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8-2 Trigonometric Functions
Unit 8 – Trigonometric and Circular Functions
Concepts and Objectives
� Definitions of Trigonometric and Circular Functions
(Obj. #27)
� Find the values of the six trigonometric functions of
angle θ.
� Find the function values of quadrantal angles.� 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
Trigonometric Ratio Review
� In Geometry, we learned that for any given right triangle,
there are special ratios between the sides.
=opposite
sinhypotenuse
A
A
op
po
site
adjacent
=sinhypotenuse
A
=adjacent
coshypotenuse
A
=opposite
tanadjacent
A
Trigonometric Functions
� Consider a circle centered at the origin with radius r:
� The equation for this circle is x2 + y2 = r2
� A point (x, y) on the circle creates a right triangle whose
sides are x, y, and r.
� The trig ratios are now� The trig ratios are now(x, y)
r
x
yθ=siny
rθ
=cosx
rθ
=tany
xθ
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:
= =1
cscr
θ(x, y)
r
x
yθ
= =1
cscsin
r
yθ
θ
= =1
seccos
r
xθ
θ
= =1
cottan
x
yθ
θ
Finding Function Values
� Example: The terminal side of an angle θ in standard
position passes through the point (15, 8). Find the
values of the six trigonometric functions of angle θ.
We know that x = 15 and y = 8, but
8
15
(15, 8)
θ
We know that x = 15 and y = 8, but
we still have to calculate r:
Now, we can calculate the values.
= +2 2r x y
= + =2 215 8 17 17
Finding Function Values
� Example: The terminal side of an angle θ in standard
position passes through the point (15, 8). Find the
values of the six trigonometric functions of angle θ.
= =8
sin17
y
rθ = =
17csc
8
r
yθ
8
15
(15, 8)
θ
17
= =sin17r
θ
= =15
cos17
x
rθ
= =8
tan15
y
xθ
= =csc8y
θ
= =17
sec15
r
xθ
= =15
cot8
x
yθ
The Unit Circle
� Angles in standard position whose terminal sides lie on
the x-axis or y-axis (90°, 180°, 270°, etc.) are called
quadrantal angles.
� To easily find function values of quandrantal angles, we use a circle with a radius of 1, which
� Notice what the different x, y,
and r values are at the quadrantal
angle points (x and y are either 0,
1, or –1; r is always 1).
use a circle with a radius of 1, which
is called a unit circle.
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.
−° = = −
1sin270 1
−° = = −
1sin270 1
1
° = =0
cos270 01
−° = =
1tan270 undefined
0
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 1° = = −−
1csc270 1
1
° = =1
sec270 undefined0
° = =−
0cot270 0
1
Identifying an Angle’s Quadrant
� To identify the quadrant of an angle given certain
conditions, note the following:
� In the first quadrant, x and y are both positive.
� In QII, x is negative and y is positive.
In QIII, both are negative.� In QIII, both are negative.
� In QIV, x is positive and y is
IVIII
II I
(+,+)(+,–)
(–,–)
negative.
(+,–)
Identifying an Angle’s Quadrant
� Example: Identify the quadrant (or possible quadrants)
of an angle θ that satisfies the given conditions.
a) sin θ > 0, tan θ < 0 b) cos θ < 0, sec θ < 0
I, II II, IV
II
II, III II, III
II, III
Homework
� College Algebra (brown book)
� Page 512: 30-102 (×3s, omit 63)
� Turn in: 42, 54, 72, 78, 96, 102
Classwork: Algebra & Trigonometry (green book)� Classwork: Algebra & Trigonometry (green book)
� Page 728: 77-80