Transcript Slide 1
1.3 Definition II: Right Triangle Trigonometry
If is an acute angle of a right triangle, then
opp
adj
opp
sin
, cos
, tan
,
hyp
hyp
adj
hyp
hyp
adj
csc
, sec
, cot
.
opp
adj
opp
Trigonometric Functions
• Let (x, y) be a point other the origin on the terminal side of an angle
in standard position. The distance from the point to the origin is
The six trigonometric functions of are defined as follows.
r x2 y 2 .
y
sin
r
r
csc ( y 0)
y
x
cos
r
y
tan (x 0)
x
r
sec ( x 0)
x
x
cot
(y 0)
y
Example: Finding Function
Values
• The terminal side of angle in standard
position passes through the point (12, 16).
Find the values of the six trigonometric
functions of angle .
r x 2 y 2 122 162
(12, 16)
16
144 256 400 20
12
Example: Finding Function
Values continued
• x = 12
y 16 4
sin
r 20 5
x 12 3
cos
r 20 5
y 16 4
tan
x 12 3
y = 16
r = 20
r 20 5
csc
y 16 4
r 20 5
sec
x 12 3
x 12 3
cot
y 16 4
Example: Finding Function
Values
• Find the six
trigonometric function
values of the angle
in standard position, if
the terminal side of
is defined by
x + 2y = 0, x 0.
• We can use any point
on the terminal side of
to find the
trigonometric function
values.
Example: Finding Function
Values continued
• Choose x = 2
x 2y 0
2 2y 0
2 y 2
y 1
• The point (2, 1) lies
on the terminal side,
22 (1) 2 5.
andr the
corresponding value
of r is
• Use the definitions:
y 1 1 5
5
sin
r
5
5
5 5
x
2
2
5 2 5
r
5
5
5 5
y
1
r
tan
csc 5
x
2
y
cos
sec
r
5
x
2
cot
x
2
y
Example: Function Values
Quadrantal Angles
• Find the values of the six trigonometric functions for
an angle of 270.
• First, we select any point on the terminal side of a
270 angle. We choose (0, 1). Here x = 0, y = 1
and r = 1. 1
0
sin 270
1
1
1
tan 270
undefined
0
1
sec 270 undefined
0
cos 270
0
1
1
csc 270
1
1
0
cot 270 0
1
Commonly Used Function
Values
sin cos
tan
cot
sec
1
0
0
1
0
undefined
90
1
0
undefine
d
0
180
0
1
0
undefined
270
1
0
0
360
0
1
undefine
d
0
undefined
csc
undefine
d
undefined
1
1
undefine
d
undefined
1
1
undefine
d
Reciprocal Identities
•
1
sin
csc
1
cos
sec
1
tan
cot
1
csc
sin
1
sec
cos
1
cot
tan
Example: Find each function
value.
2
• cos if sec =
3
• Since cos is the
reciprocal of sec
1
1 3
cos
2
sec 3 2
15
• sin if csc
3
1
3
sin
15
15
3
3 • 15
3 • 15
15
15 • 15
15
5
Example: Identify Quadrant
• Identify the quadrant (or quadrants) of any
angle that satisfies tan > 0, cot > 0.
• tan > 0 in quadrants I and III
• cot > 0 in quadrants I and III
• so, the answer is quadrants I and III
Ranges of Trigonometric
Functions
• For any angle for which the indicated
functions exist:
• 1. 1 sin 1 and 1 cos 1;
• 2. tan and cot can equal any real
number;
• 3. sec 1 or sec 1 and
csc 1 or csc 1.
(Notice that sec and csc are never
between 1 and 1.)
Identities
• Pythagorean
sin 2 cos 2 1,
tan 2 1 sec 2 ,
1 cot 2 csc 2
• Quotient
sin
tan
cos
cos
cot
sin
Example: Other Function Values
• Find sin and cos if tan = 4/3 and is in
quadrant III.
• Since is in quadrant III, sin and cos will both
be negative.
• sin and cos must be in the interval [1, 1].
Example: Other Function Values
continued
• We use the identity
tan 2 1 sec 2
2
4
2
1
s
ec
3
16
1 sec 2
9
25
sec 2
9
5
sec
3
3
cos
5
tan 2 1 sec 2
Since sin 2 1 cos 2 ,
3
sin 2 1
5
9
sin 2 1
25
16
sin 2
25
4
sin
5
2
Example: If
and is in quadrant II,
find each function value.
5
tan
3
tan 2 1 sec 2
• a) sec
2
Look for an identity that
relates tangent and
secant.
tan 1 sec
2
2
5
2
1
sec
3
25
1 sec 2
9
34
sec 2
9
sec
34
9
34
sec
3
Example: If
and is in quadrant II,
find each function value continued
5
tan
3
• b) sin
sin
tan
cos
cos tan sin
1
sec
tan sin
3 34 5
sin
34 3
5 34
si n
34
• c) cot ()
1
cot( )
tan( )
1
cot( )
tan
1
3
cot( )
53 5
Example: Express One Function
in Terms of Another
1 cot 2 x csc 2 x
1
1
2
1 cot x csc 2 x
1
2
sin
x
2
1 cot x
• Express cot x in
terms of sin x.
1
2
sin
x
2
1 cot x
1
sin x
1 cot 2 x
1 cot 2 x
sin x
1 cot 2 x
Example: Rewriting an Expression in Terms
of Sine and Cosine
• Rewrite cot tan in terms of sin and cos .
•
cos sin
cot tan
sin cos
cos 2
sin 2
sin cos sin cos
cos 2 sin 2
sin cos
(cos sin )(cos sin )
sin cos
Example: Working with One Side
• Prove the identity
(tan 2 x 1)(cos2 x 1) tan 2 x
• Solution: Start with the left side.
(tan 2 x 1)(cos 2 x 1) tan 2 x
sin 2 x
2
2
1
(cos
x
1)
tan
x
cos 2 x
sin 2 x
2
2
sin x
cos
x
1
tan
x
2
cos x
2
sin
x
2
2
2
sin x cos x
1
tan
x
2
cos x
2
sin 2 x
2
1
1
tan
x
2
cos x
sin 2 x
2
tan
x
2
cos x
tan 2 x tan 2 x
Example: Working with One Side
• Prove the identity
1
csc x sin x
sec x tan x
• Solution—start with the
right side
1
csc x sin x
sec x tan x
1
sin x
sin x
1
sin 2 x
sin x sin x
• continued
1
1 sin 2 x
sec x tan x
sin x
cos 2 x
sin x
cos x cos x
sin x 1
cot x cos x
1
1
tan x sec x
1
1
sec x tan x sec x tan x