Function Match (pp. 1 of 4) - Bob Hope School

Function Match (pp. 1 of 4)
©2010, TESCCC 08/01/10
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sketch the graph of each function. Then, determine the pairs of functions (one from the left side, and
one from the right) that “match” (or, have identical graphs).
f ( x)
sin( x)
f ( x)
sin( x ) 2
f ( x)
cos( x)
f ( x)
cos( x)
2
1. A)
2. B)
3. C)
4. D)
f ( x)
sin x
f ( x)
sin x
f ( x)
cos x
f ( x)
cos x
Pick a pair of matching functions, and explain why they are equivalent. If appropriate, describe how
each is a transformation of a parent trigonometric function.

Function Match (pp. 2 of 4)
©2010, TESCCC 08/01/10
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sketch the graph of each function. Then, determine the pairs of functions (one from the left side, and
one from the right) that “match” (or, have identical graphs).
f ( x)
tan x
f ( x)
tan( x)
f ( x)
cot x
f ( x)
cot( x)
5. E)
6. F)
7. G)
8. H)
f ( x)
tan x
f ( x)
tan( x)
2
f ( x)
tan( x ) 2
f ( x)
tan( x )
Pick a pair of matching functions, and explain why they are equivalent. If appropriate, describe how
each is a transformation of a parent trigonometric function.

Odd and Even Function Identities
sin( x)
sin x
csc( x)
csc x
Function Match (pp. 3 of 4)
cos( x)
cos x
sec( x)
sec x
©2010, TESCCC 08/01/10
tan( x)
tan x
cot( x)
cot x
(odd functions) (even functions) (odd functions)
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Use the function properties above to simplify each expression below, and match it with an expression
from the answer box.
9) sin( x) sec x
A) 1
10)
11)
12)
13)
14)
15)
16)
17)
sin( x) csc x
B) 1
Answer Box
cos( x) csc x
C) tan x
cos( x) sec x
D) cot x
1
sin( x)
1
sin( x)
1
cos( x)
1
cos( x)
cos( x)
sin( x)
E) cot x
F) sec x
G) sec x
H) csc x
I)
csc x

Function Match (pp. 4 of 4)
18) Write the trigonometric ratios for angles A and B in the right triangle below.
sin A = sin B =
©2010, TESCCC 08/01/10
cos A = cos B =
tan A = tan B =
cot A = cot B =
sec A = sec B =
csc A = csc B =
19) Explain how the ratios for A are related to those for B.
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
20) What is the relationship between the measures of the two angles? Find their degree measures
to confirm.
21) Points P and Q lie on the unit circle, as shown. Find the values of the six trigonometric functions
for rotation angles passing through these points.
sin P = sin Q =
(.28, .96)
cos P = cos Q =
Q
P
(.96, .28)
(1, 0)
Cofunction Identities
sin( x)
cos x
cos(
2
2
13
B
12
A 5 C
x)
sin x
tan P = tan Q =
cot P = cot Q =
sec P = sec Q =
csc P = csc Q =
tan(
cot(
2
2
x)
cot x
x)
tan x
sec(
csc(
2
2
x)
csc x
x)
sec x
22) The cofunction identities listed here apply to P and Q. Why? What is special about the radian
measures of these two rotation angles for the identities to apply?

Function Match (pp. 1 of 4) KEY
©2010, TESCCC 08/01/10
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sketch the graph of each function. Then, determine the pairs of functions (one from the left side, and
one from the right) that “match” (or, have identical graphs).
f ( x)
sin( x)
f ( x)
sin( x ) 2
f ( x)
cos( x)
f ( x)
cos( x)
2
1.
2.
3.
4.
Graph #1
matches
Graph B
Graph #2
matches
Graph D
Graph #3
matches
Graph C
Graph #4
matches
Graph A
A)
B)
C)
D)
f ( x)
sin x
f ( x)
sin x
f ( x)
cos x
f ( x)
cos x
Pick a pair of matching functions, and explain why they are equivalent. If appropriate, describe how
each is a transformation of a parent trigonometric function.
Samples: 1) sin(-x) = -sin x because sine is an odd function. 2) Translating a sine curve /2 to the
right results in a cosine function reflected over the x-axis. 3) cos(-x) = cos x because cosine is an
even function.

