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PreCalculus Honors: Functions and Their Graphs
Semester 2 , Unit 4: Activity 21
Resources:
SpringBoardPreCalculus
Online
Resources:
PreCalculus
Springboard Text
Unit 4
Vocabulary:
Identity
Pythagorean Identity
Trigonometric Identity
Cofunction Identity
Sum and Difference
Identities
Law of Cosines
Oblique Triangle
Law of Sines
Ambiguous Case (SSA)
Page 1 of 21
Unit Overview
In this unit students will extend their knowledge of trigonometry as they study
trigonometric identities, equations, and formulas. Students will explore the Law of
Cosines and the Law of Sines, and apply them to solve non-right triangles.
Student Focus
Main Ideas for success in lessons 21-1:
 Define the reciprocal and quotient identities.
 Use and transform the Pythagorean Identity.
Example
Lesson 21-1:
Page 2 of 21
PreCalculus Honors: Functions and Their Graphs
Semester 2 , Unit 4: Activity 22
Resources:
SpringBoardPreCalculus
Online
Resources:
PreCalculus
Springboard Text
Unit 4
Vocabulary:
Identity
Pythagorean Identity
Trigonometric
Identity
Cofunction Identity
Sum and Difference
Identities
Law of Cosines
Oblique Triangle
Law of Sines
Ambiguous Case
(SSA)
Unit Overview
In this unit students will extend their knowledge of trigonometry as they study
trigonometric identities, equations, and formulas. Students will explore the Law of
Cosines and the Law of Sines, and apply them to solve non-right triangles.
Student Focus
Main Ideas for success in lessons 22-1 and 22-2:
 Use the unit circle to write equivalent trigonometric equations.
 Write cofunction identities for Sine and Cosine.
 Solve trigonometric equations using identities and by graphing.
Example
Lesson 22-1:
Math Tip:
To convert degrees to radians, multiply the degree measure by
Page 3 of 21
Example
Lesson 22-2:
Page 4 of 21
PreCalculus Honors: Functions and Their Graphs
Semester 2 , Unit 4: Activity 23
Resources:
SpringBoardPreCalculus
Online
Resources:
PreCalculus
Springboard Text
Unit 4
Vocabulary:
Identity
Pythagorean Identity
Trigonometric
Identity
Cofunction Identity
Sum and Difference
Identities
Law of Cosines
Oblique Triangle
Law of Sines
Ambiguous Case
(SSA)
Page 5 of 21
Unit Overview
In this unit students will extend their knowledge of trigonometry as they study
trigonometric identities, equations, and formulas. Students will explore the Law
of Cosines and the Law of Sines, and apply them to solve non-right triangles.
Student Focus
Main Ideas for success in lessons 23-1, 23-2, and 23-3:
 Use Sum and Difference Identities.
 Use Half Angle Identity.
 Derive the identities and use them to find the exact values of
trigonometric functions.
 Use trigonometric identities to solve equations.
Example
Lesson 23-1:
Page 6 of 21
Example
Lesson 23-2:
Sum and Difference Identities
Page 7 of 21
Page 8 of 21
Example
Lesson 23-3:
Page 9 of 21
PreCalculus Honors: Functions and Their Graphs
Semester 2 , Unit 4: Activity 24
Resources:
SpringBoardPreCalculus
Online
Resources:
PreCalculus
Springboard Text
Unit 4
Vocabulary:
Identity
Pythagorean Identity
Trigonometric
Identity
Cofunction Identity
Sum and Difference
Identities
Law of Cosines
Oblique Triangle
Law of Sines
Ambiguous Case
(SSA)
Unit Overview
In this unit students will extend their knowledge of trigonometry as they study
trigonometric identities, equations, and formulas. Students will explore the Law of
Cosines and the Law of Sines, and apply them to solve non-right triangles.
Student Focus
Main Ideas for success in lessons 24-2:
 Write equations for the Law of Cosines using a standard angle.
 Apply Law of Cosines in a real world problem.
Example
Lesson 24-2:
Law of Cosines.
The Law of Cosines is useful in many applications involving non-right triangles,
also known as oblique triangles.
Page 10 of 21
Page 11 of 21
PreCalculus Honors: Functions and Their Graphs
Semester 2 , Unit 4: Activity 25
Resources:
SpringBoardPreCalculus
Online
Resources:
PreCalculus
Springboard Text
Unit 4
Vocabulary:
Identity
Pythagorean Identity
Trigonometric
Identity
Cofunction Identity
Sum and Difference
Identities
Law of Cosines
Oblique Triangle
Law of Sines
Ambiguous Case
(SSA)
Unit Overview
In this unit students will extend their knowledge of trigonometry as they study
trigonometric identities, equations, and formulas. Students will explore the Law of
Cosines and the Law of Sines, and apply them to solve non-right triangles.
Student Focus
Main Ideas for success in lessons 25-1 and 25-2:
 Discover mathematical relations to derive the Law of Sines.
 Find unknown sides and angles of an oblique triangle by using the Law of
Sines.
Example
Lesson 25-1:
Like the Law of Cosines, the Law of Sines relates the sides and angles in
an oblique triangle, and these can be used to find unknown sides or angles
given at least three known measures that are not all angle measures.
Law of Sines
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Example
Page 13 of 21
Lesson 25-2:
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Page 15 of 21
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Precalculus Unit 4 Practice
Lesson 21-1
5. Make sense of problems. Suppose you know that
cos u 5 20.25.
1. Make use of structure. Express each
trigonometric function as a reciprocal to write
an identity.
a. Using the Pythagorean identity, what can you
conclude about sin u?
u 
a. sec  
 2
b. How does your answer to part a change if you
know that u lies in Quadrant III?
b. tan ( x 1 2p )
c. What is the approximate value of u, in degrees,
given that u lies in Quadrant III?
2. Make use of structure. Express each
trigonometric function as a quotient to write
an identity.
 2p 
a. cot 
 3 
Lesson 21-2
6. Which expression is equivalent to tan u csc u?
b. tan (x 2 308)
3. Make use of structure. Express each
trigonometric function using the Pythagorean
Theorem to write an identity.
A.
cos u
sin 2 u
B.
1
sec u
C.csc u
p 
a. tan   1 1
 7
2
D.sec u
b.csc2 (x 1 41.258)
sec 2 t 2 1
7. Reason abstractly. Simplify
.
sec 2 t
4. Consider the function f ( x )5 sin 2 x 1 cos 2 x .
23 
What is f 
?
 181 
A.0.99
B.1.00
C.1.12
D.1.25
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SpringBoard Precalculus, Unit 4 Practice
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13. Reason abstractly. Use the unit circle to find the
value of each of the following.
Attend to precision. For Items 8 and 9, verify
each identity.
sin x
8.
5 cos x
tan x
P (c, d )
1
9.(1 2 cos2 a)(1 1 cot2 a) 5 1
10. Simone attempted to verify the identity
sin2 u csc u sec u 5 tan u by entering
sin2 u csc u sec u 2 tan u as function Y1 in the
graphing calculator. Is her method correct? Explain.
a. cos (2p 2 u)
b. sin (3608 2 u)
Lesson 22-1
For Items 11 and 12, complete each statement.
p
≈ 0.34, name three additional
9
angles whose sines are approximately equal to 0.34.
14. Given that sin
p

