algebra final

Topics: Trigonometry, Euler's formula, Trigonometric functions Pages: 8 (555 words) Published: May 30, 2014
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Algebra 2 Final Exam

Multiple Choice
Identify the choice that best completes the statement or answers the question.  
 
 
Simplify the trigonometric expression.
 
 1. 

a.

b.

c.

d.

Answer B

 
 
In , is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.  
 2. 
a = 3, c = 19
a.
= 9.1°, = 80.9°, b = 18.8
c.
= 14.5°, = 75.5°, b = 18.8
b.
= 80.9°, = 9.1°, b = 18.8
d.
= 75.5°, = 14.5°, b = 18.8

Answer A

 
 
3. 
What is the simplified form of sin(x + p)?
a.
cos x
b.
sin x
c.
–sin x
d.
–cos x

 Answer C
 
Rewrite the expression as a trigonometric function of a single angle measure.  
 4. 

a.

b.

c.

d.

 Answer A

Short Answer
 
 5. 
Consider the sequence 1, , , , ,...
a.
Describe the pattern formed in the sequence.
b.
Find the next three terms.

 
 6. 
Consider the sequence 16, –8, 4, –2, 1, ...
a.
Describe the pattern formed in the sequence.
b.
Find the next three terms.

 

7. 
Consider the graph of the cosine function shown below.

a. Find the period and amplitude of the cosine function.
b. At what values of for do the maximum value(s), minimum values(s), and zeros occur?

 
 
Verify the identity. Justify each step.
 
 8. 

sinΘ/cosΘ+cosΘ/sinΘ
sin^20+cos^2Θ/sinΘcosΘ
1/sinΘcosΘ
 
 9. 
Verify the identity .

cot ( Θ - π / 2 )= cos ( Θ - π / 2 ) / sin ( Θ - π / 2 )

cos ( Θ - π / 2 )= cos(Θ) cos (π/2) + sin(Θ) sin (π/2) = cosΘ (0) + sinΘ (1)
= sinΘ

sin ( Θ - π / 2 )= sinΘ cos(π/2) - cosΘ sin(π/2)
= sinΘ (0) - cosΘ (1)
= -cosΘ
so you get,
cot ( Θ - π / 2 )= sinΘ / -cosΘ
cot ( Θ - π / 2 )= -tanΘ
 
 10. 
Use the definitions of the trigonometric ratios for a right triangle to derive a cofunction identity for tan(90° - A). Show your steps.

 
 11. 
Use an angle sum identity to verify the identity .

Lets set θ = 45°
LHS = cos2θ = cos 2(45°) = cos 90° = 0
RHS = 2 cos²45° - 1 = 2(1/√2)² - 1 = 2(1/2) - 1 = 1- 1 = 0  
 12. 
Without graphing, determine whether the function represents exponential growth or exponential decay.

 
 13. 
Without graphing, determine whether the function represents exponential growth or exponential decay.

 
 14. 
The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10 years.

a. Write an exponential function for the graph.
b. Use the function in part a to find the value of the boat after 9.5 years.

 
 15. 
Use a graphing calculator to solve the equation in the interval from 0 to 2p. Round to the nearest hundredth.

  16. 
The equation models the height h in centimeters after t seconds of a weight attached to the end of a spring that has been stretched and then released.

a. Solve the equation for t.
b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position. Round your answers to the nearest hundredth. c. Find the times at which the weight is at a height of 1 cm, of 3 cm, and of 5 cm below the rest position for the second time. Round your answers to the nearest hundredth.

A. h=7cos[(pi/3)t]
h/7=cos[(pi/3)t]
arccos(h/7)=πt/3
t=3arccos(h/7)/π

B. h=1cm
t=3arccos(1/7)/π≈1.36 sec
..
h=3cm
t=3arccos(3/7)/π≈1.08 sec
..
h=5cm
t=3arccos(5/7)/π≈0.74 sec

C. -1cm: 1.64 sec, -3cm: 1.92, -5cm: 2.26

 
 17. 
Consider the infinite geometric series.
a.
Write the first four terms of the series.
b.
Does the series diverge or converge?
c.
If the series has a sum, find the sum.

 
 18. 
Use the graph of the sine function shown below.

a. How many cycles occur in the graph?
b. Find the period of the graph.
c. Find the amplitude of...
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