Download PTG 0116 Tutorial 2

PTG0116:
Foundation Maths on Trigonometry and Coordinate Geometry
Tutorial for Topic 2:Complex Number
2.1 General Q:
Write each expression in the standard form; a+bi
No.
1
2
3
4
5
(2 − 5𝑖) − (8 + 6𝑖)
(3 − 4𝑖)(2 + 𝑖)
6−𝑖
1+𝑖
(1 + 𝑖)3
√−25
Ans
−6 − 11𝑖
10 − 5𝑖
5 7
− 𝑖
2 2
−2 + 2𝑖
5𝑖
Solve each equation in the complex number system
No.
6
7
2
𝑥 +4=0
𝑥 2 − 6𝑥 + 10 = 0
Ans
{−2𝑖, 2𝑖}
{3 − 𝑖, 3 + 𝑖}
Without solving, determine the character of the solutions of each equation in the complex
number system
No.
8
3𝑥 2 − 3𝑥 + 4 = 0
Ans
Two complex solutions that
are conjugates of each other
PTG0116:Foundation Maths:Trigonometry & Coordinates Geometry: Complex Number
Page 1/4
2.2 Polar Coordinates:
Convert polar coordinates(r,θ) to rectangular coordinates(x,y)
No.
Ans
1
(6, 150°)
(−3√3, 3)
3𝜋
2
(√2, −√2)
(−2, )
4
2𝜋
3
(1,−√3)
(−2, )
3
4
𝜋
(−1, − )
3
5
6
(-2, -180°)
(6.3, 3.8)
1 √3
(− , )
2 2
(2,0)
(−4.98, −3.85)
Convert reactangular coordinates (x,y) to rectangular coordinates(r,θ)
No.
Ans
7
(3, 0)
(3,0)
𝜋
8
(1, -1)
(√2, − )
4
𝜋
9
(√3, 1)
(2, )
6
10
(4, 240° )
(−2, −2√3)
11
(-2.3, 0.2)
(2.3087, 175.033° )
x,y in the following equations represent rectangular coordinates. Write their expressions in
polar coordinates(r,θ)
No.
Ans
2
2
3
12
2𝑥 + 2𝑦 = 3
𝑟2 =
2
13
𝑥 2 = 4𝑦
𝑟 2 𝑐𝑜𝑠 2 𝜃 − 4𝑟 sin 𝜃 = 0
14
2𝑥𝑦 = 1
𝑟 2 sin 2𝜃 = 1
(r,θ) in the following equations represent polar coordinates. Write each equation using
rectangular coordinates(x,y)
No.
Ans
2
15
𝑟 = 𝑐𝑜𝑠𝜃
𝑥 + 𝑦2 − 𝑥 = 0
2
2
16
𝑟 = 𝑐𝑜𝑠𝜃
(𝑥 + 𝑦 2 )3/2 − 𝑥 = 0
17
𝑟=2
𝑥2 + 𝑦2 = 4
PTG0116:Foundation Maths:Trigonometry & Coordinates Geometry: Complex Number
Page 2/4
2.3 Complex numbers in Polar Form
Plot each complex number in rectangular form(x,y) in the complex palne and write it in
polar form(r,θ)
No.
1
2
3
4
Ans
√2(cos 45° + 𝑖 sin 45° )
3(cos 270° + 𝑖 sin 270° )
4√2(cos 315° + 𝑖 sin 315° )
√13(cos 123.7° + 𝑖 sin 123.7° )
1+𝑖
−3𝑖
4 − 4𝑖
−2 + 3𝑖
Write each complex number in rectangular form (x,y)
No.
5
6
7
8
7𝜋
7𝜋
4 (𝑐𝑜𝑠
+ 𝑖 𝑠𝑖𝑛 )
4
4
3𝜋
3𝜋
3 (𝑐𝑜𝑠
+ 𝑖 𝑠𝑖𝑛 )
2
2
0.2(𝑐𝑜𝑠 100° + 𝑖 sin 100° )
𝜋
𝜋
2 (𝑐𝑜𝑠
+ 𝑖 𝑠𝑖𝑛 )
18
18
Ans
2√2 − 2√2𝑖
−3𝑖
−0.035 + 0.197𝑖
1.970 + 0.347𝑖
Find zw and z/w
No.
9
10
𝜋
𝜋
𝑧 = 2 (𝑐𝑜𝑠 + 𝑖 𝑠𝑖𝑛 )
8
8
𝜋
𝜋
𝑤 = 2 (𝑐𝑜𝑠
+ 𝑖 𝑠𝑖𝑛 )
10
10
𝑧 = 2 + 2𝑖
𝑤 = √3 − 𝑖
Ans
9𝜋
9𝜋
𝑧𝑤 = 4 (cos
+ 𝑖 𝑠𝑖𝑛 )
40
40
𝑧
𝜋
𝜋
= [cos ( ) + 𝑖 sin( )]
𝑤
40
40
𝑧𝑤 = 4√2(cos 15° + 𝑖 𝑠𝑖𝑛15° )
𝑧
= √2(cos 75∘ + 𝑖 sin 75∘ )
𝑤
Write each expression in the standard form: a+bi
No.
11
12
13
14
°
°
[√3(cos 10 + 𝑖 sin 10 )]
[√5 (cos
(1 − 𝑖)5
6
(√2 − 𝑖)
6
3𝜋
3𝜋 4
+ 𝑖 sin )]
16
16
Ans
27 27√3
+
𝑖
2
2
25√2 25√2
−
+
𝑖
2
2
−4 + 4𝑖
−23 + 14.142𝑖
PTG0116:Foundation Maths:Trigonometry & Coordinates Geometry: Complex Number
Page 3/4
Derivation
Two complex numbers are:
𝑧1 = 𝑟1 (cos 𝜃1 + 𝑖 sin 𝜃1 )
𝑧2 = 𝑟2 (cos 𝜃2 + 𝑖 sin 𝜃2 )
Show that the magnitude and argument for the product of the complex numbers in the (𝑟, 𝜃)form
are given by
𝑧1 𝑧2 = 𝑟1 𝑟2 [cos(𝜃1 + 𝜃2 ) + 𝑖 sin(𝜃1 + 𝜃2 )]
Argument =𝑡𝑎𝑛−1 (𝜃1 + 𝜃2 )
Show that the magnitude and argument for the quotient of the complex numbers in the(𝑟, 𝜃)form.
𝑧2
𝑧1
𝑟
= 𝑟2 𝑐𝑜𝑠 (𝜃2 − 𝜃1 ) + 𝑖 sin(𝜃2 −𝜃1 )
1
Argument = (𝜃1 − 𝜃2 )
PTG0116:Foundation Maths:Trigonometry & Coordinates Geometry: Complex Number
Page 4/4