### Basic Trigonometry

There are three basic trigonometric functions defined for a right-angled triangle:

- sine
- cosine
- tan

Let a, and b be the two sides and c be the hypotenuse. Then the functions takes the following form:

- sin x = opposite / hypotenuse = a /c
- cos x = = adjacent / hypotenuse = b / c
- tan x = opposite / adjacent = a / b

**The following examples are based on this triangle.**

**E.g.1**

In the above triangle, AB = 3cm and x = 30^{0}. Find AC and BC.

sin 30 = 3 /AC

0.5 = 3 / AC

AC = 6cm.

cos 30 = b /AC

0.8660 = b / 6

AC = 5.1cm.

**E.g.2**

In the above triangle, BC = 4cm and x = 60^{0}. Find AC and AB.

cos 60 = 4 /AC

0.5 = 4 / AC

AC = 8cm.

tan 60 = AB /BC

1.7 = AB / 4

AB = 6.8cm.

**E.g.3**

In the above triangle, AB = 3cm and BC = 5cm. Find x.

tan x = 3 / 5 = 0.6

x = tan^{-1}0.6 = 28.6

x = 28.6^{0}

#### The Sine Rule

**sin A / a = Sin B /b = Sin C /c **

*Note the blinking letters to see the relationship.*

**E.g.1**

A = 30^{0} ; B = 50^{0}; b = 9cm; find c.

c / sin 50 = 9 / sin 30

c = 9 x sin 50 / sin 30

c = 13.7 cm.

**E.g.2**

A = 110^{0} ; b = 5cm; a = 9cm; find B.

5 / sin B = 9 / sin 110

sin B = 5*sin 110 / 9

B = 31.5^{0}

#### The Cosine Rule

**c ^{2} = a^{2} + b^{2} - 2abcosC**

*Note the blinking letters to see the relationship.*

**E.g.1**

^{0}; find c.

c

^{2}= 4

^{2}+ 5

^{2}- 2x4x5xcos60

c

^{2}= 16 + 25 - 20

c

^{2}= 21

c= 4.6 cm.

**E.g.2**

^{0}; find c.

c

^{2}= 5

^{2}+ 12

^{2}- 2x5x12xcos90

c

^{2}= 25 + 144 - 0

c

^{2}= 169

c= 13cm.

**E.g.3**

^{0}; find c.

c

^{2}= 8

^{2}+ 7

^{2}- 2x8x7xcos120

c

^{2}= 64 + 49 + 56

c

^{2}= 169

c= 13 cm.

**E.g.4**

8

^{2}= 7

^{2}+ 6

^{2}- 2x4x5xcosC

64 = 49 + 36 - 84cosC

-84cosC = -21

cosC = 0.25

C = 75.5

^{0}

#### Area of a Triangle

**A = 1/2 absinx**

*Proof:*

A = 1/2 X h X b

sin x = h / a => h = a sinx

A = 1/2 absinx

**E.g.**

^{0}

A = 1/2 * 4 * 8 * sin 30

A = 8cm

^{2}.

#### Deriving Trigonometric Values of 0^{0}, 30^{0}, 45^{0}, 60^{0}, 90^{0}, 180^{0}, 360^{0}

Trigonometric values of major angles, such as 0^{0}, 90^{0}, 180^{0}, 270^{0} and 360^{0} can be remembered
with the aid of the trigonometric curves, if you can visualize them. The three major curves are as follows:

The following image shows how to obtain the values of 0^{0}, 180^{0}, and 360^{0} from the basic definitions

The following image shows how to obtain the values of 90^{0} and 270^{0} from the basic definitions

In addition, the values of 30^{0}, 45^{0} and 60^{0} can be derived in the following way:

The trigonometric values of 45^{0} is calculated from an *right-angled isosceles triangle with each equal side being 1 unit.*

The values of 30^{0} and 60^{0} are calculated from an *equilateral triangle* of each side 2 unit.

#### Trigonometric Values in the Four Quadrants

The angles are measured from the border between the *first* and the *fourth* quadrants - and anticlockwise, by convention. If it is measured in clockwise, it is considered as negative.

The following animation explicitly demonstrates it:

**Now, in order to complement what you have just learnt, work out the following questions:**

- ABC is a triangle. AB = 6cm and angle ABC = 45
^{0}. Find the perimeter and the area of the triangle. - Prove that Pythagoras Theorem is correct using the Cosine Rule.
- The height of a man is 1.2m. When he looks at the top of a tree, standing 20m away from it, the angle of elevation is 32
^{0}. Calculate the height of the tree. - Points P and Q are due south and West of a pillar of height 20m. The angle of elevation of the top from these points are 22
^{0}and 42^{0}respectively. Find the distance between P and Q. - A ship is sailing a distance of 200 miles from port X to port Y, on a bearing 020
^{0}. At Y, it changes its course on a bearing 110^{0}and reaches port Z, after travelling 360 miles. Find the distance between port X and Port Z. Find the bearing of Z from X as well. - A jet is flying a distance of 120 miles from town P to town Q, on a bearing 030
^{0}. At Q, it changes its course on a bearing 60^{0}and reaches town R, after travelling 240 miles. Find the distance between town P and town R. Find the bearing of R from P as well. - A man is looking down from a cliff at another man on the beach. The angle of
depression is 20
^{0}. The second man walks 200m away from the cliff and the new angle of depression for the man at the top becomes 15^{0}. Find the vertical height of the cliff. - The two sides of a right-angled triangle are 5cm, 12cm. Find its size of hypotenuse and the rest of the angles of the triangle.
- The three sides of a triangle are 8cm, 7cm and 13cm. Find its area. Calculate the shortest distance between a vertex and the longest side as well.
- ABC is a triangle with AB = 4cm and AC = 5cm. AD is at an right angle to BC. AD = 3cm. Find BC.