Name: Date:
Chapter 5: Combine Sinusoids Team Test
Part 1: No Calculators Allowed
1. If possible, rewrite each of the following expressions in an equivalent form.
a. sin3xcos2x + sin3xcos2x b. sin2(4x) + cos2(4x)
None 1
c. 2cos2(25º) – 1 d. sin2(5p) – cos2(5p)
cos (50) -cos(10p)
2. If f(x) = 2x + 8, find f -1(x). Then determine if your answer is correct using 2 of the 3 methods: graphically, algebraically, or numerically.
f-1x=x-82=12x-4
Graphically: reflections over y =x
Numerically: Show two table of values where the x and y values are switched
Algebraically: Show that f(g(x) = x and g(f(x) = x.
3. Write exact answers for the following:
a. = -√22 b. = -√32
c. = -3 d. sin= sinπ3=√32
4. Carefully sketch the principal branch of y = tan x that creates the principal branch of y = tan-1x. Then draw the principal branch of y = tan-1x. Write the domain and range for both functions.
D: -π2≤x≤π2 R: -1≤y≤1 D: -1≤x≤1R: -π2≤y≤π2
6. Prove the identities below.
cosx+1cosx-1+sin2x=0
cos2x-1+sin2x=0
1-1=0
0 = 0
1) sinxsinxsinx1+cosx+1+cosxsinx1+cosx1+cosx=2cscx 2) sin2xsinx(1+cosx)+1+2cosx+cos2xsinx(1+cosx)=2cscx
3) sin2x +cos2x+1+2cosxsinx(1+cosx)=2cscx 4) sin2x +cos2x+1+2cosxsinx(1+cosx)=2cscx
5) 1+1+2cosxsinx(1+cosx)=2cscx 6) 2+2cosxsinx(1+cosx)=2cscx
7) 2(1+cosx)sinx(1+cosx)=2cscx 8) 2sinx=2cscx
9) 2cscx=2cscx
cosa+bcosa-b=cos2b-sin2a
(cosacosb-sinasinb)(cosacosb+sinasinb)=cos2b-sin2a
cos2acos2b-sin2asin2b=cos2b-sin2a
(1-sin2a)cos2b-sin2a(1-cos2b)=cos2b-sin2a
cos2b-sin2acos2b-sin2a-sin2acos2b=cos2b-sin2a
cos2b-sin2a=cos2b-sin2a
Type equation here.
Chapter 5: Combined Sinusoids Team Test
Part 2: Calculators Allowed
7. Draw an appropriate sketch and use it to write each of the following equations as a cosine with a phase displacement. Show some work.
a. y = 3 cos q + 5 sin q b. y = 5 cos q - 12 sin q
A=(32+52)≈5.83 tan-1(53)≈59.04° A=(52+(-12)2)≈13 tan-1(-125)≈-67.38°
y=5.83cos(∅-59.04°) 4th quadrant so 360º - 67.38º = 292.62º
y=13cos(∅-292.62°)
c. Describe how to verify that your new equations are equivalent to the original ones. Then use your calculator to do so!
Graph the original and your rewrite. If they overlap, they are equivalent.
8. Use the composite argument property to write y = 4 cos ( - 35) as a linear combination of
cos and sin .
y=4(cos∅cos35°+sin∅sin35°)
y=4cos35°cos∅+4sin35°sin∅
y≈3.28cos∅+2.29sin∅
9. If sinq = ¼ and 90° < q < 180°, find the exact values of sin2q and cos½q. Show all your work!
This is bonus…don’t worry about it.
10. A pendulum hangs from a ceiling and swings back and forth toward a wall. Harry starts timing and at t=4 seconds the pendulum is closest to the wall, 25 cm away. Three seconds later the pendulum is farthest from the wall (83 cm).
a) Find an equation for the distance the pendulum is from the wall at any time t.
One example: y=54+29cos2π6x-7
b) Find out how far the pendulum is away from the wall at t = 8 seconds.
y=54+29cos2π68-7 y≈68.5 cm
c) Find the first time when the pendulum is 33 cm away from the wall.
33=54+29cos2π6x-7 x=9.27+6n
The first time will be 3.27 cm.
11. Chris was solving the equation cos2x = 2 + cosx. Here is what he wrote:
c. Solve the equation correctly using the appropriate cos2x expansion.
cos2x=2+cosx
2cos2x-1=2+cosx
2cos2x-cosx-3=0
2m2-m-3=0
2m-3m+1=0
m=1.5 m=-1
cosx=1.5 cosx=-1 (1.5 is out of the range of cos)
x=arccos-1=cos-1-1+2πn=±π+2πn
12. Solve using algebra: for q Î [0°, 360°]
tan2∅+3∅=√33
tan5∅=√33
5∅=arctan(33)
5∅=tan-1(33)+180°n
∅=15tan-1(33)+180°n
∅=15tan-1(33)+180°n
∅=6°+36°n=6°, 42°, 78°, 114°, 150°, 186°, 222°, 258°, 294°,330°