent and need opposite, or the reverse).

If opposite = 5 and hypotenuse = 10:

sin θ = 5 / 10 = 0.5 θ = sin⁻¹(0.5) = 30°

Use inverse functions on the calculator: sin⁻¹, cos⁻¹, tan⁻¹ (often labeled 2nd sin, 2nd cos, 2nd tan). Always confirm the calculator is in degree mode unless a problem specifies radians.

Common field errors:

1. Degree vs radian mode — wrong mode gives nonsense angles.
2. Wrong ratio — using tan when you need sin.
3. Wrong side — labeling opposite/adjacent from the wrong angle.

Build a habit: sketch the triangle, mark θ, label O-A-H, then pick SOH, CAH, or TOA before touching keys.

Memorize or keep a small table handy:

Notice 45°: opposite and adjacent are equal, so sin and cos are both 0.707 (same number you will see when converting peak voltage to RMS in a later lesson).

When two AC quantities are out of phase, their combined effect is found with vector addition—which is trigonometry in disguise. A 30° phase shift is not abstract; it is a fixed ratio of sides on a right triangle. Master O-A-H now and phasor diagrams later will feel like labeling a triangle you already understand.

Situation. A trainee must find the vertical rise of a conduit run on a 30° offset. The hypotenuse (actual conduit length along the angle) is 20 ft per the drawing.

Work. Vertical rise = opposite side. opposite = 20 × sin 30° = 20 × 0.5 = 10 ft

Applied lesson. The same sin/cos logic applies when you later split an AC voltage into horizontal (in-phase) and vertical (out-of-phase) components. The triangle is either physical (conduit offset) or graphical (phasor)—the math is identical.