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Lesson 7.3 Exercises, pages 611–618
A
3. Write each expression in terms of a single trigonometric function.
cos u
sin2u
a)
b) 2
sin u
cos u
ⴝ cot U
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ⴝ tan2U
7.3 Reciprocal and Quotient Identities—Solutions
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c) sin2u sec u cos u csc u
d)
1
1
b
b (cos U) a
cos U
sin U
ⴝ (sin2U) a
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sin2u
tan2u
ⴝ
ⴝ sin U
sin2U
sin2U
cos2U
ⴝ cos2U
4. Determine the non-permissible values of u.
a) sec u
tan U ⴝ
non-permissible values
occur when cos U ⴝ 0,
values occur when cos U ⴝ 0,
␲
so U ⴝ ⴙ ␲k, k ç ⺪
so U ⴝ
c)
b) tan u
1
sec U ⴝ
, so
cos U
csc U ⴝ
d)
1
, so
sin U
sec u
sin u
sec U ⴝ
non-permissible values occur
when sin U ⴝ 0, so U ⴝ ␲k,
k ç ⺪; or when cos U ⴝ 0,
so U ⴝ
2
␲
ⴙ ␲k, k ç ⺪
2
csc u
cos u
␲
ⴙ ␲k, k ç ⺪
2
sin U
, so non-permissible
cos U
1
, so non-permissible
cos U
values occur when cos U ⴝ 0, so
␲
U ⴝ ⴙ ␲k, k ç ⺪; or when
2
sin U ⴝ 0, so U ⴝ ␲k, k ç ⺪
5. Verify each identity for the given value of u.
a) tan u csc u sec u ⫽ sec2u; u ⫽ 150°
Substitute: U ⴝ 150°
L.S. ⴝ tan U csc U sec U
ⴝ (tan 150°)(csc 150°)(sec 150°)
ⴝ (ⴚtan 30°) a
1
1
ba
b
sin 150° cos 150°
R.S. ⴝ sec2U
ⴝ sec2(150°)
ⴝ
1
cos (150°)
2
2
ⴚ1
b
cos 30°
ⴚ1
1
ⴝ (ⴚtan 30°) a
ba
b
sin 30° cos 30°
ⴝ a
ⴚ1
ⴚ2
ⴝ a√ b (2) a√ b
3
3
4
ⴝ
3
ⴝ a√ b
ⴚ2
3
ⴝ
2
4
3
The left side is equal to the right side, so U ⴝ 150° is verified.
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b)
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4␲
tan u csc2u
⫽ cot u; u ⫽ 3
sec2u
Substitute: U ⴝ
L.S. ⴝ
ⴝ
ⴝ
4␲
3
tan U csc2U
sec2U
R.S. ⴝ cot U
ⴝ cot
4␲
2 4␲
atan b acsc a bb
3
3
sec2 a
atan
tan
␲
3
1
ⴝ√
␲
␲ 2
b aⴚcos b
3
3
aⴚsin
1
ⴝ
4␲
b
3
4␲
3
3
␲ 2
b
3
√
1 2
( 3)aⴚ b
2
ⴝ
√ 2
3
b
aⴚ
2
1
3
ⴝ√
The left side is equal to the right side, so U ⴝ
4␲
is verified.
3
B
6. Prove each identity in question 5.
a) tan U csc U sec U ⴝ sec2U
L.S. ⴝ tan U csc U sec U
1
sin U
1
ⴝ a
ba
ba
b
cos U sin U cos U
ⴝ
1
cos2U
ⴝ sec2U
ⴝ R.S.
The left side is equal to the right
side, so the identity is proved.
b)
tan U csc2U
ⴝ cot U
sec2U
L.S. ⴝ
ⴝ
ⴝ
ⴝ
tan U csc2U
sec2U
a
1 2
sin U
b
ba
cos U sin U
a
1 2
b
cos U
1
(sin U)(cos U)
1 2
b
a
cos U
cos U
sin U
ⴝ cot U
ⴝ R.S.
The left side is equal to the right
side, so the identity is proved.
©P DO NOT COPY.
7.3 Reciprocal and Quotient Identities—Solutions
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7. For each identity:
i) Verify the identity using graphing technology.
ii) Prove the identity.
a) 1 ⫺ sin u ⫽ (sin u)(csc u ⫺ 1)
b) ⫺cot u ⫽
1
1ⴚ
tan U
ⴚ1
i) Graph: y ⴝ
and y ⴝ
tan U
1 ⴚ tan U
i) Graph: y ⴝ 1 ⴚ sin U and
y ⴝ sin U a
1
ⴚ 1b
sin U
The graphs coincide, so the identity
is verified.
The graphs coincide, so the
identity is verified.
1 ⴚ cot U
1 ⴚ tan U
1
1ⴚ
tan U
ⴝ
1 ⴚ tan U
tan U ⴚ 1
ⴝ
(tan U)(1 ⴚ tan U)
ⴚ1
ⴝ
tan U
ii) R.S. ⴝ
ii) R.S. ⴝ (sin U)(csc U ⴚ 1)
ⴝ (sin U) a
1 - cot u
1 - tan u
1
ⴚ 1b
sin U
ⴝ 1 ⴚ sin U
ⴝ L.S.
The left side is equal to the
right side, so the identity is
proved.
ⴝ ⴚcot U
ⴝ L.S.
The left side is equal to the right side,
so the identity is proved.
8. For each identity:
i) Verify the identity for u ⫽ 45°.
ii) Prove the identity.
a)
cot u
⫺ csc u ⫽ 0
cos u
i) Substitute: U ⴝ 45°
cot U
L.S. ⴝ
ⴚ csc U
cos U
cot 45°
ⴝ
ⴚ csc 45°
cos 45°
√
1
ⴝ
ⴚ 2
1
√
2
√
√
ⴝ 2ⴚ 2
ⴝ0
ⴝ R.S.
The left side is equal to the
right side, so U ⴝ 45° is
verified.
b) tan2u cos2u ⫹ sin2u ⫽
2
csc2u
i) Substitute: U ⴝ 45°
L.S. ⴝ tan2U cos2U ⴙ sin2U
ⴝ (tan 45°)2(cos 45°)2 ⴙ (sin 45°)2
ⴝ (1) a b ⴙ a b
1
2
1
2
ⴝ1
2
csc2U
2
ⴝ
(csc 45°)2
R.S. ⴝ
2
2)2
ⴝ √
(
ⴝ
2
2
ⴝ1
The left side is equal to the right side,
so U ⴝ 45° is verified.
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cot U
ⴚ csc U
cos U
cos U
sin U
1
ⴝ
ⴚ
cos U
sin U
1
1
ⴝ
ⴚ
sin U
sin U
ii) L.S. ⴝ tan2U cos2U ⴙ sin2U
ii) L.S. ⴝ
ⴝa
sin2U
b (cos2U) ⴙ sin2U
cos2U
ⴝ sin2U ⴙ sin2U
ⴝ 2 sin2U
ⴝ
ⴝ0
ⴝ R.S.
The left side is equal to the
right side, so the identity is
proved.
9. For each identity:
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2
csc2U
ⴝ R.S.
The left side is equal to the
right side, so the identity is
proved.
7␲
i) Verify the identity for u ⫽ 6 .
ii) Prove the identity.
a) csc u ⫽
csc u - 1
1 - sin u
7␲
i) Substitute: U ⴝ
6
L.S. ⴝ csc U
7␲
6
␲
ⴝ ⴚcsc
6
ⴝ csc
ⴝ ⴚ2
csc U ⴚ 1
1 ⴚ sin U
7␲
csc
ⴚ1
6
ⴝ
7␲
1 ⴚ sin
6
ⴚ2 ⴚ 1
R.S. ⴝ
ⴝ
1
1 ⴚ aⴚ b
2
ⴝ ⴚ2
The left side is equal to the
7␲
right side, so U ⴝ
is
6
verified.
csc U ⴚ 1
1 ⴚ sin U
1
ⴚ1
sin U
ⴝ
1 ⴚ sin U
1 ⴚ sin U
ⴝ
(sin U)(1 ⴚ sin U)
1
ⴝ
sin U
ii) R.S. ⴝ
ⴝ csc U
ⴝ L.S.
