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Obj. 16 Trigonometric Functions
Unit 5 Trigonometric and Circular Functions
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Concepts and Objectives
Definitions of Trigonometric and Circular Functions
(Obj. #16) Find the values of the six trigonometric functions of
angle .
Find the function values of quadrantal angles.
Identify the quadrant of a given angle.
Find the other function values given one value and
the quadrant
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Trigonometric Ratio Review
In Geometry, we learned that for any given right triangle,
there are special ratios between the sides.
A
opposite
adjacent
=opposite
sin
hypotenuse
A
=adjacent
coshypotenuse
A
=opposite
tanadjacent
A
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Trigonometric Functions
Consider a circle centered at the origin with radius r:
The equation for this circle isx2 +y2 = r2
A point(x,y) on the circle creates a right triangle whose
sides arex,y, and r.
The trig ratios are now (x,y)r
x
y
=siny
r
=cos
x
r
=tany
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Trigonometric Functions
There are three other ratios in addition to the three we
already know : cosecant, secant, and cotangent. These ratios are the inverses of the original three:
(x,y)r
x
y
= =1
csc sin
r
y
= =1
seccos
r
x
= =
1cot tan
x
y
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Finding Function Values
Example: The terminal side of an angle in standard
position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .
8
15
(15, 8)
We know thatx= 15 andy= 8, but
we still have to calculate r:
Now, we can calculate the values.
= +2 2
r x y
= + =2 2
15 8 1717
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Finding Function Values
Example: The terminal side of an angle in standard
position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .
8
15
(15, 8)
17
= =8
sin
17
y
r
= =15
cos17
x
r
= =
8tan 15
y
x
= =17
csc
8
r
y
= =17
sec15
r
x
= =
15cot 8
x
y
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The Unit Circle
Angles in standard position whose terminal sides lie on
thex-axis ory-axis (90, 180, 270, etc.) are calledquadrantal angles.
To find function values of quandrantal angles easily, we
Notice that at the quadrantal
angle pointsxandyare either
0, 1, or 1 (ris always 1).
use a circle with a radius of 1, which
is called a unit circle.
90
(0, 1)
(0, 1)
270
180
(1, 0)0/360
(1, 0)
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Values of Quadrantal Angles
Example: Find the values of the six trigonometric
functions for an angle of 270.At 270,x= 0,y= 1, r= 1.
= =
1
sin270 11
= =0
cos270 01
= =
1tan270 undefined
0(0, 1)
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Values of Quadrantal Angles
Example: Find the values of the six trigonometric
functions for an angle of 270.At 270,x= 0,y= 1, r= 1.
= =
1
csc270 11
= =1
sec270 undefined0
= =
0cot 270 0
1
(0, 1)
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Identifying an Angles Quadrant
To identify the quadrant of an angle given certain
conditions, note the following: In the first quadrant,xandyare both positive.
In QII,xis negative andyis positive.
In QIII, both are negative. In QIV,xis positive andyis
IVIII
II I
(+,+)(,+)
(,)
negative.
(+,)
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Identifying an Angles Quadrant
Example: Identify the quadrant (or possible quadrants)
of an angle that satisfies the given conditions.
a) sin > 0, tan < 0 b) cos < 0, sec < 0
I, II II, IV
II
II, III II, III
II, III
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Homework
College Algebra
Page 512: 30-78 (6), 93-102 (3)
Classwork: Algebra & Trigonometry(green book)
Page 728: 77-78