Law of Tangents — Definition, Formula & Examples

The Law of Tangents is a trigonometric formula that relates two sides of a triangle to the tangent of half the sum and half the difference of their opposite angles. It provides an alternative to the Law of Cosines for solving triangles when you know two sides and the included angle (SAS).

For a triangle with sides $a$ and $b$ opposite angles $A$ and $B$ respectively, the Law of Tangents states that $\dfrac{a - b}{a + b} = \dfrac{\tan\!\left(\frac{A - B}{2}\right)}{\tan\!\left(\frac{A + B}{2}\right)}$ . This identity can be derived from the Law of Sines combined with sum-to-product identities.

Key Formula

\frac{a - b}{a + b} = \frac{\tan\!\left(\dfrac{A - B}{2}\right)}{\tan\!\left(\dfrac{A + B}{2}\right)}

Where:

$a$ = Side opposite angle A
$b$ = Side opposite angle B
$A$ = Angle opposite side a
$B$ = Angle opposite side b

How It Works

You typically use this law when you know two sides and the included angle (SAS). First, note that

A + B = 180° - C

, so

\frac{A+B}{2}

is known. Then plug the known sides into the formula to find

\frac{A-B}{2}

. Once you have both

\frac{A+B}{2}

and

\frac{A-B}{2}

, you recover

A

and

B

individually by adding and subtracting. Finally, use the Law of Sines to find the remaining side.

Worked Example

Problem: In triangle ABC, a = 7, b = 3, and C = 60°. Find angles A and B using the Law of Tangents.

Find (A + B)/2: Since A + B + C = 180°, we have A + B = 120°, so (A + B)/2 = 60°.

\frac{A + B}{2} = \frac{180° - 60°}{2} = 60°

Apply the Law of Tangents: Substitute a = 7, b = 3, and (A + B)/2 = 60° into the formula.

\frac{7 - 3}{7 + 3} = \frac{\tan\!\left(\frac{A - B}{2}\right)}{\tan(60°)} \implies \frac{4}{10} = \frac{\tan\!\left(\frac{A - B}{2}\right)}{\sqrt{3}}

Solve for (A − B)/2: Multiply both sides by √3 to isolate the tangent term, then take the inverse tangent.

\tan\!\left(\frac{A - B}{2}\right) = 0.4\sqrt{3} \approx 0.6928 \implies \frac{A - B}{2} \approx 34.7°

Find A and B: Add and subtract the two half-angle results.

A = 60° + 34.7° = 94.7°, \quad B = 60° - 34.7° = 25.3°

Answer: A ≈ 94.7° and B ≈ 25.3°.

Why It Matters

Before calculators, the Law of Tangents was preferred over the Law of Cosines for SAS problems because it only requires addition and table lookups of tangent values, avoiding the square root step. It still appears in surveying, navigation, and advanced trigonometry courses as a classic technique for solving oblique triangles.

Common Mistakes

Mistake: Mixing up which side is a and which is b, leading to a sign error in (a − b).

Correction: Always let a be the larger side so that (a − b) is positive, which keeps (A − B)/2 positive and easier to interpret. The formula works either way, but consistent labeling avoids confusion.