1
Chapter 1 Functions
1.1 Functions
1 Determine which of the following define a function.
1
(b) 𝑔 ∶ 𝑥 ⟼ ±√4 + 𝑥 2 ; 𝑥 ∈ ℝ
(a) 𝑓 ∶ 𝑥 ⟼ (1 + 𝑥)3 ; 𝑥 ∈ ℝ
2 Sketch the graph for each of the following linear functions and determine its range.
(a) 𝑓 ∶ 𝑥 ⟼ 3𝑥 − 4 , 𝑥 ∈ ℝ , 1 ≤ 𝑥 ≤ 3
(b) 𝑔 ∶ 𝑥 ⟼ 2𝑥 + 5 , 𝑥 ∈ ℝ , −1 < 𝑥 ≤ 4
3
(c) ℎ ∶ 𝑥 ⟼ 6 − 𝑥 , 𝑥 ∈ ℝ , 0 ≤ 𝑥 < 6
2
3 Sketch the graph for each of the following quadratic functions and determine its range.
(a) 𝑓 ∶ 𝑥 ⟼ 𝑥 2 + 2𝑥 + 3 , 𝑥 ∈ ℝ , −2 < 𝑥 ≤ 1
(b) 𝑔 ∶ 𝑥 ⟼ −2𝑥 2 − 4𝑥 + 5 , 𝑥 ∈ ℝ , −3 ≤ 𝑥 < 2
4 Sketch the graph for each of the following reciprocal functions and determine its range.
1
1
(a) 𝑓 ∶ 𝑥 ⟼ 𝑥−1 , 𝑥 ∈ ℝ , 𝑥 ≠ 1
(b) 𝑔 ∶ 𝑥 ⟼ 𝑥 2 +1 , 𝑥 ∈ ℝ
5 Sketch the graph for each of the following root functions and determine its range.
1
3
(a) 𝑓 ∶ 𝑥 ⟼ √2𝑥 + 1 , 𝑥 ∈ ℝ , − 2 ≤ 𝑥 < 4 (b) 𝑔 ∶ 𝑥 ⟼ −√3 − 4𝑥 , 𝑥 ∈ ℝ , −2 ≤ 𝑥 ≤ 4
2𝑥
6 Sketch the function defined by 𝑓 ∶ 𝑥 ⟼ 𝑥 + |𝑥| and find its domain and range.
7 Sketch the graph of f defined by 𝑓 ∶ 𝑥 ⟼ |𝑥 − 1| + |𝑥 − 2| and determine its domain and range.
8 Sketch the graph for each of the following functions and state its range.
(a) 𝑓(𝑥) = {
2𝑥 + 3 for 0 ≤ x ≤ 3
𝑥 2 for 3 < 𝑥 ≤ 4
(b) 𝑔(𝑥) = {
𝑥 2 for 0 ≤ x ≤ 2
10 − 3𝑥 for 2 < 𝑥 ≤ 4
Theorem 1 : Let 𝑥1, 𝑥2 ∈ 𝑋 , if 𝑓(𝑥1 ) = 𝑓(𝑥2 ) ⟹ 𝑥1 = 𝑥2 then f is a one-to-one function.
OR Let 𝑥1, 𝑥2 ∈ 𝑋 , if 𝑥1 ≠ 𝑥2 ⟹ 𝑓 (𝑥1 ) ≠ 𝑓(𝑥2 ) then f is a one-to-one function.
9 Determine which of the following functions are one-to –one.
(a) 𝑓 ∶ 𝑥 ⟼ 2𝑥 , 𝑥 ∈ ℝ
(b) 𝑔 ∶ 𝑥 ⟼ 𝑥 2 + 1 , 𝑥 ∈ ℝ (c) ℎ ∶ 𝑥 ⟼
2𝑥+3
𝑥−1
, 𝑥 ∈ ℝ, 𝑥 ≠ 1
Theorem 2 : Horizontal Line Test
A function is one-to-one function if and only if each horizontal line intersects the graph of the
function at most one point.
2
10 Determine which of the following functions are one-to –one.
(a) 𝑓 ∶ 𝑥 ⟼ √𝑥 , 𝑥 ∈ ℝ, 𝑥 ≥ 0 (b) 𝑔 ∶ 𝑥 ⟼ |𝑥| , 𝑥 ∈ ℝ
(c) ℎ ∶ 𝑥 ⟼ 𝑥 3 , 𝑥 ∈ ℝ , 𝑥 ≤ 0
𝑥 ⟼ −𝑥 , 𝑥 ∈ ℝ , 𝑥 > 0
Theorem 3 : Increasing and decreasing Functions
If a function is increasing or decreasing throughout its domain, then f is a one-to-one function.
Let 𝑥1, 𝑥2 ∈ 𝑋 , if 𝑥2 > 𝑥1 and 𝑓(𝑥2 ) > 𝑓(𝑥1 ) that is, as the value of x increase, the value of f (x)
also increases. Therefore, f is an increasing function.
Let 𝑥1, 𝑥2 ∈ 𝑋 , if 𝑥2 > 𝑥1 and 𝑓(𝑥2 ) < 𝑓(𝑥1 ) that is, as the value of x increase, the value of f (x)
decreases. Therefore, f is a decreasing function.
