Sine rule, cosine rule and triangle area

Extend sine and cosine to obtuse angles

For an obtuse angle θ, the unit-circle point lies left of the vertical axis. Its vertical coordinate is positive and its horizontal coordinate negative. Thus sin(180°-α) = sin α and cos(180°-α) = - cos α for acute α. For example, sin 150° = 0.5 while cos 120° = -0.5.

Opposite pairs matter in the sine rule

For a triangle, name a,b,c opposite angles A,B,C. Then a/sin A = b/sin B = c/sin C. A perpendicular height gives h = b sin A = a sin B, which explains the relationship between the opposite pairs.

If A = 40°, B = 65° and a = 8 cm, then b = 8 sin 65°/sin 40° ≈ 11.28 cm. The side opposite the larger angle is longer, a useful check.

If you find sin B = 0.7, both B ≈ 44.4° and B ≈ 135.6° are possible before checking the other data. Test whether each leaves a positive third angle and satisfies the side conditions. In a right-triangle task the acute condition chooses one; an arbitrary triangle can have two valid configurations.

Two sides and their included angle

The cosine rule is c² = a² + b²-2ab cos C. If a = 7, b = 9 and included angle C = 60°, then c² = 49 + 81-63 = 67, so c = square root of 67 ≈ 8.19. For C = 90°, it reduces to Pythagoras because cos 90° = 0.

With all three sides, rearrange to cos C = (a² + b²-c²)/(2ab), then use inverse cosine.

Area uses the included angle too

Triangle area is (1/2)ab sin C, since its perpendicular height is b sin C. For the 7-by-9 triangle with included angle 60°, area is 31.5 sin 60° ≈ 27.28 square units. An unrelated angle cannot replace the included angle between those two sides.

Check your understanding 1 Select the information pattern

A triangle has sides 5 cm and 8 cm with included angle 120°. Find the third side and its area.

Your answer or drawing on paper

Show answer

The third side satisfies c² = 25 + 64-80 cos 120° = 129, so c = square root of 129 ≈ 11.36 cm. Area is (1/2)(5)(8) sin 120° = 10 square root of 3 ≈ 17.32 cm². The obtuse cosine is negative; the sine remains positive.

Check an ambiguous case

Given A = 30°, a = 5 cm and b = 8 cm, the sine rule gives sin B = 0.8. Both B ≈ 53.1° and B ≈ 126.9° leave a positive third angle with A. Both triangles fit the stated information. Do not discard the second merely because a calculator returns the principal inverse-sine value.