This document contains a tutorial on hyperbolic functions and inverse hyperbolic functions with various examples and exercises to evaluate. It begins by defining hyperbolic functions like sinh, cosh, tanh and evaluating expressions in terms of them. It then covers identities involving hyperbolic functions, using them to solve equations, and expressing hyperbolic functions in terms of exponents. The document also explores inverse hyperbolic functions, expressing them in terms of logarithms, and evaluating inverse function expressions. Finally, it covers inverse trigonometric functions, evaluating expressions and solving equations involving inverse trig and hyperbolic functions.

1. POLITEKNIK KOTA BHARU
JABATAN MATEMATIK, SAINS DAN KOMPUTER
B5001 MATEMATIK KEJURUTERAAN 5
TUTORIAL 1
Fungsi Hiperbola
1) Dengan menggunakan fungsi hiperbola, dapatkan nilai bagi setiap yang di bawah
a. Sinh (– 3.678) d. Cosech 1.4
b. Tanh (– 0.01432 ) e. Coth 0.38
c. Cosh 2.397 f. Sech (– ln2 )
2) Tunjukkan bahawa
a. 1 – tanh2 x = sech2 x
1 1
b. cosh x = cosh(2 x ) +
2
2 2
n
c. (cosh x + sinh x) = cosh (nx) + sinh (nx)
d. sinh 3u = 3 sinh u + 4 sinh3 u
1 1
e. kosh u sinh v = sinh(u + v) − sinh(u − v)
2 2
3) Dengan menggunakan identiti fungsi hiperbola, dapatkan nilai bagi
a. Cosh x, jika sinh x = 4 c. Tanh x, jika sinh x = –100
b. Sinh x, jika cosh x = 3 d. Coth x, jika cosh x = 4
4) Setiap yang berikut memberi nilai salah satu dari 6 fungsi hiperbolik bagi u.
Gunakan takrif dan identiti fungsi hiperbolik untuk menentukan 5 fungsi
hiperbolik yang tertinggal.
3 13
a. sinh u = − d. koth u =
4 12
17 3
b. kosh u = , u > 0 e. sekh u =
15 5
7 5
c. tanh u = − f. ko sec h u =
25 12
5) Tulis ungkapan di bawah dalam sebutan eksponen. Permudahkan jawapan anda
seberapa yang boleh.
a. 2 kosh (ln x) e. kosh 5x + sinh 5x
b. sinh (2 ln x) f. ( sinh x + kosh x)4
c. tanh (ln x) g. kosh 3x – sinh 3x
1 h. ln (kosh x + sinh x) + ln (kosh x – sinh x)
d.
kosh x − sinh x
1

2. 6) Selesaikan persamaan yang diberi untuk x
1 c. sinh x = 3
a. kosh x = sinh x +
2
3 d. 2.6 cosh x + 5.1 sinh x = 8.73
b. tanh x =
5
e. 2 cosh x = 3
x
7) A telegraph wire hangs so that its shapes is described by y = 50 cosh . Evaluate
50
correct to 4 significant figure the value of y when x = 25.
8) v 2 = 0.55 L tanh ( 6.3d/L ) is a formula for velocity ,v of wares over the bottom of
shallow water, where d is the depth and L is the warelength. If d = 8.0 and L = 96,
calculate the value of v.
 L
9) The length l of a heavy cable hanging under gravity is given by l = 2c sinh   .
 2c 
Find the value of l when c = 40 and L = 30.
 x 
10) A chain hangs in the form given by y = 40 cosh  . Determine correct to 4
 40 
significant figures
a. The value of y when x = 25
b. The values of x when y = 54.30
−x
11) Diberi Ae + Be ≡ 4 cosh x − 5 sinh x . Dapatkan nilai A dan B.
x
−x
12) Given Re − Me ≡ 6 cosh x − 2 sinh x . Find R and M.
x
13) If 4e x − 3e − x ≡ P sinh x + Q cosh x , determine the values of P and Q.
14) If 5e x − 4e − x ≡ A sinh x + B cosh x , find A and B.
Fungsi Hiperbola Songsang
15) Use logarithmic equivalents of inverse hyperbolic functions to evaluate correct to
4 decimal places.
−1 1 −1 5 −1 5
a. sinh c. cosh e. tanh
2 4 8
b. sinh-1 0.9 d. cosh-1 4.3 f. tanh-1 0.7
2

3. −1  x  1 a+ x
16) Show that tanh   = ln  and evaluate, correct to 4 decimal places ,
a 2 a− x
3
tanh −1 .
5
17) State the following expressions in terms of logarithms.
a. −1 1 c. sinh-1 (x2 – 1)
−1 b. cosh
sec h x x
18) Solve the following equations
a. Sinh-1 x = ln 2
b. Cosh-1 5x = sinh-1 4x.
Fungsi Trigonometri Songsang
19) Find the principal values of the following expressions
a. cos-1(0.5) d. tan-1(-1)
−1 
b. sin  −
1
2
 (
e. ko sec −1 − 2 )
 2  f. kot −1 3
−1  1 
c. cos  − 3
 2 
−1  1
20) Given α = kos  −  , find sin α, tan α, sec α and cosec α.
 2
21) Nilaikan ungkapan-ungkapan berikut
 2  −1  1  
a. sin  kos
−1
 c. tan sin  −  
 
 2    2 
 
 −1  1    −1 1 
b. kot  sin  −   d. sec kos 
   2
  2 
e. ko sec(sec −1 2)
f. kos(tan −1 (− 3 )
22) Permudahkan ungkapan berikut
 −1  x    −1  x  
a. sec tan  −  
  c. ko sec sin  −  
 
  2    2 
b. tan(sec −1 x ) d. sin 2( cot -1x)
e. kosek2 (tan-1x)
3

4. 23) Evaluate correct to 3 decimal places
a. P = 2 tan −1 1.64 + sec −1 2.43 − 3 cos ec −1 3.85
−1  1  −1  4  −1  8 
b. x = sin   + cos   − tan  
3 5 9
24) Evaluate y correct to 4 significant figures
y = 3 sec −1 2 − 4 cos ec −1 2 + 5 cot −1 2
25)
65o Cari sudut α.
21 β
50
α ( petunjuk α + β = 65o )
4