Trigonometry for Any Angle
opposite
sin  
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
hypotenuse
csc  
opposite
hypotenuse
sec  
adjacent
adjacent
cot  
opposite
Sine
Abbreviation
Reciprocal
Function
Co-function
Right Triangle
Definition
Unit Circle
Definition
Any Angle
Definition
Positive
Quadrants
Negative
Quadrants
Odd or Even
Domain
Range
Period
Inverse
Inverse
Domain
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
Reciprocal
Function
Co-function
Right
Triangle
Definition
Unit Circle
Definition
Any Angle
Definition
Positive
Quadrants
Negative
Quadrants
Odd or Even
Sine
sin
Cosine
cos
Tangent
tan
Cosecant
csc
Secant
sec
Cotangent
cot
csc
sec
cot
sin
cos
tan
cos
sin
cot
sec
csc
tan
Abbreviation
Reciprocal
Function
Co-function
Right
Triangle
Definition
Unit Circle
Definition
Any Angle
Definition
Positive
Quadrants
Negative
Quadrants
Odd or Even
Sine
sin
Cosine
cos
Tangent
tan
Cosecant
csc
Secant
sec
Cotangent
cot
csc
sec
cot
sin
cos
tan
cos
sin
cot
sec
csc
tan
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
y
x
y/x
1/ y
1/x
x/y
Abbreviation
Reciprocal
Function
Co-function
Right
Triangle
Definition
Unit Circle
Definition
Any Angle
Definition
Positive
Quadrants
Negative
Quadrants
Odd or Even
Sine
sin
Cosine
cos
Tangent
tan
Cosecant
csc
Secant
sec
Cotangent
cot
csc
sec
cot
sin
cos
tan
cos
sin
cot
sec
csc
tan
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
y
x
y/x
1/ y
1/x
x/y
1 and 2
1 and 4
1 and 3
1 and 2
1 and 4
1 and 3
3 and 4
2 and 3
2 and 4
3 and 4
2 and 3
2 and 4
An even function:
f(x) = f(-x)
cos(30o) = cos(-30o)?
cos(135o) = cos(-135o)?
The cosine and its
reciprocal are even
functions.
7
An odd function:
f(-x) = -f(x)
sin(-30o) = -sin(30o)?
sin(-135o) = -sin(135o)?
The sine and its
reciprocal are odd
functions.
8
An odd function:
f(-x) = -f(x)
tan(-30o) = -tan(30o)?
tan(-135o) = -tan(135o)?
The tangent and its
reciprocal are odd
functions.
9
Cosine and secant functions are even
cos (-t) = cos t
sec (-t) = sec t
Sine, cosecant, tangent and cotangent
are odd
sin (-t) = - sin t
csc (-t) = - csc t
tan (-t) = - tan t
cot (-t) = - cot t
Add these to your worksheet
10
How can we memorize it?



Symmetry
For Radians
the
denominators
help!
Knowing the
quadrant gives
the correct
+ / - sign
 ON
YOUR OWN try these…
 Write the question and the answer
1. cos

3
13
2. sin

6
14
5
3. sin
6
15
2
4. cos
3
16
5. cos 

2
17
6. sin

3
18
13
7. cos
6
19
3
8. sin
2
20
9. sin ( )
21
3
10. tan
4
22
23
1
2
1. cos

3
24
1
2
2. sin

6
25
1
2
5
3. sin
6
26
1

2
2
4. cos
3
27
0
5. cos 

2
28
3
2
6. sin

3
29
3
2
13
7. cos
6
30
1
3
8. sin
2
31
0
9. sin(  )
32
1
3
10. tan
4
33
Let θ be an angle in standard position. Its
reference angle is the acute angle θ’ (called
“theta prime”) formed by the terminal side of θ
and the horizontal axis.
Let θ be an angle in standard position and its
reference angle has the same absolute value
for the functions, the sign ( +/ - ) must be
determined by the quadrant of the angle.
 Quadrant II θ’ = π – θ (radians)
= 180o – θ (degrees)
 Quadrant III θ’ = θ – π (radians)
= θ – 180o (degrees)
 Quadrant IV θ’ = 2π – θ (radians)
= 360o – θ (degrees)
Given a point on the terminal side
Let  be an angle in standard position with
(x, y) a point on the terminal side of  and r
be the length of the segment from the origin
to the point
(x,y)
r
Then….
θ
r x y 0
2
2
 The
six trigonometric functions can be
defined as
sin   y
r
csc   r , y  0
y
cos  x
r
sec   r , x  0
x
tan   y , x  0
x
cot   x , y  0
y
Add these definitions to summary worksheet
Abbreviation
Reciprocal
Function
Co-function
Right
Triangle
Definition
Unit Circle
Definition
Any Angle
Definition
Positive
Quadrants
Negative
Quadrants
Odd or Even
Sine
sin
Cosine
cos
Tangent
tan
Cosecant
csc
Secant
sec
Cotangent
cot
csc
sec
cot
sin
cos
tan
cos
sin
cot
sec
csc
tan
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
y
x
y/x
1/ y
1/x
x/y
y/r
x/r
y/x
r/y
r/x
x/y
1 and 2
1 and 4
1 and 3
1 and 2
1 and 4
1 and 3
3 and 4
2 and 3
2 and 4
3 and 4
2 and 3
2 and 4
Odd
Even
Odd
Odd
Even
Odd
Find sin, cos and tan given (-3, 4) is a point on
the terminal side of an angle.
Find r
2. Find the ratio of the sides of the reference angle
3. Make sure you have the correct sign based upon
quadrant
1.
Find r. (-3)2 + (4)2= r2
r =5
sin θ = 4/5
cos θ = -3/5
tan θ = -4/3
(-3, 4)
r
θ
The cosine and sine of the angle are positive
1
The cosine and sine are negative
3
The cosine is positive and the sine is negative.
4
The sine is positive and the tangent is negative
2
The tangent is positive and the cosine is negative.
3
The secant is positive and the sine is negative.
4
Given tan  = -5/4 and the cos  > 0, find the
sin  and sec .
Which quadrant is it in?
The tangent is negative,
and the cosine is positive
Quadrant IV
at point (4, -5)
θ
r
(4, -5)
Find r and use the
triangle to find the sine
and secant
 be an angle in quadrant II such that
sin  = 1/3 find the cos  and the tan .
 Let
Set up a triangle based
upon the information
given .
Calculate the other side
Find the other
trigonometric functions
(x, 1)
3
θ