Unformatted text preview: MAC1114  Homework 11
Due Tuesday  March 29
cot u cot v − 1
(hint: use the sum identity for tangent)
cot u + cot v
1
1−
1
1 − tan u tan v
cot u cot v
1
cot(u + v ) =
= tan u+tan v =
= 1 cot u cot v
1
tan(u + v )
tan u + tan v
cot u cot v
1−tan u tan v
cot u + cot v 1. Show that cot(u + v ) = = cot u cot v − 1
cot v + cot u 2. Let f (x) = sin x. Verify the following identity used in calculus: (note the typo)
f (x + h) − f (x)
= sin x
h cos h − 1
h + cos x sin h
h f (x + h) − f (x)
sin(x + h) − sin x
1
=
=
sin x cos h + sin h cos x − sin x
h
h
h
1
1
=
sin x cos h − sin x + sin h cos x
h
h
sin h
cos h − 1
+ cos x
= sin x
h
h
2
3. Let sin α = 1 and cos β = 5 . Evaluate each of the six trig functions at α + β , i.e., nd sin(α + β ),
3
cos(α + β ), etc. (hint: only use sum identities for sin and cos, get the rest of them in an easier way).
Note thatsin(α + β ) = sin α cos β + cos α sin β , and cos(α + β ) = cos α cos β − sin α sin β , so we need to
nd sin β and cos α, then just plug everything in.
To get sin β , draw a triangle with angle √ with adjacent√side 2 and hypotenuse 5. Then by the
β
√
21
Pythagorean theorem, the opposite side is 21, so sin β = 5 . Similarly, get that cos α = 38 . So: √√
√
8 21
2 + 2 42
12
+
=
35
35
15
√
√
√
√
8 2 1 21
4 2 − 21
cos(α + β ) = cos α cos β − sin α sin β =
−
=
35 3 5
15
sin(α + β ) = sin α cos β + cos α sin β = Once you have sin(α + β ) and cos(α + β ), it's easy to get the rest:
sin(α + β )
tan(α + β ) =
=
cos(α + β ) √
2+2 42
15√
√
4 2− 21
15 √
2 + 2 42
√
=√
4 2 − 21 √
√
1
4 2 − 21
√
cot(α + β ) =
=
tan(α + β )
2 + 2 42
1
15
√
sec(α + β ) =
=√
cos(α + β )
4 2 − 21
1
15
√
csc(α + β ) =
=
sin(α + β )
2 + 2 42 It's ne if you rationalized these, but not necessary (denitely don't spend time doing that on a test). 1 ...
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 Spring '11
 Gentimis
 Algebra

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