Download ATHS FC –Math Department -Al Ain (2013

ATHS FC –Math Department -Al Ain (2013-2014)
Homework Sheet (Term I)
Grade 11 Core
Name
Date
Section
Lessons
12.7 Graphing Trigonometric Functions
Find the amplitude, if it exists, and period of each function. Then graph the
function.
1.
2.
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3. y = cos 5θ
4.
5. WEATHER The function y = 60 + 25 sin
, where t is in months and t = 0
corresponds to April 15, models the average high temperature in degrees
Fahrenheit in Centerville.
a. Determine the period of this function. What does this period represent?
b. What is the maximum high temperature and when does this occur?
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12.8 Translations of Trigonometric Graphs
State the amplitude, period, phase shift, and vertical shift for each function. Then
graph the function.
1.
2. y = 2 cos (θ + 30º) + 3
3. y = 3 sin (2θ + 60º) − 2.5
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4.
5. y = 3 cos 2 (θ + 45º) + 1
6. ECOLOGY The population of an insect species in a stand of trees follows the
growth cycle of a particular tree species. The insect population can be modeled
by the function y = 40 + 30 sin 6t, where t is the number of years since the stand
was first cut in November, 1920.
a. How often does the insect population reach its maximum level?
b. When did the population last reach its maximum?
c. What condition in the stand do you think corresponds with a minimum
insect population?
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12.9 Inverse Trigonometric Functions
Find each value. Write angle measures in degrees and radians.
1. Arcsin 1
2.
3.
4.
Find each value. Round to the nearest hundredth if necessary.
5.
6.
7. cos [Arctan (-1)]
8.
Solve each equation. Round to the nearest tenth if necessary.
9.Tan θ = 10
10. Sin θ = 0.7
11.Sin θ = -0.5
12.Cos θ = 0.05
13. PULLEYS The equation cos θ = 0.95 describes the angle through which pulley A
moves, and cos θ = 0.17 describes the angle through which pulley B moves.
Which pulley moves through a greater angle?
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13.1 Trigonometric Identities
Find the exact value of each expression if 0° < θ < 90°.
1. If cos
, find sin θ.
2.If tan θ = 4, find sec θ.
Find the exact value of each expression if 180° < θ < 270°.
3.If sin
, find sec θ.
4.If csc
, find cot θ.
Find the exact value of each expression if 270° < θ < 360°.
5.If cos
, find cot θ.
6.If csc θ = -8, find sec θ.
Simplify each expression.
7. csc θ tan θ
8.
9.sin2 θ cot2 θ
10.cot2 θ + 1
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11.
12.
13.sin θ + cos θ cot θ
14.
15.sec2 θ cos2 θ − tan2 θ
13.2 Verifying Trigonometric Identities
Verify that each equation is an identity.
1.
2.
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3. (1 + sin θ)(1 − sin θ) = cos2 θ
4. tan4 θ + 2 tan2 θ + 1 = sec4 θ
5. cos2 θ cot2 θ = cot2 θ − cos2 θ
6. (sin2 θ)(csc2 θ + sec2 θ) = sec2 θ
7. PROJECTILES The square of the initial velocity of an object launched from the
ground is
, where θ is the angle between the ground and the initial
path h is the maximum height reached, and g is the acceleration due to gravity.
Verify the identity
.
8. LIGHT The intensity of a light source measured in candles is given by I = ER2
sec θ, where E is the illuminance in foot candles on a surface, R is the distance
in feet from the light source, and θ is the angle between the light beam and a line
perpendicular to the surface. Verify the identity ER2(1 + tan2 θ) cos θ = ER2 sec
θ.
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13.3 Sum and Difference of Angles Identities
Find the exact value of each expression.
1. cos 75°
2.sin (-165°)
3.sin 150°
4.cos 240°
Verify that each equation is an identity.
5.cos (180° − θ) = -cos θ
6. sin (360° + θ) = sin θ
7. sin (45° + θ) − sin (45° − θ) =
sin θ
8.
9. ELECTRICITY In a certain circuit carrying alternating current, the formula c = 2
sin (120t) can be used to find the current c in amperes after t seconds.
a. Rewrite the formula using the sum of two angles.
b. Use the sum of angles formula to find the exact current at t = 1 second.
