Trignometry

What is TRIG?

Trigonometry is a branch of mathematics that developed from simple geometry, and surveying.

Trigonometry was probably invented for use in astronomy. The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.

Coordinate Systems

Used to describe the position of a point in space

Coordinate system consists of– a fixed reference point called the origin– specific axes with scales and labels– instructions on how to label a point relative to the

origin and the axes

Types of Coordinate Systems

Cartesian Plane polar

Cartesian coordinate system

Also called rectangular coordinate system

x- and y- axes Points are labeled

(x,y)

Plane polar coordinate system

Origin and reference line are noted

Point is distance r from the origin in the direction of angle , ccw from reference line

Points are labeled (r,)

Trigonometry Review

sin

cos

tan

opposite side

hypotenuse

adjacent side

hypotenuse

opposite side

adjacent side

More Trigonometry

Pythagorean Theorem

To find an angle, you need the inverse trig function– for example,

Be sure your calculator is set appropriately for degrees or radians

2 2 2r x y

1sin 0.707 45

Example

You walk 6 blocks east and then 13 blocks north. As the crow flies, how far are you from home?

Home

6 blks

13 blks

Example - Answer

Home

6 blks

13 blks

a2 b2 c 2

62 132 c 2

c 2 36 169

c 2 205

c 205

c 14.32 blocks from home

Example

You walk 6 blocks east and then 13 blocks north. At what angle would you need to walk to go straight to your destination? (NOTE - several ways to solve…)

Home

6 blks

13 blks

Example - Answer #1

Home

6 blks

13 blks

14.32 blks

sin() opposite

hypotenuse

sin 1 opposite

hypotenuse

sin 1 13

14.32

65.2

Example - Answer #2

Home

6 blks

13 blks

14.3 blks

cos() adjacent

hypotenuse

cos 1 adjacent

hypotenuse

cos 1 6

14.32

65.2

Example - Answer #3

Home

6 blks

13 blks

14.3 blks

tan() opposite

adjacent

tan 1 opposite

adjacent

tan 1 13

6

65.2

Example

A telescope pointed directly at the top of a distant flagpole makes an angle of 31° with the ground. If the scope is low to the ground and 40 m from the base of the pole, how tall is the pole?

Flagpole

Distance = 40 m

31°

Example - Answer

Flagpole

Distance = 40 m

31°

tan() opposite side

adjacent side

tan(31 ) flagpole height

40 metersFlagpole Height 40 meters tan(31 )

Flagpole height 24 meters tall