Solving Triangles Using Sine Rule
Trigonometry allows solving non-right-angled triangles with the sine rule. Understand how to find side lengths and angles using this rule, making triangle calculations simpler.
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4 March 2025 The sine rule LO: Using the sine rule to solve non-right- angled triangles www.mathssupport.org www.mathssupport.org
The sine rule You can use trigonometry to solve triangles that are not right-angled. Look at the triangle ABC. perpendicular to AC. In the right-angled triangle ABD. ? ? This gives h = c sin A In the right-angled triangle BCD. ? ? This gives h = a sin C Equate the values of h to give = a sin C c sin A The altitude (height), h of the B triangle is BD a c h ???? = C A b D Rearranging this equation gives sin? ? The ratios of the sine of each angle to the length of the opposite side are equal ???? = =sin? ? www.mathssupport.org www.mathssupport.org
sin? ? =sin? =sin? The sine rule ? ? You can use trigonometry to solve triangles that are not right-angled. Look at the triangle ABC. perpendicular to AB. In the right-angled triangle ACE. ? ? This gives h = b sin A In the right-angled triangle BCE. ? ? This gives h = a sin B Equate the values of h to give = a sin B b sin A Now draw the altitude (height), h B from C to E E a c h ???? = C A b Rearranging this equation gives sin? ? The ratios of the sine of each angle to the length of the opposite side are equal ???? = =sin? ? www.mathssupport.org www.mathssupport.org
The sine rule The sine rule is a method to calculate sides and angles of any triangle. It is useful for finding the length of one side when all the angles and one other side are known, or finding an angle when two sides and one other angle are known To find a side, use the sine rule written as For any triangle ABC: C b c a = = b a sin B sin C sin A or A B To find an angle, use the sine rule written as c Where ais the length of the side opposite ? bis the length of the side opposite ? cis the length of the side opposite ? sin A a sin B b sin C c = = www.mathssupport.org www.mathssupport.org
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 B Using the sine rule: 39 7 a a = sin 39 sin 118 118 7 sin 118 sin 39 a = C A 7 cm a = 9.82 cm (to 3 s.f.) www.mathssupport.org www.mathssupport.org
Using the sine rule to find side lengths 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 A Using the sine rule: sin A 9.4 C sin 98 = 14 14 cm 9.4 cm 9.4 sin 98 14 sin A = 9.4 sin 98 14 98 A B sin 1 A = A = 41.7 (to 3 s.f.) www.mathssupport.org www.mathssupport.org
Using the sine rule to find side angles Find the missing angles and sides in this triangle. C Finding angle B Using the sine rule: sin B 8 60.4 sin 46 8 cm = 6 cm 6 8 sin 46 6 sin B = 73.6 46 A B 8 sin 46 6 c 7.25 cm sin 1 B = Finding side c B = 73.6 (to 3 s.f.) Finding angle C Using the sine rule: c = 6 sin 46 6 sin 60.4 sin 46 c =7.25 cm sin 60.4 C = 180 46 - 73.6 C = 60.4 c = www.mathssupport.org www.mathssupport.org
Using the sine rule to find side angles Find the missing angles and sides in this triangle. Finding angle B C 27.6 Using the sine rule: sin B 8 sin 46 8 cm = 6 6 cm 8 sin 46 6 sin B = 106.4 46 c B A 8 sin 46 6 B must be obtuse = 106.4 3.86 cm sin 1 B = Finding side c B = 73.6 B = 180 - 73.6 Using the sine rule: c = 6 Finding angle C sin 46 6 sin 27.6 sin 46 c =3.86 cm sin 27.6 C = 180 46 - 106.4 C = 27.6 c = www.mathssupport.org www.mathssupport.org
Using the sine rule to find side lengths What do you notice in these triangles. C C 27.6 60.4 8 cm 8 cm 6 cm 6 cm 106.4 73.6 46 c 46 B A B A c 7.25 cm 3.86 cm At the beginning they have the same values for given angles an sides, but they are two different triangles This known as the ambiguous case. When using the sine rule there can be an ambiguous case if: You are given two sides and a non-included acute angle. The side opposite the given acute angle is the shorter of the two given sides www.mathssupport.org www.mathssupport.org
Using the sine rule to solve problems A ship S is on a bearing of 120 from port A and 042 from port B. Port B is on a bearing of 180 from port A. Ports A and B are 35 km apart.Find the distance between the ship and each port. The ship is at S, it is on the bearing 120 from A bearing 042 from B To find the angle ? ?? 180 - 60 - 42 = 78 Angles in a triangle Using the sine rule: a sin 60 = , and on a . B is on a bearing of 180 from A N 35 b 120 A sin 42 = b sin 78 60 S a =35 sin 60 sin 78 b =35 sin 42 sin 78 78 35 km a 31.0 km N 42 a b 23.9 km B www.mathssupport.org www.mathssupport.org
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