Spherical Trigonometry Intro

Key idea

Triangles on a sphere and the spherical law of cosines.

In depth

On a sphere, triangles have sides measured as angles and use laws such as cos c = cos a cos b + sin a sin b cos C. It is used for navigation and astronomy because Earth is nearly spherical.

Worked example

For a right spherical triangle, cos c = cos a cos b.

Real-world uses

• Calculating distances and bearings for aircraft and ships on Earth’s curved surface

Common mistake

Applying plane formulas, where angles sum to 180°.

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).