Law of Tangents

Key idea

Relates differences and sums of sides and angles.

In depth

(a − b)/(a + b) = tan((A − B)/2) / tan((A + B)/2). It is an alternative to the law of cosines when two sides and the included angle are known, and was convenient with logarithm tables.

Worked example

With a = 7, b = 5, C = 60°, A + B = 120°, find A − B from the equation and then each angle.

Formula

Real-world uses

• Solving triangles when two sides and the included angle are known in surveying

Common mistake

Mixing the half angles with whole angles.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Trigonometry

• Half Angle Formulas – Trig of θ/2.
• Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
• Product-to-Sum Formulas – Convert products into sums.
• Sum-to-Product Formulas – Convert sums into products.
• Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
• Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).