Trig Values via Pythagoras Triples

Key idea

Use 3-4-5 triangles to find exact trig ratios.

In depth

A right triangle with integer sides gives rational sine, cosine and tangent. For example 3-4-5, 5-12-13, 8-15-17 provide exact trig values for the corresponding angles.

Worked example

In the 5-12-13 triangle, sin θ = 5/13, cos θ = 12/13, tan θ = 5/12.

Real-world uses

• Finding exact sine and cosine values quickly in exams and on site without a calculator

Common mistake

Assuming these angles are nice values like 30° or 45°.

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 intermediate topics in Trigonometry

• Radians – Angle measured by arc length on a unit circle; 2π rad = 360°.
• Converting Degrees and Radians – Multiply by π/180 or 180/π.
• Sine Ratio – Opposite over hypotenuse.
• Cosine Ratio – Adjacent over hypotenuse.
• Tangent Ratio – Opposite over adjacent.
• Reciprocal Trig Functions – Cosecant, secant and cotangent.