Algebra Answers

0.2 Let z = cos θ + isin θ.
Then z
n = cos(nθ) + isin(nθ) for all n ∈ N (by de Moivre) and z
−n = cos(nθ) − isin(nθ).
(a) Show that 2 cos(nθ) = z
n + z
−n
and 2isin(nθ) = z
n − z
−n
. (2)
(b) Show that 2
n
cosn
θ =

z +
1
z

n
and (2i)
n
sinn
θ =

z −
1
z

n
. (2)
(c) Use (b) to express sin7
θ in terms of multiple angles. (6)
(d) Express cos3
θ sin4
θ in terms of multiple angles. (6)
(e) Eliminate θ from the equations 4x = cos(3θ) + 3 cos θ; 4y = 3 sin θ − sin(3θ). (5

Herbert and Dranreb can thoroughly clean a house in 3 hours. If each work, one of them would do the job in 1 hour less time than the other. How long would it take each person to do the cleaning?

Jan purchased a number of pencils for PHP 180.00 from Christian and sold all except 6 at a profit of PHP 2.00 per pencil. With the total amount received, he could buy 30 more items than before. Find the number of items he bought and the cost of each in pesos.

1. Fault analysis in diagnostic assessment is a powerful way to establish whether a teacher understands the development of number sense and number concepts. Go through the following (true) case study of a Grade 7 learner with an extra mathematics teacher who tries to make sense of his ever increasing backlog:
“John, what is three fifths plus three fifths?” 3/5 +3/5
“I think it is six tenths Mam.”
“John, show me on this number line where three fifths is”
|______|______|______|______|______|______|______|__
0 1 2 3 4 5 * 6 7
“It is somewhere over here I think Mam.”

Analyse the learner’s fault(s) and try to establish which concepts are not in place.

Design an activity that you will give to your learners so that they can apply the following properties of numbers:
1.1. Commutative properties
1.2. Associative properties
1.3. Distributive property
1.4. Density property
1.5. Identity property
2. Provide solutions for the activity in above

1. Discuss the following concepts which are important in the development of number sense:
1. Counting
2. Subitising
3. Relational skill

3.1. Mothers and pre-school teachers often boasted that a four-year old child could already
read numbers. Is this an ideal situation? Motivate your answer.

1. Differentiate between rational and irrational numbers (give examples).
2. Discuss any five special facts to remember about the number zero.
3. What is an absolute value? Discuss the reasons why it is necessary.

1. Express 54069,269 in words; place value; and in terms of the powers of 10.
2. Differentiate between real and non-real numbers (give an examples).

How much longer in percentage is a one mile race than 1500 m race?