Sine and Cosine rules Trigonometry applied to triangles

Sine and Cosine rules Trigonometry applied to triangles without right angles. © D R Martin 1

Introduction You have learnt to apply trigonometry to right angled triangles. 2

Now we extend our trigonometry so that we can deal with triangles which are not right angled. 3

First we introduce the following notation. We use capital letters for the angles, and lower case letters for the sides. In ABC The side opposite angle A is called a. The side opposite angle B is called b. In PQR The side opposite angle P is called p. And so on 4

There are two new rules. 5

1. The sine rule Draw the perpendicular from C to meet AB at P. Using BPC: PC = a sin. B. Using APC: PC = b sin. A. Therefore a sin. B = b sin. A. B Putting both results together: The proof needs some changes to deal with obtuse angles. 6

Example 1 Find the length of BC. Substitute A = 35 o, B = 95 o, b = 6. 2: Multiply by sin 35 o: 7

Example 2 Find the size of angle R. Substitute: P = 100 o, p = 18, r = 15: leading to Multiply by 15: leading to 8

When using the sine rule to find an angle: Try to avoid finding the largest angle as you probably do not know whether you want the acute or the obtuse angle. The largest angle is opposite the longest side, the smallest angle opposite the shortest side. When there are two possible angles they add up to 180 degrees. 9

Now do: Page 121 : Exercise 3 N Check your answers. 10

2. The cosine rule 11

Press to skip proof Proof of the cosine rule Applying Pythagoras’ Theorem to APC gives: h 2 = b 2 – x 2 Applying Pythagoras’ Theorem to BPC gives: a 2 = h 2 + (c – x)2 = h 2 + c 2 – 2 cx + x 2. Substituting from equation into equation gives: a 2 = b 2 – x 2 + c 2 – 2 cx + x 2 = b 2 + c 2 – 2 cx. Using APC again: x = bcos. A. Substituting this into gives: a 2 = b 2 + c 2 – 2 cb cos. A. a 2 i. e. = b 2 + c 2 – 2 bc cos. A Again the proof needs some changes to deal with obtuse angles. 12

There are two main ways of writing the cosine rule one for finding a side, one for finding an angle. 13

The cosine rule for finding a side. The formula starts and ends with the same letter, one lower case, one capital. The square of a side is the sum of the squares of the other 2 sides minus twice the product of the 2 known sides and the cosine of the angle between them. 14

The cosine rule for finding an angle. Add 2 bc cos. A and subtract a 2 getting Divide both sides by 2 bc: The cosine of an angle of a triangle is the sum of the squares of the sides forming the angle minus the square of the side opposite the angle all divided by twice the product of first two sides. 15

Example 3 Find the length of BC. (You need to show what you are calculating, but you do not need to show intermediate results. ) 16

Example 4 Find the size of angle D. Substitute d = 8, r = 4, m = 6 into getting There is no need to show intermediate results. leading to The cosine rule automatically takes care of obtuse angles. 17

Now do: Page 123: questions 1 - 5 Check your answers. 18

How do I know whether to use the sine rule or the cosine rule? To use the sine rule you need to know an angle and the side opposite it. You can use it to find a side (opposite a second known angle) or an angle (opposite a second known side). To use the cosine rule you need to know either two sides and the included angle or all three sides. 19