More: www.PinoyBIX.org Lesson Objectives Trigonometric Functions of Angles Trigonometric Function Values Could find the Six Trigonometric Functions Learn the signs of functions in different Quadrants Could easily determine the signs of each Trigonometric Functions Solve problems involving Quadrantal Angles Find Coterminal Angles Learn to solve using reference angle Solve problems involving Trigonometric Functions of Common Angles Solve problems involving Trigonometric Functions of Uncommon Angles

1. www.PinoyBIX.org
Presents:
Trigonometric Function of General Angles
credit: Shawna Haider

2. OF GENERAL ANGLES

3. Our method of using right triangles only works for
acute angles. Now we will see how we can find the trig
function values of any angle. To do this we'll place
angles on a rectangular coordinate system with the
initial side on the positive x-axis.

HINT: Since it is 360° all the way
around a circle, half way around
(a straight line) is 180°
If  is 135°, we can find the angle
formed by the negative x-axis and
the terminal side of the angle.
This is an acute angle and is
called the reference angle.
reference
angle
What is the measure of
this reference angle?
 =135°
180°- 135° = 45°
Let's make a right triangle by drawing a line perpendicular to
the x-axis joining the terminal side of the angle and the x-axis.

4. Let's label the sides of the triangle according to a 45-45-90
triangle. (The sides might be multiples of these lengths but
looking as a ratio that won't matter so will work)
45°
 =135°
The values of the trig functions of angles and their reference
angles are the same except possibly they may differ by a
negative sign. Putting the negative on the 1 will take care of this
problem.
-1
1
2
211 
Now we are ready to find the 6 trig
functions of 135°
This is a Quadrant II angle. When
you label the sides if you include
any signs on them thinking of x & y
in that quadrant, it will keep the
signs straight on the trig functions.
x values are negative in quadrant II
so put a negative on the 1

5. -1
45°
 =135°1
2
1 2
sin135
22
o
h
   
Notice the -1 instead of 1 since the
terminal side of the angle is in quadrant II
where x values are negative.
1
1
10
135tan 


a
We are going to use this method to find
angles that are non acute, finding an acute
reference angle, making a triangle and
seeing which quadrant we are in to help
with the signs.
2
2
2
1
135cos




h
a

6. Let  denote a nonacute angle that lies in a quadrant.
The acute angle formed by the terminal side of  and
either the positive x-axis or the negative x-axis is called
the reference angle for .
Let's use this idea to find the 6 trig functions for 210°
First draw a picture and label  (We know
that 210° will be in Quadrant III)
Now drop a perpendicular line from the
terminal side of the angle to the x-axis
The reference angle will be the angle
formed by the terminal side of the angle
and the x-axis. Can you figure out it's
measure?
=210°
210°-180°=30°
The reference angle is the
amount past 180° of 
30°
Label the sides of the 30-60-90
triangle and include any negative
signs depending on if x or y values
are negative in the quadrant.
2
-1
3

7. 30°
210°
2
-1
3
You will never put a negative on the
hypotenuse. Sides of triangles are
not negative but we put the negative
sign there to get the signs correct on
the trig functions.
210csc
You should be thinking csc is the reciprocal of sin
and sin is opposite over hypotenuse so csc is
hypotenuse over opposite.
2
1
2


210tan
3
3
3
1



210cos
2
3

8. Using this same triangle idea, if we are given a point on the
terminal side of a triangle we can figure out the 6 trig
functions of the angle.
Given that the point (5, -12) is on the terminal side of an angle ,
find the exact value of each of the 6 trig functions.
First draw a picture
(5, -12)
Now drop a perpendicular
line from the terminal side
to the x-axis
Label the sides of the triangle
including any negatives. You
know the two legs because they
are the x and y values of the point
5
-12
Use the Pythagorean
theorem to find the
hypotenuse
    222
125 h
13h
13

9. Given that the point (5, -12) is on the terminal side of an angle ,
find the exact value of each of the 6 trig functions.
(5, -12)
5
-12
13






cottan
seccos
cscsin


We'll call the reference angle . The trig
functions of  are the same as  except
they possibly have a negative sign.
Labeling the sides of triangles with
negatives takes care of this problem.
12
13
o
h


13
5

h
a
12
5
o
a


12
13


o
h
5
13

a
h
12
5


o
a

10. The Signs of Trigonometric Functions
Since the radius is always positive (r > 0), the signs of the trig
functions are dependent upon the signs of x and y.
Therefore, we can determine the sign of the functions by
knowing the quadrant in which the terminal side of the angle
lies.

11. In quadrant I both the x
and y values are positive
so all trig functions will be
positive
+
+
All trig
functions
positive
In quadrant II x is negative
and y is positive.
_
+ 
We can see from this that any value that
requires the adjacent side will then have a
negative sign on it.
Let's look at the signs of sine,
cosine and tangent in the other
quadrants. Reciprocal functions will
have the same sign as the original
since "flipping" a fraction over
doesn't change its sign.
sin is +
cos is -
tan is -

12. _
_

In quadrant IV, x is positive
and y is negative .
_
+

So any functions using opposite
will be negative.
Hypotenuse is always positive so if
we have either adjacent or opposite
with hypotenuse we'll get a
negative. If we have both opposite
and adjacent the negatives will
cancel
sin is -
cos is +
tan is -
In quadrant III, x is
negative and y is negative.
sin is -
cos is -
tan is +

13. All trig
functions
positive
sin is +
cos is -
tan is -
sin is -
cos is +
tan is -
sin is -
cos is -
tan is +
To help
remember
these sign
we look at
what trig
functions
are
positive in
each
quadrant.
AS
T C
Here is a mnemonic
to help you
remember.
(start in Quad I and
go counterclockwise)
AllStudents
Take Calculus