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13.4 Double-Angle and Half-Angle Identities
Find the exact values of sin 2θ, cos 2θ, sin
1. cos
, 0° < θ < 90°
2. sin
, 90° < θ < 180°
, and cos
for each of the following.
Find the exact value of each expression.
3. tan 105°
4.cos 67.5°
Verify that each equation is an identity.
5.sin
6.sin 4θ = 4 cos 2θ sin θ cos θ
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7.AERIAL PHOTOGRAPHY In aerial photography, there is a reduction in film exposure
for any point X not directly below the camera. The reduction Eθ is given by Eθ = E0 cos4
θ, where θ is the angle between the perpendicular line from the camera to the ground
and the line from the camera to point X, and E0 is the exposure for the point directly
below the camera. Using the identity 2 sin2 θ = 1 − cos 2θ, verify that E0 cos4
.
8.IMAGING A scanner takes thermal images from altitudes of 300 to 12,000 meters. The
width W of the swath covered by the image is given by W = 2H′ tan θ, where H′ is the
height and θ is half the scanner's field of view. Verify that
tan θ.
13.5 Solving Trigonometric Equations
Solve each equation for the given interval.
1. sin 2θ = cos θ, 90° ≤ θ < 180°
2.
3.cos 4θ = cos 2θ, 180° ≤ θ < 360°
4.cos θ + cos (90 − θ) = 0, 0 ≤ θ < 2π
5.2 + cos θ = 2 sin2 θ, π ≤ θ ≤
6.tan2 θ + sec θ = 1,
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cos θ = sin 2θ , 0° ≤ θ , 360°
≤θ<π
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Solve each equation for all values of θ if θ is measured in radians.
7. cos2 θ = sin2 θ
9.
sin3 θ = sin2 θ
11.2cos 2θ = 1 − 2 sin2 θ
8.cot θ = cot3 θ
10.cos2 θ sin θ = sin θ
12.sec2 θ = 2
Solve each equation for all values of θ if θ is measured in degrees.
13. sin2 θ cos θ = cos θ
14.csc2 θ − 3 csc θ + 2 = 0
15.
16.
cos2 θ = cos2 θ
Solve each equation.
17. 4 sin2 θ = 3
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18.4 sin2 θ − 1 = 0
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19.2
sin2 θ − 3 sin θ = -1
20.cos 2θ + sin θ − 1 = 0
21.WAVES Waves are causing a buoy to float in a regular pattern in the water. The
vertical position of the buoy can be described by the equation h = 2 sin x. Write an
expression that describes the position of the buoy when its height is at its midline.
22.ELECTRICITY The electric current in a certain circuit with an alternating current
can be described by the formula i = 3 sin 240t, where i is the current in amperes and
t is the time in seconds. Write an expression that describes the times at which there
is no current.
5.1 Operations with Polynomials
1.
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2.
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3.
4.
5.-(4w-3z-5)(8w)2
6.(m4n6)4(m3n2p5)6
7.
8.
9.
10.
11.(3n2 + 1) + (8n2 − 8)
12.(6w − 11w2) − (4 + 7w2)
13.(w + 2t)(w2 − 2wt + 4t2)
14.(x + y)(x2 − 3xy + 2y2)
15.GEOMETRY The area of the base of a rectangular box measures 2x2 + 4x − 3 square
units. The height of the box measures x units. Find a polynomial expression for the
volume of the box.
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5.2 Dividing Polynomials
Simplify.
1.
2. (-30x3y + 12x2y2 − 18x2y) ÷ (-6x2y)
3.
4.(a3 − 64) ÷ (a − 4)
5.(3w3 + 7w2 − 4w + 3) ÷ (w + 3)
6.(x4 − 3x3 − 11x2 + 3x + 10) ÷ (x − 5)
7.(2r3 + 5r2 − 2r − 15) ÷ (2r − 3)
8.
9.GEOMETRY The area of a rectangle is 2x2 − 11x + 15 square feet. The length of
the rectangle is 2x − 5 feet. What is the width of the rectangle?
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