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UNIVERSITI TEKNOLOGI MALAYSIA
SSCE 1693 ENGINEERING MATHEMATICS I
TUTORIAL 1
1.
By using the following identities
sinh (x + y) = sinh x cosh y + cosh x sinh y,
cosh (x + y) = cosh x cosh y + sinh x sinh y.
Show that
(i)
2.
3.
4.
5.
6.
sinh 2x = 2 sinh x cosh x
and
(ii)
cosh 2x = cosh2 x + sinh2 x.
Solve the following equations.
(a)
4 cosh x − sinh x = 8.
(b)
11 cosh x − 5 sinh x = 10.
(c)
2 cosh 2x − sinh 2x = 2.
(d)
cosh x = 5 − sinh x.
Solve the following equations
(a)
sinh x + 2 = cosh x.
(b)
cosh x = 2e−x .
(c)
4
1
sinh 2x − cosh 2x + 1 = 0.
2
5
(d)
tanh x = e−x .
Evaluate
(a)
cosh−1 2.
(b)
1
tanh−1 (− ).
2
(c)
1
sech−1 ( ).
3
(d)
cosech−1 (−1).
Express the following expression in logarithm form.
(a)
cosh−1 4.
(b)
sinh−1 2.
(c)
3
tanh−1 (− ).
4
(d)
cosech−1 (2).
Express the following expression in logarithm form.
(a)
cosh−1 (3x).
(c)
tanh−1 (−
3x
).
4
(b)
sinh−1 (−2x).
(d)
cosech−1 (2x).
7.
If sin−1 p = 2 cos−1 q, show that p2 = 4q 2 (1 − q 2 ).
8.
Given that 2 sin−1 x + sin−1 2x =
´
π
1 ³√
3−1 .
, show that x =
2
2
9.
If all the angles are acute, show that
µ ¶
µ ¶
µ ¶
1
1
π
1
−1
−1
−1
+ tan
+ tan
= .
tan
2
5
8
4
10. Show that
(a)
1
= cosh 2x + sinh 2x.
cosh 2x − sinh 2x
(b)
cosh 3x = 4 cosh3 x − 3 cosh x.
(c)
cosh x + cosh y = 2 cosh
(d)
1 + tanh2 x
= cosh 2x.
1 − tanh2 x
x+y
x−y
cosh
.
2
2
11. Solve the following equations and give your answers in logarithmic form.
(a)
cosh x = 4 + sinh x.
(b)
3 sinh x − cosh x = 1.
12. Find the values of R and α so that
5 cosh x − 3 sinh x = R cosh(x − α).
Hence, deduce
(a)
the minimum value of 5 cosh x − 3 sinh x,
(b)
the roots of 5 cosh x − 3 sinh x = 7.
13. Solve the following equations:
(a)
sinh−1 x = ln 2.
(b)
cosh−1 5x = sinh−1 4x.
14. Show that the points (a cosh x, b sinh x) is on the hyperbola
y2
x2
−
= 1.
a2
b2
15. State 4 cosh x + 5 sinh x in the form of r sinh(x + y). Hence, evaluate r and tanh y.
16. State the following expressions in terms of logarithms:
√
(a) sech−1 x.
(b) cosh−1 (1/x).
(c) sinh−1 (x2 − 1).
17. Express tanh x in terms of ex and e−x . Hence, prove that
(a)
(b)
2 tanh x
= tanh 2x.
1 + tanh2 x
¶
µ
1
1+x
−1
.
tanh x = ln
2
1−x
If tanh 2y = − 54 , show that y = − 21 ln 3. Hence, evaluate tanh y.
18. Express cosech−1 x in terms of logarithm. Hence, solve the equation
cosech−1 x + ln x = ln 3.
19. Solve the following equations simultaneously and give your answer in terms of logarithms.
cosh x − 3 sinh y = 0,
2 sinh x + 6 cosh y = 5.
20. Find the positive root of the equation
sin−1 x + cos−1 (4x) =
π
.
6
21. Show that sinh−1 (−x) = − sinh−1 x. Hence, find x if
4
3
sinh−1 x = sinh−1 ( ) + sinh−1 (− )
5
4
µ
¶
1+x
1
1
1
= p, tanh−1
= q and tanh−1
= r.
. If tanh−1
1−x
17
19
49
1
Show that 7p + 9q − 4r can be written as ln k, where k is a positive integer. Hence, find
2
k.
22. Show that tanh−1 x =
1
ln
2
23. Show that the value of
cosh
24. If y = sin−1
µ
1
√
10
¶
µ
1
5
cosh−1 ( )
2
4
¶
=
3√
2.
4
¶
1
√
, find and simplify sin y and cos y. Deduce that
5
µ
¶
µ
¶
1
1
π
−1
−1
√
√
sin
+ sin
= .
4
10
5
+ sin−1
µ
25. Show that
1 + tanh x
= e2x .
1 − tanh x
26. Solve the equation 2ex sinh x = 3 + a, where a is a constant.
27. Show that
tanh
−1
µ
x2 − 1
x2 + 1
¶
= ln x.
28. Evaluate
(a)
cos{π sinh(ln 2)}.
(b)
cosh−1 {π coth(ln 3)}.
29. If f (x) = tan−1 x, show that
f (x) + f (y) = f
µ
x+y
1 − xy
¶
.
30. If x, y, and α are positive numbers such that xy = α2 + 1, then show that
tan−1
1
1
1
+ tan−1
= tan−1
α+x
α+y
α
for all acute angles. Hence deduce that
tan−1
1
π
1
+ tan−1 =
2
3
4
and
tan−1
1
1
1
π
1
+ tan−1 + tan−1 + tan−1 = .
3
5
7
8
4
31. By using the definition of cosh x, show that
cosh 3x = 4 cosh3 x − 3 cosh x.
Deduce that one of the roots of the equation
4y 3 − 3y = 2
is
√ 1i
√ 1
1h
(2 + 3) 3 + (2 − 3) 3 .
2