Download 5-4 Sum and Difference Identities

ANSWER: 5-4 Sum and Difference Identities
Find the exact value of each trigonometric
expression.
4. cos
SOLUTION: SOLUTION: Write
6. tan
as the sum or difference of angle Write
as the sum or difference of angle measures with cosines that you know.
measures with cosines that you know.
ANSWER: 6. tan
SOLUTION: Write
as the sum or difference of angle measures with cosines that you know.
ANSWER: 2−
10. TECHNOLOGY A blind mobility aid uses the
same idea as a bat’s sonar to enable people who are
visually impaired to detect objects around them. The
sound wave emitted by the device for a certain
patient can be modeled by b = 30(sin 195 )t, where
t is time in seconds and b is air pressure in pascals.
a. Rewrite the formula in terms of the difference of
two angle measures.
b. What is the pressure after 1 second?
SOLUTION: a. b.
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The pressure after 1 second is –7.8 pascals.
ANSWER: ANSWER: a. Sample answer: b =
b. about –7.8 pascals
5-4 Sum and Difference Identities
2−
Find the exact value of each expression.
cos – cos sin 10. TECHNOLOGY A blind mobility aid uses the
same idea as a bat’s sonar to enable people who are
visually impaired to detect objects around them. The
sound wave emitted by the device for a certain
patient can be modeled by b = 30(sin 195 )t, where
t is time in seconds and b is air pressure in pascals.
a. Rewrite the formula in terms of the difference of
two angle measures.
b. What is the pressure after 1 second?
14. sin
SOLUTION: SOLUTION: a. ANSWER: b.
15. cos 40 cos 20 – sin 40 sin 20
SOLUTION: ANSWER: The pressure after 1 second is –7.8 pascals.
16. ANSWER: SOLUTION: a. Sample answer: b =
b. about –7.8 pascals
Find the exact value of each expression.
cos – cos sin 14. sin
SOLUTION: ANSWER: Simplify each expression.
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ANSWER: ANSWER: ANSWER: ANSWER: 5-4 Sum and Difference Identities
Simplify each expression.
17. sin 4y
Write each trigonometric expression as an
algebraic expression.
24. sin (arcsin x + arccos x)
SOLUTION: SOLUTION: ANSWER: If we let
and
, then
. Sketch one right triangle
and with an acute angle and another with an acute
angle . Label the sides such that
and
. Then use the Pythagorean Theorem to
express the length of each third side.
tan θ
18. cos
cos x + sin
sin x
SOLUTION: ANSWER: sin x
19. sin 3y cos y + cos 3y sin y
Using these triangles, we find that
SOLUTION: and
values in
ANSWER: . Substitute these
and simplify.
sin 4y
Write each trigonometric expression as an
algebraic expression.
24. sin (arcsin x + arccos x)
SOLUTION: ANSWER: If we let
1
and
, then
. Sketch one right triangle
and with an acute angle and another with an acute
angle . Label the sides such that
and
. Then use the Pythagorean Theorem to
express the length of each third side.
26. SOLUTION: eSolutions Manual - Powered by Cognero
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ANSWER: 26. 5-4 Sum and Difference Identities
SOLUTION: Verify each cofunction identity using one or
more difference identities.
32. tan
= cot x
SOLUTION: If we let
and
and , then
. Sketch one right triangle
with an acute angle and another with an acute
angle . Label the sides such that
and
. Then use the Pythagorean Theorem to
ANSWER: express the length of each third side.
Using these triangles, we find that
,
, and
. Substitute these values
in
and
33. sec
= csc x
simplify.
SOLUTION: ANSWER: ANSWER: Verify each cofunction identity using one or
more difference identities.
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5-4 Sum and Difference Identities
33. sec
= csc x
38. sin (90 +
) = cos SOLUTION: SOLUTION: ANSWER: Find the solution to each expression on the
interval [0, 2π).
ANSWER: 40. cos
– sin
= 0
SOLUTION: when x =
On the interval [0, 2 ),
and x =
Verify each reduction identity.
+ ) = cos
36. cos (2
ANSWER: ,
SOLUTION: 42. cos
ANSWER: 38. sin (90 +
.
+ sin = 0
SOLUTION: ) = cos SOLUTION: =
ANSWER: when x =
On the interval [0, 2 ),
and x
.
ANSWER: ,
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43. sin
Find the solution to each expression on the
+ sin = =
.
ANSWER: 5-4 Sum and Difference Identities
,
43. sin
+ sin = SOLUTION: On the interval [0, 2 ),
=
when x =
and x
.
ANSWER: ,
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