Transcript PPT Chapter 7 Review-1x

Chapter 7 Review
Solve for 0° ≤ θ ≤ 90°
1.) If tan θ = 2, find cot θ
½
2.) if sin θ = ⅔, find cos θ
5
3
3.) If cos θ = ¼, find tan θ
15
4.) If tan θ = 3, find sec θ
10
5.) if sin θ = 7/10, find cot θ
51
7
6.) If tan θ = 7/2, find sin θ
7 53
53
Express each value as a function of an angle
in Quadrant I
1.) sin 458°
sin 82°
3.) tan (-876°)
tan 24°
2.) cos 892°
-cos 8°
4.) csc 495°
csc 45°
Simplify
1.)
2.)
cot A
tan A
sin 2 b cot b
cos b
sin b
cot 2 A
3.) sin2 q cos2 q - cos2 q
-cos4 q
4.)
cos x + sin xtan x
sec x
Find a numerical value of one trig function.
1.) sin x = 3 cos x
tan x = 3
2.) cos x = cot x
csc x = 1
Or
sin x = 1
Use the sum and difference identities to find the exact value of
each function:
1.) cos 75°
2.) cos 375°
3.) sin (-165°)
6+ 2
4
6- 2
4
4.) sin (-105°)
2- 6
4
5.) sin 95° cos 55° + cos 95° sin 55°
- 6- 2
4
6.) tan (135° + 120°)
1
2
2+ 3
7.) tan 345°
-2 + 3
If α and β are the measures of two first quadrant angles,
find the exact value of each function.
1.) if sin α = 12/13 and cos β = 3/5,
find cos (α – β)
= cos a cos b + sin a sin b
æ 5 öæ 3 ö æ 12 öæ 4 ö
= ç ÷ç ÷ + ç ÷ç ÷
è13 øè 5 ø è 13 øè 5 ø
15 48 63
=
+
=
65 65 65
=
2.) if cos α = 12/13 and cos β = 12/37,
find tan (α – β)
tan a - tan b
1+ tan a tan b
5
35
(
)
(
12
12)
=
1+ ( 512)( 3512)
=
-30
12 = -30 · 12
-30
=
12
187
187
187
12
If α and β are the measures of two first quadrant angles,
find the exact value of each function.
3.) if cos α = 8/17 and tan β = 5/12, find cos (α + β)
= cos a cos b - sin a sin b
æ 8 öæ 12 ö æ 15 öæ 5 ö
= ç ÷ç ÷ - ç ÷ç ÷
è17 øè 13 ø è 17 øè13 ø
96
75
21
=
=
221 221 221
4.) if csc α = 13/12 and sec β = 5/3, find sin (α – β)
If sin A = 12/13, and A is in the first quadrant, find each
value.
-119
169
2.) sin 2A
3.) tan 2A
-120
119
4.) cos A/2
5.) sin A/2
2 13
13
6.) tan A/2
1.) cos 2A
120
169
3 13
13
2
3
Use a half-angle identity to find the value of each
1.) tan
p
2.)
8
5p
cos
8
2- 2
2
2 -1
3.)
19p
sin
12
2+ 3
2
4.)
cos67.5°
2- 2
2
Solve for 0° ≤ x ≤ 180°
1.) 2sin2 x - 5sin x + 2 = 0
2sin 2 x - 4 sin x -1sin x + 2 = 0
2sin x(sin x - 2) -1(sin x - 2) = 0
(sin x - 2)(2sin x -1) = 0
sin x = 2
1
sin x =
30°, 150°
2
3.) 3cos2x - 5cos x =1
120°
2.) sin2 x - 2sin x - 3 = 0
(sin x - 3)(sin x +1) = 0
sin x = 3
sin x = -1
No solution, 270° is not in our domain
4.) 2tan xcos x + 2cos x = tan x +1
60°
Solve for 0° ≤ x ≤ 180°
1.) 2sin2 x -1 = 0
2.)