14. To find the sine, cosine, tangent, etc. of angles whose terminal side
falls on one of the axes , we will use the
unit circle.
3
(..., , , 0, , , , 2 ,...)
2 2 2
  
   
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
0
2

3
2


Unit Circle:
 Center (0, 0)
 radius = 1
 x2 + y2 = 1
Trigonometric Functions Quadrantal Angle

15. What about quadrantal angles?
We can take a point on the terminal side of quadrantal
angles and use the x and y values as adjacent and
opposite respectively. We use the x or y value that is not
zero as the hypotenuse as well.
Try this with 90°
(0, 1)
We can take a point on the terminal side of quadrantal
angles and use the x and y values as adjacent and
opposite respectively. We use the x or y value that is not
zero as the hypotenuse as well (but never with a negative).
90sin 
h
o
1
1
1

90cos 
h
a
0
1
0

90tan 
a
o
0
1
dividing by 0 is
undefined so the
tangent of 90° is
undefined
90cosec 1
1
1

90sec undef
0
1
90cot 0
1
0


16. Let's find the trig functions of 
(-1, 0)
sin 
h
o
0
1
0

cos 
h
a
1
1
1


tan 
a
o
0
1
0

cosec undef
0
1
sec 1
1
1


cot undef
0
1
Remember
x is adjacent,
y is opposite
and
hypotenuse
here is 1

17. Coterminal angles are angles that have the same terminal side.
62°, 422° and -298° are all coterminal because graphed,
they'd all look the same and have the same terminal side.
62°
422°
-298°
Since the terminal
side is the same, all of
the trig functions
would be the same so
it's easiest to convert
to the smallest
positive coterminal
angle and compute
trig functions.

18. Reference Angles
for The reference angles in Quadrants II, III, and IV.
 ′ =  –  (radians)
 ′ = 180 –  (degrees)
 ′ =  –  (radians)
 ′ =  – 180 (degrees)
 ′ = 2 –  (radians)
 ′ = 360 –  (degrees)

19. Finding Reference Angles
angle ′.Find the reference
a.  = 300
b.  = 2.3
c.  = –135

20. Trigonometric Functions of Real Numbers

21. Trigonometric Functions of Any Angle
Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and
2 2
r x y 
sin csc
cos sec
tan cot
y r
r y
x r
r x
y x
x y
 
 
 
 
 
 
y
x

(x, y)
r

22. Evaluate each trigonometric function using Reference
Angle.
a. cos
b. tan(–210)
c. csc
Evaluating Trigonometric Functions

23. (a) – Solution
Because  = 4/3 lies
is
in Quadrant III, the reference angle
As shown in Figure (a).
Moreover, the cosine is negative
Quadrant III, so
in
Figure (a)

24. (b) – Solution
Because –210 + 360 = 150, it follows that –210 is
150.coterminal with the second-quadrant angle
is ′ = 180 – 150 = 30,So, the reference angle as shown
in Figure (b)
Figure (b)

25. cont’d
Finally, because the tangent is
have
negative in Quadrant II, you
tan(–210) =
=
(–) tan 30
.
(b) – Solution

26. (c) – Solution
Because (11/4) – 2 = 3/4, it follows that 11/4 is
coterminal with the second-quadrant angle 3/4.
–is ′ =  (3/4) /4,So, the reference angle = as shown
in Figure (c)
Figure (c)

27. cont’d
Because the cosecant is positive in Quadrant II, you have
(c) – Solution

28. Evaluating Trigonometric Functions
of .Let (–3, 4) be a point on the terminal
.
side Find the
sine, cosine, and tangent of
Solution:
x = –3, y = 4,

29. Solution

30. Try this:
13
-5
y
x

(-12, -5)
-12
sin
cos
tan
csc
sec
cot
y
r
x
r
y
x
r
y
r
x
x
y






 
 
 
 
 
 

31. Given and , find the values of the five
other trig function of .
Next Problems...
8
cos
17
   cot 0 
Given and , find the values of the five
other trig functions of .
3
cot
8
   2   

32. Find the value of the six trig functions for
For Quadrantal Angles
2

  
sin
2
cos
2
tan
2
1
csc
2
1
sec
2
cot
2
y
x
y
x
y
x
x
y






 
   
 
 
   
 
 
   
 
 
   
 
 
   
 
 
   
 
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
0
3
2


2





33. Find the value of the six trig functions for
For Quadrantal Angles
7 
 
 
 
 
 
 
sin 7
cos 7
tan 7
1
csc 7
1
sec 7
cot 7
y
x
y
x
y
x
x
y






 
 
 
 
 
 

34. Find one positive and one negative coterminal angle of
For Coterminal Angles
3
4

  

35. Using Coterminal Angle to Find the Exact Value of a
Trigonometric Funcion

36. Using reference angles and the special reference triangles, we can
find the exact values of the common angles.
To find the value of a trig function for any common angle 
1. Determine the quadrant in which the angle lies.
2. Determine the reference angle.
3. Use one of the special triangles to determine the function value
for the reference angle.
4. Depending upon the quadrant in which  lies, use the
appropriate sign (+ or –).
Trig Functions of Common Angles

37. Examples
Give the exact value of the trig function (without using a calculator).
1. 2.
5
sin
6
 3
cos
4
 
 
 

38. Trig Functions of “Uncommon” Angles
To find the value of the trig functions of angles that do NOT
reference 30°, 45°, or 60°, and are not quadrantal, we will use the
calculator. Round your answer to 4 decimal places, if necessary.
 Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree
 To find the trig functions of csc, sec, and cot, use the reciprocal
identities.

39. Examples
Evaluate the trig functions to four decimal places.
1. 2.sec( 2.5)  csc 23 38' 45"

40. Online Notes and Presentations
www.PinoyBIX.org
Visit: