Download CLEP-Precalculus - Problem Drill 11: Trigonometric Identities

CLEP-Precalculus - Problem Drill 11: Trigonometric Identities
Question No. 1 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
1. Which of the following equalities is a conditional equation?
(A)
x =
5x
5
(B) 3x – x = 2x
Question
(C)
x
+ 1 = −4
5
(D) cos(-x) = cos x
(E)
cot x =
1
tan x
A. Incorrect!
This equality is true for all values of x. Therefore, this is an identity and not a
conditional equation.
B. Incorrect!
This equality is true for all values of x. Therefore, this is an identity and not a
conditional equation.
C. Correct!
The only number that makes this equality true is -25. Therefore, this equality is a
conditional equation and not an identity.
Feedback
D. Incorrect!
This equality is true for all values of x. Therefore, this is an identity and not a
conditional equation.
E. Incorrect!
This equality is true for all values of x. Therefore, this is an identity and not a
conditional equation.
An identity is true for all defined values of x, whereas only certain numbers satisfy
a conditional equation. The only number that makes
x
+ 1 = −4
5
Therefore, it is a conditional equation and not an identity.
(C)
Solution
x
+ 1 = −4
5
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true is -25.
Question No. 2 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
2. Which of the following equalities is an identity?
Question
(A) cos2 x + sin2 x = 1
(B) sec x = 2
(C) 4x + 2 = 6x
(D) x4 = 16
(E) x2 + 2x – 3 = 0
A. Correct!
Every number belonging to the domains of cosine and sine satisfy this equality.
Therefore, this equality is an identity.
B. Incorrect!
This equality has an infinite number of solutions, but not every number belonging
to the domain of secant will satisfy this equality.
C. Incorrect!
The only number that satisfies this equality is the number 1. Therefore, this
equality is a conditional equation and not an identity.
Feedback
D. Incorrect!
The only numbers that satisfy this equality are the -2 and 2. Therefore, this
equality is a conditional equation and not an identity.
E. Incorrect!
The only numbers that satisfy this equality are the -3 and 1. Therefore, this
equality is a conditional equation and not an identity.
An identity is true for all defined values of x, whereas only certain numbers satisfy
a conditional equation. Every number belonging to the domains of cosine and sine
satisfy cos2 x + sin2 x = 1. Therefore, this equality is an identity.
(A) cos2 x + sin2 x = 1
Solution
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Question No. 3 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
3. Which of the following is a Pythagorean identity?
Question
(A)
csc x =
(B)
cos
(
1
sin x
π
− x
2
)
= sin x
(C) cos2 x + sin2 x = 1
(D) tan(-x) = -tan x
(E)
cot x =
cos x
sin x
A. Incorrect!
This is one of the reciprocal identities.
B. Incorrect!
This is one of the complementary identities.
C. Correct!
This is one of the Pythagorean identities.
Feedback
D. Incorrect!
This is one of the even/odd identities.
E. Incorrect!
This is one of the quotient identities.
The three Pythagorean identities are:
cos2 x + sin2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
(C) cos2 x + sin2 x = 1
Solution
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Question No. 4 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
4. Which of the following is a reciprocal identity?
Question
(A)
sec x =
(B)
sec
(
1
cos x
π
− x
2
)
= csc x
(C) 1 + cot2 x = csc2 x
(D) cos(-x) = cos x
(E)
tan x =
sin x
cos x
A. Correct!
This is one of the reciprocal identities.
B. Incorrect!
This is one of the complementary identities.
C. Incorrect!
This is one of the Pythagorean identities.
Feedback
D. Incorrect!
This is one of the even/odd identities.
E. Incorrect!
This is one of the quotient identities.
The three reciprocal identities are:
1
sin x
1
sec x =
cos x
1
cot x =
tan x
csc x =
Solution
(A) sec x = 1 / cos x
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Question No. 5 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper
as needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
5. Given cos 1.39 = 0.18, find sin 1.39.
Question
(A) 0.18
(B) 0.58
(C) 0.68
(D) 0.78
(E) 0.98
A. Incorrect!
Find sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
B. Incorrect!
Find sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
C. Incorrect!
Find sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
Feedback
D. Incorrect!
Find sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
E. Correct!
You found sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
Find sin 1.39 by using the Pythagorean identity cos2 x + sin2 x = 1.
cos2 1.39 + sin2 1.39 = 1
0.182 + sin2 1.39 = 1
0.0324 + sin2 1.39 = 1
sin2 1.39 = 1 - 0.0324
sin2 1.39 = 0.9676
sin 1.39 = ±√0.9676
sin 1.39 ≈ ±0.98
Solution
The angle 1.39 radians lies in the first quadrant because 0 < 1.39 < π/2 (0 < 1.39
< 1.57). Therefore, keep only the positive sign because the trigonometric function
sine is positive within the first quadrant.
sin 1.39 = 0.98
(E)0.98
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Question No. 6 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
6. sin 0.18 = 0.18. Evaluate tan 0.18.
Question
(A) 0.08
(B) 0.18
(C) 0.28
(D) 0.38
(E) 0.48
A. Incorrect!
Find tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using
the related quotient identity.
B. Correct!
You found tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then
using the related quotient identity,
tan x =
sin x
cos x
.
C. Incorrect!
Find tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using
the related quotient identity.
Feedback
D. Incorrect!
Find tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using
the related quotient identity.
E. Incorrect!
Find tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using
the related quotient identity.
First calculate cos 0.18 and then calculate tan 0.18 by making use of the quotient
identity tan x
=
sin x
cos x
.
The value of cos 0.18 may be obtained by making use of the Pythagorean identity
cos2 0.18 + sin2 0.18 = 1
cos2 0.18 + 0.182 = 1
cos2 0.18 + 0.0324 = 1
cos2 0.18 = 1 - 0.0324
cos2 0.18 = 0.9676
cos 0.18 = ±√0.9676
cos 0.18 ≈ ±0.98
Solution
The angle 0.18 radians lies within the first quadrant because 0 < 0.18 < π/2 (0 <
0.18 < 1.57). Therefore, keep only the positive solution because the trigonometric
function cosine is positive within the first quadrant.
cos 0.18 = 0.98
Now use a quotient identity.
sin0.18
cos 0.18
0.18
=
0.98
tan 0.18 =
≈ 0.18
(B)
0.18
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Question No. 7 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
7. csc x · tan x is equal to which of the following expressions?
Question
(A) cos x
(B) cos2 x
(C) sec x
(D) sin2 x
(E) cot x
A. Incorrect!
The given expression can be rewritten using a sequence of reciprocal and quotient
identities.
B. Incorrect!
The given expression can be rewritten using a sequence of reciprocal and quotient
identities.
C. Correct!
The given expression can be rewritten as sec x by using a sequence of reciprocal
and quotient identities.
Feedback
D. Incorrect!
The given expression can be rewritten using a sequence of reciprocal and quotient
identities.
E. Incorrect!
The given expression can be rewritten using a sequence of reciprocal and quotient
identities.
One of the reciprocal identities states:
csc x =
1
sin x
One of the reciprocal identities states:
tan x =
sin x
cos x
Rewrite and simplify the original expression using the reciprocal identity
sec x =
1
cos x
:
Solution
csc x ⋅ tan x =
=
1
sin x
⋅
sin x cos x
1
cos x
= sec x
(C)
sec x
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Question No. 8 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
8.
Question
1
−1
sin2 x
is equal to which of the following expressions?
(A) cot x
(B) cot2 x
(C) tan x
(D) tan2 x
(E) cos2 x
A. Incorrect!
The original expression can be rewritten using a combination of a reciprocal identity
and a Pythagorean identity.
B. Correct!
You rewrote the original expression using
csc x =
1
sin x
and 1 + cot2 x = csc2 x.
C. Incorrect!
The original expression can be rewritten using a combination of a reciprocal identity
and a Pythagorean identity.
Feedback
D. Incorrect!
The original expression can be rewritten using a combination of a reciprocal identity
and a Pythagorean identity.
E. Incorrect!
The original expression can be rewritten using a combination of a reciprocal identity
and a Pythagorean identity.
One of the reciprocal identities states:
csc x =
1
sin x
Therefore,
2
csc x =
1
sin2 x
The original expression can be written as:
1
2
− 1 = csc x − 1
sin2 x
One of the Pythagorean identities states:
Solution
1 + cot2 x = csc2 x
→
cot2 x = csc2 x – 1
Therefore, the original expression can be rewritten:
1
2
− 1 = csc x − 1
sin2 x
2
= cot x
(B)
cot2 x
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Question No. 9 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
9. The expression
and
Question
sin x
tan x
sec x + csc x
can be written as an expression containing only cos x
by using a combination of:
(A) Pythagorean and reciprocal identities.
(B) Pythagorean and quotient identities.
(C) Quotient and complementary identities.
(D) Quotient and reciprocal identities.
(E) Complementary and reciprocal identities.
A. Incorrect!
Pythagorean identities are most useful if an expression contains the square or other
even power of a trigonometric function.
B. Incorrect!
Pythagorean identities are most useful if an expression contains the square or other
even power, of a trigonometric function.
C. Incorrect!
Complementary identities are most useful in the evaluation of trigonometric
expressions containing complementary angles.
Feedback
D. Correct!
Use the quotient and reciprocal identities to rewrite the expression containing only
cos x and sin x .
E. Incorrect!
Complementary identities are most useful in the evaluation of trigonometric
expressions containing complementary angles.
The original expression can be rewritten using the quotient identity
two of the reciprocal identities:
Solution
sec x =
1
cos x
and csc x
=
1
sin x
tan x =
sin x
cos x
.
sin x
tan x
cos x
=
sec x + csc x
sec x + csc x
sin x
cos x
=
1
1
+
cos x sin x
The original expression has been rewritten using only sin x and cos x. Thus, we
have accomplished our goal making use of quotient and reciprocal identities.
(D)Quotient and reciprocal identities.
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and
Question No. 10 of 10
Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as
needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.
10. Let f(x) = sin x · cos x. Which of the following statements is true?
Question
(A) f(x)
(B) f(x)
(C) f(x)
(D) f(x)
(E) f(x)
is
is
is
is
is
even for all values of x.
odd for all values of x.
neither even nor odd for all values of x.
even for positive values of x, but odd for negative values of x.
odd for positive values of x, but even for negative values of x.
A. Incorrect!
Review the definitions for even and odd functions then try again.
B. Correct!
f(x) is an odd function.
C. Incorrect!
Review the definitions for even and odd functions then try again.
Feedback
D. Incorrect!
This function is even, odd, or neither, not a combination of even or odd.
E. Incorrect!
This function is even, odd, or neither, not a combination of even or odd.
Determine whether a function is even, odd, or neither by replacing its argument (x)
with the negative of its argument (-x).
f(-x) = sin(-x) · cos(-x)
The even/odd identities state that sin(-x) = -sin x and cos(-x) = cos x. Therefore,
f ( − x ) = sin( − x ) ⋅ cos( − x )
= − sin x ⋅ cos x
Solution
= −(sin x ⋅ cos x )
= −f ( x )
The function f(x) is odd for all values of x.
(B)
f(x) is odd for all values of x.
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