45°, 135°
3.) tan x = sin x
0°, 180°
cos x = 3cos x - 2
0°
4.)
cos xsin2x = 0
0°, 90°, 180°
Solve for 0° ≤ x ≤ 180°
1.) sec x =1+ tan x
0°
3.) sin2x = 2cos x
90°
2.) 4sin2 x - 4sin x +1 = 0
30°, 150°
4.) tan2 x + tan x = 0
0°, 135°, 180°
Solve for 0° ≤ x ≤ 180°
1.)
2sin2x =1
15°, 75°
2.)
cos2x + sin x =1
0°, 30°, 150°, 180°
Write each equation in normal form. Then find the measure of
the normal, p, and ϕ, the angle that the normal makes with the
positive x-axis.
1.) 3x – 2y – 1 = 0
3x
2y
1
=0
13
13
13
2.) 5x + y – 12 = 0
x
y
5
+
=0
2
2
2
f = 11°
f = 45°
12
p=
» 2.35units
26
5
p=
» 3.54units
2
Write each equation in normal form. Then find the measure of
the normal, p, and ϕ, the angle that the normal makes with the
positive x-axis.
3.) y = x + 5
x
y
5
+
=0
2
2
2
4.) y = x - 2
x
y
2
=0
2
2
2
f = 135°
f = 315°
5
p=
» 3.54units
2
2
p=
» 1.41units
2
Write each equation in normal form. Then find the measure of
the normal, p, and ϕ, the angle that the normal makes with the
positive x-axis.
5.) x + y – 5 = 0
x
y
5
+
=0
2
2
2
6.) 2x + y – 1 = 0
x
y
1
=0
5
5
5
f = 45°
f = 27°
5
p=
» 3.54units
2
1
p=
» 0.45units
5
Write the standard form of the equation of the
each line given “p”, and ϕ.
1.) p = 4, ϕ = 30°
3x + 3y + 8 3 = 0
2.) p = 2, ϕ = 45°
x+ y -2 2 = 0
Write the standard form of the equation of the
each line given “p”, and ϕ.
3.) p = 3, ϕ = 60°
x + 3y - 6 = 0
4.) p = 12, ϕ = 120°
x - 3y + 24 = 0
Write the standard form of the equation of the
each line given “p”, and ϕ.
5.) p = 8, ϕ = 150°
3x - 3y +16 3 = 0
2.) p = 15, ϕ = 225°
x + y +15 2 = 0
Find the distance between the point with the given
coordinates and the line with the given equation.
1.) (-1, 5), 3x – 4y – 1 = 0
d = 4.8
3.) (1, -4), 12x + 5y – 3 = 0
11
d = » 0.85
13
2.) (2, 5), 5x – 12y + 1 = 0
49
d=
» 3.77
13
4.) (-1,-3), 6x + 8y – 3 = 0
d = 3.3
Find the distance between each equation.
1.) 2x – 3y + 4 = 0
y = ⅔x + 5
(0, 4/3)
11 13
d=
» 3.05
13
3.) x + 3y – 4 = 0
x + 3y + 20 = 0
(0, 4/3)
12 10
d=
» 7.59
5
2.) 4x – y + 1 = 0
4x – y – 8 = 0
(0, 1)
9
d = » 0.53
17
4.) 3x – 2y = 6
3x – 2y + 30 = 0
(0, -3)
36 13
d=
» 9.98
13
Find an equation of the line that bisects the acute
angle formed by the graphs of the equations
x + 2y - 3 = 0 and x – y + 4 = 0
( 2 - 5)x + (2 2 + 5)y - (3 2 + 4 5) = 0
Find an equation of the line that bisects the acute
angle formed by the graphs of the equations
x + y – 5 = 0 and 2x – y + 7 = 0
( 5 - 2 2)x + ( 5 - 2)y - (5 5 - 7 2) = 0
Find an equation of the line that bisects the acute
angle formed by the graphs of the equations
2x + y – 3 = 0 and x – y + 5 = 0
(2 2 - 5)x + ( 2 + 5)y - (3 2 + 5 5) = 0