11. If cos x 5 0.8, then sin  2 x  5____.
2

A. 0.8
B. 20.8
C.1.25
15. Verify the identity cos (1808 2 u) tan u 5 2sin u.
Be sure to justify each step of your argument with a
valid reason.
D. 21.25
12. Make use of structure. If tan x 5 1.2, then
tan (p 2 x) 5
.
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SpringBoard Precalculus, Unit 4 Practice
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Lesson 22-2
Lesson 23-1
16. Attend to precision. Solve each equation over the
given interval without a calculator.
1
3p 

a. 2 cot x 2 cos x 5 0,  p ,

2
2 
The sound of a single musical note can be represented
using the function
y 5 a sin (2p ft),
where a is the amplitude (volume) of the sound
measured in decibels (dB), f is the frequency (pitch)
of the note measured in hertz (Hz), and t is time.
 p
b. 4 tan2 a 5 sin2 a,  0, 
 2 
21. Model with mathematics. The frequency of the
musical note middle G is about 208 Hz. Write the
equation for the sine wave that represents middle G
played at a volume of 55 dB.
17. What is the solution of cos2 x 1 5 5 0 over the
interval [08, 3608)?
A.58
B.5.99978
22. Write the equation for the sine wave for the note
one octave below middle G played at a volume of
80 dB.
C. infinite number of solutions
D. no solution
23. Make use of structure. Write the equation for
the sine wave that represents the sound when
both notes in Items 21 and 22 are played at the
same time.
18. Use appropriate tools strategically. Solve the
equation over the given interval. You may use a
calculator.
3 tan2 u 2 1 5 0, [08, 3608)
24. Use a calculator to graph the functions
1
3
y1 =
sin x , y 2 52 cos x , and y1 1 y2 on the
2
2
interval [22p, 2p], and then make a sketch in the
19. Evan solved the equation cos x 5 21 sin x on the
p
interval  , p . His work is shown below.
2

