Download Will those signs be true for any angle that has a terminal ray in the 1

7.4 Evaluating and Graphing Sine
and Cosine
Using Reference Angles to Evaluate
Trig Functions at Special Angles
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
2n
1
2
n
3
2
n 𝟑
What does this
picture look like if
our radius is 1?
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.
1
𝟏
So what is the sin(30)? cos(30)?
𝟐
𝟑
𝟐
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.
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
Sine is positive
T
Tangent is
positive
A
All are positive
C
Cosine is
positive
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)
0
sin(120 )
ϴ
α
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.
Practice Creating Reference angles
sin(135o )
cos(330o )
tan(210o )
 5 
sin 

 4 
 5 
cos 

 6 
 2 
tan 

 3 
Use reference angles to find the
following values
 2 
cos 

 3 
Use reference angles to find the
following values
 19 
sin 

 4 
Use reference angles to find the
following values
 
tan   
 3
P.280 11-18