Function Match (pp. 2 of 4) KEY
©2010, TESCCC 08/01/10
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sketch the graph of each function. Then, determine the pairs of functions (one from the left side, and
one from the right) that “match” (or, have identical graphs).
f ( x)
tan x
f ( x)
tan( x)
f ( x)
cot x
f ( x)
cot( x)
5.
6.
7.
8.
Graph #5
matches
Graph H
Graph #6
matches
Graph E
Graph #7
matches
Graph F
Graph #8
matches
Graph G
E)
F)
G)
H)
f ( x)
tan x
f ( x)
tan( x)
2
f ( x)
tan( x ) 2
f ( x)
tan( x )
Pick a pair of matching functions, and explain why they are equivalent. If appropriate, describe how
each is a transformation of a parent trigonometric function.
Samples: 5) tan x translated to the right is the same as tan x because its period is .
6) tan(-x) = -tan x because tangent is an odd function. 8) Reflecting a cotangent function over the yaxis
results in the same graph as a tangent function translated /2 to the right.

Odd and Even Function Identities
sin( x)
sin x
csc( x)
csc x
Function Match (pp. 3 of 4) KEY
cos( x)
cos x
sec( x)
sec x
©2010, TESCCC 08/01/10
tan( x)
tan x
cot( x)
cot x
(odd functions) (even functions) (odd functions)
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Use the function properties above to simplify each expression below, and match it with an expression
from the answer box.
9) sin( x) sec x
C) tan x A) 1
10) sin( x) csc x
B) 1 B) 1
Answer Box
11) cos( x) csc x
D) cot x C) tan x
12) cos( x) sec x
A) 1 D) cot x
13)
14)
15)
16)
17)
1
sin( x)
1
sin( x)
1
cos( x)
1
cos( x)
cos( x)
sin( x)
I) csc x E) cot x
H) csc x F) sec x
F) sec x G) sec x
G) sec x H) csc x
E) cot x I)
csc x

Function Match (pp. 4 of 4) KEY
18) Write the trigonometric ratios for angles A and B in the right triangle below.
sin A = 12/13 sin B = 5/13
©2010, TESCCC 08/01/10
cos A = 5/13 cos B = 12/13
tan A = 12/5 tan B = 5/12
cot A = 5/12 cot B = 12/5
sec A = 13/5 sec B = 13/12
csc A = 13/12 csc B = 13/5
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
19) Explain how the ratios for A are related to those for B.
The sine of one angle is the cosine of the other. The tangent of one is the cotangent of the
other. The secant of one is the cosecant of the other.
20) What is the relationship between the measures of the two angles? Find their degree measures
to confirm.
The angles are complementary, or mA + mB = 90.
Here, mA = sin -1 (12/13) 67.38, and mB = cos -1 (12/13) 22.62.
67.38 + 22.62 = 90
21) Points P and Q lie on the unit circle, as shown. Find the values of the six trigonometric functions
for rotation angles passing through these points.
sin P = 0.28 = 7/25 sin Q = 0.96 = 24/25
(.28, .96)
cos P = 0.96 = 24/25 cos Q = 0.28 = 7/25
Q
P
(.96, .28)
(1, 0)
Cofunction Identities
sin( x)
cos x
cos(
13
2
2
13
B
12
A 5 C
x)
sin x
tan P = 0.2916… = 7/24 tan Q = 3.4285… = 24/7
cot P = 3.4285… = 24/7 cot Q = 0.2916… = 7/24
sec P = 1.1416… = 25/24 sec Q = 3.5714… = 25/7
csc P = 3.5714… = 25/7 csc Q = 1.1416… = 25/24
tan(
cot(
2
2
x)
cot x
x)
tan x
sec(
csc(
2
2
x)
csc x
x)
sec x
22) The cofunction identities listed here apply to P and Q. Why? What is special about the radian
measures of these two rotation angles for the identities to apply?
It must be shown that /2 P = Q, and /2 Q = P (or, that P + Q = /2). Since P = sin -1 (0.28)
0.2838, and Q = cos -1 (0.28) 1.2870, then /2 0.2838… = Q, and /2 1.2870… = P (or,
P + Q = 0.2838… + 1.2870… = /2).