The left side is equal to the
right side, so the identity is
proved.
©P DO NOT COPY.
b)
cos u - cot u
⫽ ⫺cot u
1 - sin u
7␲
6
cos U ⴚ cot U
L.S. ⴝ
1 ⴚ sin U
7␲
7␲
cos
ⴚ cot
6
6
ⴝ
7␲
1 ⴚ sin
6
√
√
3
ⴚ
ⴚ 3
2
i) Substitute: U ⴝ
ⴝ
1 ⴚ aⴚ b
√
ⴝⴚ 3
R.S. ⴝ ⴚcot U
1
2
7␲
ⴝ ⴚcot
6
√
ⴝⴚ 3
The left side is equal to the
7␲
right side, so U ⴝ
is
6
verified.
ii) L.S. ⴝ
cos U ⴚ cot U
1 ⴚ sin U
cos U
sin U
ⴝ
1 ⴚ sin U
(cos U)(sin U) ⴚ cos U
cos U ⴚ
ⴝ
ⴝ
(1 ⴚ sin U)(sin U)
(cos U)(sin U ⴚ 1)
(1 ⴚ sin U)(sin U)
ⴚ cos U
ⴝ
sin U
ⴝ ⴚcot U
ⴝ R.S.
The left side is equal to the
right side, so the identity is
proved.
7.3 Reciprocal and Quotient Identities—Solutions
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10. Use algebra to solve each equation over the domain 0 ≤ x < 2␲.
Give the roots to the nearest hundredth where necessary.
√
a) tan x ⫽ cot x
b) cos x ⫹ 3 sin x ⫽ 0
Assume tan x ⴝ 0, then divide by
tan x.
cot x
tan x
ⴝ
tan x
tan x
1
1ⴝ
tan2x
1
3
tan x ⴝ ⴚ√
tan2x ⴝ 1
tan x ⴝ — 1
5␲
␲
3␲
x ⴝ ,x ⴝ
,x ⴝ
, or
4
4
4
7␲
xⴝ
4
For tan x ⴝ 0, x ⴝ 0 or x ⴝ ␲
Verify by substitution that neither
value of x is a root of the given
equation.
Verify the roots by substitution.
c) 2 cos x ⫽ 7 ⫺ 3 sec x
3
2 cos x ⴝ 7 ⴚ cos x
Multiply by cos x, then collect
terms on one side.
2 cos2x ⴚ 7 cos x ⴙ 3 ⴝ 0
(2 cos x ⴚ 1)(cos x ⴚ 3) ⴝ 0
Either 2 cos x ⴚ 1 ⴝ 0
cos x ⴝ
xⴝ
Assume cos x ⴝ 0, then divide
by cos x.
√ sin x
cos x
cos x ⴙ 3 cos x ⴝ 0
√
1 ⴙ 3 tan x ⴝ 0
1
2
5␲
␲
or x ⴝ
3
3
Or cos x ⴚ 3 ⴝ 0
cos x ⴝ 3
This equation has no solution.
Verify the roots by substitution.
xⴝ
11␲
5␲
or x ⴝ
6
6
For cos x ⴝ 0, x ⴝ
3␲
␲
or x ⴝ
2
2
Verify by substitution that neither
value of x is a root of the given
equation.
