Labelling a Triangle In a general triangle, the vertices are labelled with capital letters. B A C B A C The angles at each vertex are labelled with that letter. c b a The sides are labelled in lower case letters. It is common to use the letter that corresponds to the opposite vertex.

The sine rule For any triangle ABC, C A B b c a a sin A = b sin B c We can use the first form of the formula to find side lengths and the second form of the equation to find angles. a sin A = b sin B c sin C sin A sin B sin C a = b c or

Otherwise, use the sine rule Sine or Cosine? sin A = sin B sin C a b c cos A = b2 + c2 – a2 2bc Which rule should you use if… You know 3 sides and want to find an angle Cosine rule You know 2 angles, 1 side and want to find another side Sine rule You know 2 sides and the included angle and want to find the third side You know 2 angles and want to find the third Neither! The angles must equal 180° Use the cosine rule if: You are given three sides or You are given two sides and the included angle Otherwise, use the sine rule

Using the sine rule to find side lengths If we are given two angles in a triangle and the length of a side opposite one of the angles, we can use the sine rule to find the length of the side opposite the other angle. For example, Find the length of side a a 7 cm 118° 39° A B C Using the sine rule, a sin 118° = 7 sin 39° When trying to find a side length it is easier to use the formula in the form a/sin A = b/sin B. Encourage pupils to wait until the last step in the equation to evaluate the sines of the required angles. This avoids errors in rounding. a = 7 sin 39° x sin 118° a = 9.82 (to 2 d.p.)

Using the sine rule to find angles If we are given two side lengths in a triangle and the angle opposite one of the given sides, we can use the sine rule to find the angle opposite the other given side. For example, Find the angle at B 6 cm 46° B 8 cm A C Using the sine rule, sin B 8 = 6 sin 46° When trying to find an angle it is easier to use the formula in the form sin A/a = sin B/b. sin B = sin 46° 6 x 8 sin–1 B = sin 46° 6 x 8 B = 73.56° (to 2 d.p.)

Using the Cosine rule to find side lengths m2 + n2 l2 = m2 + n2 – 2mn cos L 2mn cos L L l2 8cm a2 = b2 + c2 – 2bc cos A 110° l2 = 62 + 82 – 2 x 6 x 8 x cos110 = 36 + 64 – 96 cos 110 = 100 – 96 x (-0.342…) = 100 + 32.832… = 132.832… l = 11.5 (3s.f.) M 6cm N Find MN. You can change the letters to a,b and c if you wish MN = 12cm (nrst cm)

Using the cosine rule to find angles If we are given the lengths of all three sides in a triangle, we can use the cosine rule to find the size of any one of the angles in the triangle. For example, Find the size of the angle at A. 4 cm 8 cm 6 cm A B C cos A = b2 + c2 – a2 2bc cos A = 42 + 62 – 82 2 × 4 × 6 Point out that if the cosine of an angle is negative, we expect the angle to be obtuse. This is because the cosine of angles in the second quadrant is negative. We do not have the same ambiguity as with the sine rule where the sine of angles in both the first and second quadrants are positive and so two solutions between 0° and 180° exist. Angles in a triangle can only be within this range. This is negative so A must be obtuse. cos A = –0.25 A = cos–1 –0.25 A = 104.48° (to 2 d.p.)

The area of a triangle using ½ ab sin C The area of a triangle is equal to half the product of two of the sides and the sine of the included angle. A c b B C a Talk through the formula as it is written in words. Remind pupils that the included angle is the angle between the two given sides. Remind pupils, too, that when labelling the sides and angles in a triangle it is common to label the vertices with capital A, B and C. The side opposite vertex A is labelled a, the side opposite vertex B is labelled b and the side opposite vertex C is labelled c. This formula could also be written as ½ bc sin A or ½ ac sin B. Area of triangle ABC = ab sin C 1 2