Deriving the Law of Sines

Trigonometry · Axiom Academy

A geometric derivation showing how a single altitude reveals the beautiful relationship between sides and angles in any triangle. This states that the ratio of each side to the sine of its opposite angle is constant throughout the triangle. But why is this true? Let's derive it geometrically using an altitude. Consider triangle ABC with sides labeled in standard notation: side a is opposite angle A , side b is opposite angle B , and side c is opposite angle C . Now, draw an altitude from vertex B perpendicular to side b (which runs from A to C ). Call this altitude h . 3. Finding the Sine Relationships The altitude h divides our triangle into two right triangles. Let's use the definition of sine in each: In the left triangle: The sine of angle A relates the opposite side (the altitude h ) to the hypotenuse (side c ). In the right triangle: The sine of angle C relates the opposite side (still h ) to the hypotenuse (side a ). Notice that both expressions equal h . This is the key observation! Since both expressions equal the same altitude h , we can set them equal: Now divide both sides by sin A · sin C : By repeating this process with an altitude from vertex A or C , we can show that all three ratios are equal: This is the Law of Sines! Each side of the triangle divided by the sine of its opposite angle gives the same value. When you know two angles and one side (AAS or ASA) When you know two sides and an angle opposite one of them (SSA)

This is the written version of the interactive lesson above. See the full Trigonometry course.