unit 5

Further trigonometry

Chapter 12

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.              

In the right-angled triangle, the side opposite to the right-angle is labelled the hypotenuse, the side opposite θ is labelled “opposite”, the side next to θ is labelled “adjacent”. 

 Define the trigonometric ratios, sine, cosine and tangent of an angle, as follows:

sinθcosθtanθ===oppositehypotenuseadjacenthypotenuseoppositeadjacent

The sine rule

• two sides and an angle opposite to one of the two sides
• one side and any two angles

Example

Find the size of angle R.

Substituting in: 

Cross multiplying: 

Dividing by 9: 

Taking inverse sine: 

The cosine rule

If you need to find the length of a side, you need to know the other two sides and the opposite angle. You need to use the version of the Cosine Rule where a2 is the subject of the formula:      a2 = b2 + c2 – 2bc cos(A) Side a is the one you are trying to find. Sides b and c are the other two sides, and angle A is the angle opposite side a. Example

Step 1	Cosine Rule formula for finding sides:      a2 = b2 + c2 – 2bc cos(A)
Step 2	Fill in the values you know, and the unknown length:      x2 = 222 + 282 – 2×22×28×cos(97°) It doesn't matter which way around you put sides b and c – it will work both ways.
Step 3	Evaluate the right-hand-side and then square-root to find the length:      x2 = 222 + 282 – 2×22×28×cos(97°)            x2 = 1418.143.....            x = 37.7                                                                  Two possible answers Imagine we know angle A, and sides a and b. We can swing side a to left or right and come up with two possible results. BOTH ANSWERS ARE RIGHT :) *but this only happens in two sides and an angle not between