Download Reference Triangle: An acute right triangle formed with the terminal

4.4 Notes Trigonometric Functions of Any Angle
Reference Triangle: An acute right triangle formed with the terminal side of an angle,
the origin, and an altitude drawn to the x-axis.
Reference Angle: The acute angle in a reference triangle formed at the origin.
Ex 1: Sketch a reference triangle and find the reference angle.
2
a)   300
b)   225
c)  
3
d)   

4
Review: Trig ratios SOH-CAH-TOA ( 0    90 )
sin   b / c, cos  a / c, tan   b / a
c
b

a
Expanded domain for trig ratios:
Trig ratios will now include all angles  for which the ratio is defined.
Angles will be plotted in a coordinate system – Point P(x, y)
In each quadrant, set up a reference triangle (above)



x and y are the coordinates and can be positive or negative
R is the distance from the origin to a point on the terminal side
of a given angle and is always positive
Reference Triangle(right triangle) is formed by dropping a perpendicular
from Point (x, y) to the horizontal (x) axis
Ratios: (GSP: Coor.of Unit Circle)
y
csc  
sin  
R
x
cos  
sec  
R
y
cot  
tan  
x
R
y
R
x
x
y
R  x2  y 2
1
4.4 Notes Trigonometric Functions of Any Angle
EX 2:
a) Find the exact trig functions if the terminal side of  contains the point (-8, -6).
b) Find the exact trig functions if the terminal side of  contains the point (3, -4).
EX 3:
a) Determine the exact value of the other five trig functions given cos  
 is in quadrant III.
3
and
5
b) Determine the exact value of the other five trig functions for the angle 
4
in quadrant II, given tan    .
3
To determine exact values for angles that are multiples of the above:
 Form a reference triangle (drop  from P(x, y) to X axis)
 Reference angle is the acute angle (always positive) between the terminal side
of  and the X axis.
EX 4:
a) sin 30
b) cos45
c) tan 210
d) sin 315°
e) cos
3
4
f) cot(
2
4
)
3
4.4 Notes Trigonometric Functions of Any Angle
Summary of Sign Properties:
QUAD I
x y R
QUAD II
x
y
R
QUAD III
x
y
R
QUAD IV
x
y
R
Sin x = y/R
Csc x = R/y
Cos x = x/R
Sec x = R/x
Tan x = y/x
Cot x = x/y
All(All functions +) Students(Sine & rec +) Take(Tangent & rec +) Calculus(Cosine & rec +)
EX 5:
a) Use a reference triangle and your knowledge about special triangles to determine the
sin, cos, and tan for 300 .
b) Determine the sin, cos, and tan for 
3
.
4
c) Determine the sin, cos, and tan for 150 .
3
4.4 Notes Trigonometric Functions of Any Angle
Day 2
Exact values can be determined for all quadrantal angles (angles whose terminal sides lie
on an axis).
Assume P(x, y) is 1 unit out and that R is always equal to 1. Therefore x and y will
always be -1, 0, or 1 and R will always be 1. It is then simple to set up the ratios.
(GSP:Coor of Unit Circle)
EX 6:
a) sin 90 
b) cos  
y

R
x

R
cos 90° =
tan 90° =
sin π =
tan π =
c) tan( )  ________________ =
(direction)
d) cot(180)  _________________ =
Trig Ratios and Real Numbers:
Expanding the trig ratios to the set of real numbers –
sin x  sin( xrad )
cos x  cos(xrad )
tan x  tan(xrad )
can be applied to all real numbers (no connection to angles)
csc x  csc(xrad )
sec x  sec(xrad )
tan x  tan(xrad )
cos 7 = cos(7 radians)
Set calculator to radian mode and plug in
EX 7: CHECK MODE!!!!!
 5 
a) sin 0.75
b) cot(220)
c) csc(1.025)
d) tan 

 7 
e) cos 5 radians
4