Trigonometry

2

Unit 4:Mathematics

Aims• Solve oblique

triangles using sin & cos laws

Objectives• Calculate angles and

lengths of oblique triangles.

a=2.4

c=5.2

b=3.5A

B

C

The table shows some of the values of these functions for various angles.

Between 0o a 90o:

Sines increase from 0 to 1

Cosines decrease from 1 to 0

Between 0o a 90o:

Tangents increase from 0 to infinity.

Cos(90 - X) = Sin(X)Sin(90 - X) = Cos(X)

Write out the each of the trigonometric functions (sin, cos, and tan) of the following

1. 45º

2. 38º

3. 22º

4. 18º

5. 95º

6. 63º

7. 90º

8. 152º

9. 112º

10. 58º

When solving oblique triangles, simply using

trigonometric functions is not enough. You need…

The Law of Sines

C

c

B

b

A

a

sinsinsin

The Law of Cosines

a2=b2+c2-2bc cosA

b2=a2+c2-2ac cosB

c2=a2+b2-2ab cosC

a

c

bA

B

C

Whenever possible, the law of sines should be used. Remember

that at least one angle measurement must be given in order to use the law of sines.

The law of cosines in much more difficult and time consuming

method than the law of sines and is harder to memorize. This law,

however, is the only way to solve a triangle in which all sides but no

angles are given.

Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.

The triangle has three sides, a, b, and c. There are three angles, A, B, C (where angle A is opposite side a, etc). The height of the triangle is h. The sum of the three angles is always 180o. A + B + C = 180o

The area of this triangle is given by one of the following three formulae

Area = (a × b × Sin C) = (a × c × Sin B) =

2 2(b × c × Sin A)

2 = b × h

2

a2 = b2 + c2 - (2 × b × c × Cos A) b2 = a2 + c2 - (2 × a × c × Cos B) c2 = a2 + b2 - (2 × a × b × Cos C)

The relationship between the three sides of ageneral triangle is given by The Cosine Rule. There are three forms of this rule. All are equivalent.

Show that Pythagoras' Theorem is a special case of the Cosine Rule.

In the first version of the Cosine Rule, if angle A is a right angle, Cos 90o = 0. The equation then reduces to Pythagoras' Theorem.

a2 = b2 + c2 - (2 × b × c × Cos 90o) = b2 + c2 - 0 = b2 + c2

The relationship between the sides and angles of a general triangle is given by The Sine Rule.

C

c

B

b

A

a

sinsinsin

Find the missing length and the missing angles in the following triangle.

By the Cosine Rule, a2 = b2 + c2 - (2 × b × c × Cos A)

Find the missing length and the missing angles in the following triangle.

Now, from the Sine Rule,

C

c

A

a

C

c

B

b

A

a

sinsin.........

sinsinsin

This can be rearranged to 42.3

326.4sin

sin..sin

xSinC

a

AxcC

REMEMBER

Side a is opposite angle ASide b is opposite angle BSide c is opposite angle C

Solve the following oblique triangles with the dimensions given

12

22

14A

B

C

a

25

b

28 º

A

B

C

31 º

15

c

24

35 º

A

B

C

5

c

8A

B

C

168 º