Verify the roots by substitution.
d) sin2x ⫽ sin x cos x
sin2x ⴚ sin x cos x ⴝ 0
(sin x)(sin x ⴚ cos x) ⴝ 0
Either sin x ⴝ 0
x ⴝ 0 or x ⴝ ␲
Or sin x ⴚ cos x ⴝ 0
sin x ⴝ cos x
Assume cos x ⴝ 0, then divide by cos x.
tan x ⴝ 1
xⴝ
5␲
␲
or x ⴝ
4
4
For cos x ⴝ 0, x ⴝ
3␲
␲
or x ⴝ
2
2
Verify by substitution that neither value
of x is a root of the given equation.
Verify the roots by substitution.
11. Identify any errors in this proof, then write a correct algebraic proof.
sin u
1
Correct proof:
=
To prove:
1 - sin u
csc u - 1
1
sin u
R.S. ⴝ
L.S. ⫽
csc U ⴚ 1
1 - sin u
1
sin u
sin u
ⴝ
⫽ 1 1
sin u
ⴚ1
⫽ sin u ⫺ 1
1
- 1
csc u
1
⫽
csc u - 1
⫽
⫽ R.S.
22
sin U
1
ⴝ
1 ⴚ sin U
sin U
sin U
ⴝ
1 ⴚ sin U
ⴝ L.S.
From the 1st line of the proof to the 2nd line,
sin U
sin U
sin U
cannot be written as
.
ⴚ
1
1 ⴚ sin U
sin U
From the 4th line of the proof to the 5th line,
1
1
.
ⴚ 1 cannot be written as
csc U
csc U ⴚ 1
7.3 Reciprocal and Quotient Identities—Solutions
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12. Identify which equation below is an identity. Justify your answer.
Prove the identity. Solve the other equation over the domain
⫺␲ ≤ x ≤ ␲. Give the roots to the nearest hundredth.
a)
2 sin2x + 1
⫽ 2 csc2x ⫺ 1
sin x
I graphed each side of the
equation. The graphs do not
coincide, so this is an
equation.
From the graphs, the roots are
approximately: x ⴝ 0.81 and
x ⴝ 2.33
b)
1 + csc2x
sin2x + 1
⫽ csc x
sin x
I graphed each side of the
equation. The graphs appear to
coincide, so this is probably the
identity.
1 ⴙ csc2x
csc x
1
1ⴙ
sin2x
ⴝ
1
sin x
R.S. ⴝ
Multiply numerator and
denominator by sin2x.
R.S. ⴝ
sin2x ⴙ 1
sin x
ⴝ L.S.
The left side is equal to the right
side, so the identity is proved.
C
13. Here are two identities that involve the cotangent ratio:
1
cos u
cot u ⫽
and cot u ⫽
tan u
sin u
a) Show how you can derive one identity from the other.
cot U ⴝ
1
tan U
1
sin U
cos U
cos U
cot U ⴝ
sin U
cot U ⴝ
©P DO NOT COPY.
Write tan U ⴝ
sin U
.
cos U
Multiply numerator and denominator by cos U.
7.3 Reciprocal and Quotient Identities—Solutions
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b) Determine the non-permissible values of u for each identity.
Explain why these values are different. How could you illustrate
this using graphing technology?
1
tan U
sin U
Since tan U ⴝ
, then
cos U
For cot U ⴝ
sin U ⴝ 0 and
For cot U ⴝ
sin U ⴝ 0
cos U ⴝ 0
So, U ⴝ ␲k, k ç ⺪ and U ⴝ
cos U
sin U
So, U ⴝ ␲k, k ç ⺪
␲
ⴙ ␲k, k ç ⺪
2
The values are different because there are two restrictions for
and only one restriction for
cos U
.
sin U
1
tan U
␲
1
, and set the TABLE for intervals of , it shows
tan U
2
cos U
ERROR for all values of X in the table. When I graph y ⴝ
, with the
sin U
When I graph y ⴝ
same TABLE settings, it shows ERROR only for X ⴝ ␲k, k ç ⺪.
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7.3 Reciprocal and Quotient Identities—Solutions
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