Explain his error and find the correct solution.
space below.
Step 1: cos x 5 21 sin x Original equation
cos x
521 sin x
Step 3: tan x 5 21 Step 2:
Step 4: x 5
3p
4
Divide both sides by cos x.
Definition of tangent
Solve for x.
20. Explain how the answer to tan2 x 51, [08, 3608)
could not be an infinite number of solutions.
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Page 18 of 21
25. Express the function y1 1 y2 in the form
y 5 sin (x 1 c).
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SpringBoard Precalculus, Unit 4 Practice
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Lesson 23-2
Lesson 23-3
26. Make sense of problems. In Item 25, you
found the function y1 1 y2 in the form
y 5 sin (x 1 c). Use the sum identity for sine to
algebraically verify this identity.
Make use of structure. Verify each identity.
31. sin (x 2 p) 5 2sin x
27. Make use of structure. Find the exact value.
 15p 
a. sin 
 6 
32. 12tan u tan a 5
b. tan (1058)
9
3
and tan b 52 with angle
10
5
a terminating in Quadrant I and angle b
terminating in Quadrant II, what is the exact
value of sin (a 1 b)?
cos (u 1 a )
cos u cos a
28.Given sin a 5
Attend to precision. Solve each equation on the
interval [08, 3608].
7
A.
5
33. sin 2u 2 cos u 5 0
31
B. 2
25
C. 2
D.
13 181
905
6 181
905
34.cos4 u 2 sin4 u 5 cos 2u
3
9
29.Given sin A 5 and tan B 52 with A
5
10
terminating in Quadrant I and B terminating in
Quadrant II, find the exact value of tan (A2B).
35. Determine the number of solutions on the interval
1
[0, 2p ) for cos (4u) 5 2 .
2
1
p
with 0 < u < , find sin u, cos u,
6
2
sin 2u, and tan 2u.
A.2
30.Given cot θ 5
B.4
C.6
D.8
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Page 19 of 21
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SpringBoard Precalculus, Unit 4 Practice
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42. From Ghirardelli Square in San Francisco, you can
see the Golden Gate Bridge and Alcatraz Island.
The angle between the sight lines to these
landmarks is approximately 808. The approximate
distance from Ghirardelli Square to the Golden
Gate Bridge is 3.2 miles and to Alcatraz is 1.4 miles.
A surveyor precisely measured the angle between
the sight lines to be 78.28. By how many miles does
the approximate distance change from the Golden
Gate Bridge to Alcatraz?
Lesson 24-1
For Items 36–40, refer to Item 17a in Lesson 24-1.
36. Make sense of problems. Explain why the speed
of the blade is at 0 ft/sec at the value(s) you found.
Model with mathematics. For Items 37–40, find the
distance from the center of the wheel to the stirrer blade
for each angle.
37.908
38.558
39.2628
43. The sides of an isosceles triangle have lengths of
7.2, 7.2, and 10.5. What are the angles of the
triangle?
40. At which value of u does the speed of the blade
reach a maximum?
A.778
B.1038
C.2598
44. A triangle has side lengths of 8, 10, and 12. What
are the angles of the triangle?
D.2848
Lesson 24-2
41. Model with mathematics. A new courtyard at
Ghirardelli Square in San Francisco will be
triangular, as shown in the diagram below.
1
2
and and an
2
3
angle measure of 328 between them. Which of
the following is closest to the length of the
remaining side?
45. A triangle has two side lengths of
Retaining Wall
30 yd
45 yd
1058
Suppose the angle opposite of the retaining wall
needs to be decreased by 78. If the lengths of the
courtyard are to remain the same, find the length
of the retaining wall and the new angle measurement.
© 2015 College Board. All rights reserved.
Page 20 of 21
5
A.
1
2500
B.
1
200
C.
3
10
D.
1
4
SpringBoard Precalculus, Unit 4 Practice
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Lesson 25-1
Lesson 25-2
For Items 46–48, refer to Items 1 and 2 in Lesson 25-1.
51. Determine how many triangles are possible with
the given information: 428, b 5 60, a 5 112.
46. Model with mathematics. Suppose the plane
traveled on its diverted path instead of turning
course towards Honolulu after 1.5 hours.
A. no triangle
B. one triangle
a. How many more miles would the plane have
traveled after 2 hours?
C. two triangles
D. three triangles
52. Construct viable arguments. Determine how
many triangles are possible with the given
information. Draw a sketch and show any
calculations you used.
b. How far would the plane have been from
Honolulu after 2 hours?
a.33.18, b 5 237.2, a 5 159.7
47. Which equation could be used to find the bearing
of the plane if it had traveled for 2 hours and
needed to turn at that point towards Honolulu?
sin x sin 20
5
2000
1000
sin x sin 20
B.
5
2000
1114
sin x sin 20
C.
5
1000
2000
sin x sin 20
D.
5
1114
2000
b. 60 , b 5 5 3, a 5 3
A.
53. Make use of structure. Solve the two-solution
ambiguous case situation given b 5 20, a 5 25,
B 5 258.
48. Attend to precision. Find the new bearing of
the plane when it turns towards Honolulu after
2 hours.
For Items 54–55, solve each triangle using the
Law of Sines.
49. In triangle DEF, angle D is 428, angle E is 788, and
DE is 10.2. Find angle F, EF, and DF.
54. A 5 538, B 5 658, c 5 13
50. In triangle STU, ST is 25, TU is 30, and SU is 26.5.
Find all three angles.
55. B 5 298, C 5 158, b 5 10
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SpringBoard Precalculus, Unit 4 Practice