Download Chapter 8 Trigonometric Equations and Applications Inclination

By
Kevin McGowan, Frank Vasquez, and
John Giordano
Inclination-angle x where 0° ≤ x < 180° that is measured from
the positive x-axis to the line
• Theorem:
For any line with slope m and inclination x,
m = tan x if x ≠ 90°
• If x = 900, then the line has no slope. (The line is vertical.)
Period and Amplitude of Sine and Cosine Curves:
• For functions y = A sin Bx and y = A cos Bx (A ≠ 0 and B > 0)
amplitude = ∣A∣ period =
General Sine Waves
• If the graphs of y = A sin Bx and y = A cos Bx are translated
horizontally h units and vertically k units, then the resulting
graphs have the equations:
y – k = A sin B(x – h) and y – k = A cos B(x – h)
• Reciprocal Relationships
csc Ѳ=
sec Ѳ =
cot Ѳ =
• Relationships with negatives:
sin (- Ѳ) = -sin Ѳ
csc (- Ѳ) = -csc Ѳ
tan (- Ѳ) = -tan Ѳ
and
and
and
• Pythagorean Relationships:
sin2 Ѳ + cos2 Ѳ = 1
1 + tan2 Ѳ = sec2 Ѳ
1 + cot2 Ѳ = csc2 Ѳ
cos (- Ѳ) = cos Ѳ
sec (- Ѳ) = sec Ѳ
cot (- Ѳ) = -cot Ѳ
• Cofunction relationships:
sin Ѳ= cos (90° – Ѳ) and cos Ѳ= sin (90° – Ѳ)
tan Ѳ= cot (90° – Ѳ) and cot Ѳ = tan (90° – Ѳ)
sec Ѳ= csc (90° –Ѳ ) and csc Ѳ= sec (90° – Ѳ)
• Each of the trigonometric relationships is true
for all values of the variable for which each
side of the equation is defined. Such
relationships are called trigonometric
identities.
35)
4  (tanx  cotx) 2  sec 2 x  csc 2 x
4  tan 2  2tanxcotx  cot 2 x  sec 2 x  csc 2 x
2  tan x  cot x  sec x  csc x
2
2
2
2
1  tan x   1  cot x   sec x  csc x
2
2
2
sec 2 x  csc 2 x  sec 2 x  csc 2 x
2
sinx cosx

1
cosx sinx
36)
2 x  3 y  17
3 y  17  2 x
2 x 17
y

3
3
m  tan 
2
tan  
3
1
1 2
tan tan α  tan
3
  33.7
37)
tanxsinx  sinx  tanx  1  0
sinx
sinx
 sinx  sinx 
1  0
cosx
cosx
2
sin
sinx
cosx(
 sinx 
 1  0)
cosx
cosx
2
sin  sinxcosx  sinx  cosx  0
sin x  sinx  sinxcosx  cosx  0
2
sinx  sinx  1  cosx  sinx  1  0
 sinx  cosx  sinx  1  0
37 cont.
sinx  cosx  0
sinx  cosx
sinx
1
cosx
tanx  1
x
 5
,
4 4
sinx  1  0
sinx  1
x

2
FALSE
UNDEFINED
38) Find the equation of this sine wave where 2
cycles have been drawn
1
5
4
3
2
1
1
2
n( x )
3
4
5
6
7
8
9
7 10
0
x
4
f  x   4sin x  3
41)

y  3sin(4( x  ))  8
2
a. Plot 2 complete waves
3
b. What is the amplitude?
c. What is the wavelength?
2 2 


B
4
2
41 cont.
d. What are the maximum and minimum points?
Maximum:

  5
  9
  13

, 5  , 
, 5 
 , 5  ,  , 5  , 
8
 8
 8
  8

Minimum:
 3
  7
  11
  15

, 11 , 
, 11
 , 11 ,  , 11 , 
 8
 8
 8
  8

3

e. How often does the minimum value occur? Every
starting at
.
8
2