8.2.1 The Sine and Cosine Functions..slowed down UNIT CIRCLE

Remember from geometry that we used sine, cosine, and tangent while working with right triangles. Here we also want to work with right triangles. So when you are provided an angle measurement in degrees or radians, you will want to draw a picture of that angle in the coordinate plane, and then construct a line from a point on the circle of that angle down to or up to the x-axis. Thus creating a right angle. We are going to start with talking about circles of any radius, then scale everything down to the unit circle.

r So if we remember SOH CAH TOA We can say the following. Y X Learn this idea and then we can look at any angle and do the same thing (just have to pay attention to the quadrant) So if we remember SOH CAH TOA We can say the following. r Y X

Consider the angle ϴ with a terminal ray in the second quadrant. The terminal ray is shared by the right triangle and the angle ϴ. Thus what we learn about the triangle will also be true for the angle ϴ. Y X

Example If the terminal ray of an angle ϴ in standard position passes through (-3,4) find sin(ϴ) and cos(ϴ). 4 -3

If ϴ is a 4th quadrant angle and find cos(ϴ).

If we let the radius of the circle be 1 If we let the radius of the circle be 1. Then r=1 and when we think about the following functions P(x,y) 1 y ϴ x When we deal with a circle of radius 1, it is referred to as the UNIT CIRCLE.

P(x,y) = (cosϴ, sinϴ) 1 If working in the unit circle, your x value is simply the same as cosϴ and your y value is the same as your sinϴ. y ϴ x

Examples .. Not the unit circle. How would you find theta in this picture? You must create a right triangle.

Examples.. Not the unit circle

Examples.. Not the unit circle

State whether each expression is positive, negative, or zero.

What would be the equation of a circle with its center at the origin and a radius of 1?

Lets take a look at the points that we found yesterday on the Unit Circle. Did those points all satisfy the equation x2+y2=1? Now lets remember from yesterday that on the unit circle, x=cos(ϴ) and y=sin(ϴ). So does it make since then that we can take the unit circle equation and substitute the x and y values and get (cosϴ)2+(sinϴ)2=1 This equation is referred to as an IDENTITY (or an equation that is true regardless of what values are chosen) and it is the PYTHAGOREAN IDENTITY.

Evaluating Sine and Cosine using Reference Angles Using Reference Angles to Evaluate Trig Functions at Special Angles THIS ALL TAKES PLACE USING THE UNIT CIRCLE

Special right triangles While working in the unit circle we come across angles that are multiples of 30, 60, and 45. If you remember from geometry there was a need to work with 30-60-90 and 45-45-90 right triangles, that need resurfaces within the unit circle.

1st quadrant special angles 1 2 n 3 2 n 𝟑 What does this picture look like if our radius is 1?

1 𝟏 𝟐 𝟑 𝟐 What are the coordinates of this point? 𝟑 𝟐 𝟏 𝟐 What are the coordinates of this point? Remember that this is the unit circle, and in the unit circle for any P(x,y), the x value is cosine and the y value is sine. So what is the sin(30)? cos(30)?

Lets try the same thing with a 60 degree angle for ϴ

Can it be done with a 45 degree

If we look at the unit circle note that it is symmetric about both the x and y axis. Because of this symmetry we can learn a lot about the first quadrant and then use that information as a reference (or guide) for the remaining three quadrants.

Now do the same procedure for the 2nd, 3rd, and 4th quadrants. Take an angle that has a terminal ray in the 1st quadrant, what will the sign of cos(ϴ) be and what will the sin(ϴ) be? Will those signs be true for any angle that has a terminal ray in the 1st quadrant? Now do the same procedure for the 2nd, 3rd, and 4th quadrants. S A Sine is positive All are positive T C Tangent is positive Cosine is positive

A reference angle is an angle that will always be between 0 and 90 degrees and will always be coterminal with the angle ϴ you are interested in. Thus what you are able to learn about the reference angle should then be true about the interested angle ϴ. So if the terminal ray of a reference angle passes through (x,y) then the terminal ray of your initial angle ϴ should pass through (x,y)…Reference angles are only needed when the initial angle ϴ is bigger than 90 degrees. When the initial angle ϴ is less between 0-90 degrees, then the reference angle and the initial angle ϴ are the same exact angle.

To create a reference angle. Draw out the angle ϴ. Take a perpendicular line to the x axis from the terminal ray Note what quadrant the terminal ray is in. Determine the angle of the new triangle. Reproduce the same triangle in the first quadrant, label that angle α (alpha) Now if we can figure out what sin(60) is we can determine what sin(120) is. Making sure that our sign matches up to that of a terminal angle in the 2nd quadrant. ϴ α

Practice Creating Reference angles

Use reference angles to find the following values

Use reference angles to find the following values

Use reference angles to find the following values

Because we know that sin(45⁰) is the same value as sin(45⁰+360⁰) it is evident that the sine and cosine functions repeat their values every 360⁰ degrees or every 2π radians. Thus these functions are referred to as being periodic and have a fundamental period of 360⁰ or 2π radians. This is useful in analyzing repetitive phenomena such as tides, sound waves, orbital paths, etc.