Pop Quiz #1 Take out a paper and pencil (put everything else on the floor!)

Pop Quiz #1 – Put your name Pop Quiz #1 – Put your name 1.Solve the following triangle (all sides and angles): 2.Express in degrees: 3.Express in radians: 135˚ 4.Express in turns: 5.The radius of circular railroad track is 600m. What is the central angles of a train’s trajectory if it travels 1.8 km along this track? x y 10 30˚ 2π rad 5 2π rad 3

Trig Angle vertexThe trig angle is a measure of rotation occurring at the origin (aka the vertex) of the Cartesian A positive rotation is in the counter- clockwise direction A negative rotation is a clockwise movement

Trig Angle The angle has two sides: –The initial side along the positive x-axis –The terminal side, which we obtain once the rotation is complete.

Trig Angle A trig angle may be greater that 360˚ A trig angle can be measured in degrees or radians

Example 1 Approximately how many degrees? 45˚ -135˚ 405˚

Co-Terminal Two angles are co-terminal when they have different rotations but end up in the same position. Example: 10˚ and -350˚ -π and 3π 2

Trig Circle & Trig Points unit circleImagine a circle centered at the origin with a radius of 1. We call this the unit circle. trig pointAny point on this circle is called a trig point

Trig Circle & Trig Points There are infinite number of trig points, but we are mainly concerned with the points that are result of remarkable angles and their multiples.

Trig Circle & Trig Points P(x,y) is a trig point if and only if: x 2 + y 2 = 1

Example 2 Are the following trig points: 1.(1,0)? 2.(0,-1)? 3.( ½, ½)? YES NO

Example 3 What are the exact co-ordinates of the trig point if we know the angle of rotation is π? 6 sin(π) = y 6 1 y = 0.5 ( (either change your calculator to radians, or convert to degrees!) 1

Example 3 What are the exact co-ordinates of the trig point if we know the angle of rotation is π? x 2 = 1 x 2 = 0.75 x =

General Rule You can find the exact co-ordinates of any trig point P if you know the angle of rotation! P(θ)=(cosθ,sinθ) In degrees or radians

Important Points If no unit is given, radians are implied If π is used then radians is implied as well P(π/4) is an example of a way to describe a trig point. –P tells you it is a point –(π/4) tells you the rotation you went through to get to P

Example 4 What are the exact co-ordinates of P(160˚)? P(160˚) = (cos160˚, sin160˚) = (-0.94, 0.34)

Important Note You will want to add radians to your remarkable angles table. This allows you to find certain trig points without using a calculator. Angles 0˚30˚ = π/6 45˚ = π/4 60˚ = π/3 90˚ = π/2 Sine √2 2 √3 2 1 Cosine1 √3 2 √ Tangent0 √3 3 1√3 Undefined

Example 5 Without using a calculator, determine: P(π/6) P(π/6) = (cos(π/6), sin(π/6)) = (√3, 1 ) 2 2

Homework Workbook p.198 #2,3 p. 199 #5,6 p.201 #7a,b,c, #8a,b,c p.202 #10, 11a,c,e p.203 #12a, #13b, 14,15,16,17