11 Show that the function f defined by 𝑓 ∶ 𝑥 ⟼ 2𝑥 3 + 1 , 𝑥 ∈ ℝ is a one-to-one function.
12 Show that the function g defined by 𝑔 ∶ 𝑥 ⟼ −𝑥 2 , 𝑥 ∈ ℝ , 𝑥 ≥ 0 is a one-to-one function.
Onto Function : f is an onto function if range of f = codomain of f
13 If each of the following functions is defined on the set ℝ, determine which functions are onto.
(a) 𝑓 ∶ 𝑥 ⟼ ln(x − 1) , 𝑥 ∈ ℝ , 𝑥 > 1
(b) 𝑓 ∶ 𝑥 ⟼ 1 − 𝑥 2 , 𝑥 ∈ ℝ
Properties of inverse functions
If 𝑓 −1 exists, then (i) 𝑓−1 is a one-to-one and onto function.
(ii) Domain of 𝑓−1 = Range of f
(iii) Range of 𝑓−1 = Domain of f
14 The function f is defined by 𝑓(𝑥) = √𝑥 − 2 , 𝑥 ∈ ℝ, 𝑥 ≥ 2 . Find 𝑓−1 (𝑥 ) and verify that
𝑓[𝑓−1 (𝑥)] = 𝑥 and 𝑓−1 [𝑓(𝑥 )] = 𝑥.
15 The function g is defined by 𝑔 ∶ 𝑥 ⟼ 𝑥 2 − 2𝑥 , for 𝑥 ≥ 1 . Explain why 𝑔−1 exist and find
𝑔−1 . State the range of 𝑔−1.
In general, (a) 𝑓 ∶ 𝑥 ⟼ 𝑦 ⇔ 𝑓 −1 ∶ 𝑦 ⟼ 𝑥
𝑦 = 𝑓 (𝑥 ) ⇔ 𝑥 = 𝑓−1 (𝑦)
(b) Any point (b, a) on the graph of 𝑓−1 is the reflection of point (a, b) on the
graph of f in the line y = x. Geometrically, the graph of 𝑓−1 is the reflection
of the graph of f in the line y = x.
3
16 The function f is defined by 𝑓 ∶ 𝑥 ⟼ (𝑥 − 2)2 , 𝑥 ∈ ℝ , 𝑥 ≥ 2. Find the inverse function 𝑓−1
and find the coordinates of the point of intersection of the graphs of f and 𝑓−1 .
For the function 𝒈 ∘ 𝒇 to be defined, the range of f must be the subset of the domain of g .
𝑹𝒇 ⊆ 𝑫𝒈 , Domain of 𝒈 ∘ 𝒇 ⊆ 𝐃𝐨𝐦𝐚𝐢𝐧 𝐨𝐟 𝒇 , Range of 𝒈 ∘ 𝒇 ⊆ 𝐑𝐚𝐧𝐠𝐞 𝐨𝐟 𝒈
17 The functions f and g are defined by
𝑓 ∶ 𝑥 ⟼ 3𝑥 + 2 , 𝑥 ∈ ℝ 𝑔 ∶ 𝑥 ⟼
𝑥−2
3
,𝑥 ∈ℝ
Find the composite function 𝑔 ∘ 𝑓 and ∘ 𝑔 .
18 The functions f and g are defined by
𝑓 ∶ 𝑥 ⟼ √9 − 𝑥 2 , 𝑥 ∈ ℝ , −3 ≤ 𝑥 ≤ 3
𝑔 ∶ 𝑥 ⟼ √4 − 𝑥 , 𝑥 ∈ ℝ , 𝑥 ≤ 4
Find the composite function 𝑓 ∘ 𝑔 and state its domain and range.
19 A function g is defined by 𝑔 ∶ 𝑥 ⟼ 2𝑥 − 1, 𝑥 ∈ ℝ. Find the function f if
(a) (𝑔 ∘ 𝑓)(𝑥) = 3𝑥 2 + 2
(b) (𝑓 ∘ 𝑔)(𝑥) = 3𝑥 2 + 2 .
20 The functions f and g are defined by 𝑓 ∶ 𝑥 ⟼ 𝑥 − 2 , 𝑥 ∈ ℝ and 𝑔 ∶ 𝑥 ⟼ 3𝑥 + 1 , 𝑥 ∈ ℝ
(a) Write down an expression for the composite function ∘ 𝑓 .
(b) Find the expression for each of the following inverse functions.
(i) 𝑓−1 (𝑥)
(ii) 𝑔−1 (𝑥 )
(iii) (𝑔 ∘ 𝑓 )−1 (𝑥).
(c) Verify that (𝑔 ∘ 𝑓)−1 (𝑥) = (𝑓 −1 ⋄ 𝑔−1 )(𝑥).
1.2 Polynomial and Rational Functions
21 Find the degree of each of these polynomials.
(a) 4𝑥 6 − 3𝑥 5 + 2𝑥 4 − 𝑥 + 2
(b) 2𝑥 2 + 5 + 4𝑥 − 6𝑥 8
(c) 1 + 𝑥 + 3𝑥 5 + 4𝑥 3 + 7𝑥 7
In general, a polynomial F(x) divided by a polynomial D(x) can be written as
F(x) = 𝑫(𝒙) × 𝑸(𝒙) + 𝑹(𝒙)
Divisor Quotient
Remainder
22 Find the quotient and the remainder when the polynomial 𝑥 3 − 4𝑥 + 6 is divided by x - 1.
23 Find the quotient and the remainder when 2𝑥4 − 𝑥3 + 3𝑥2 − 2 is divided by (𝑥 − 2)(𝑥 + 3).
24 Show that 𝑥 = −1, x = -2 and x =3 are the zeroes of 𝑓 (𝑥) = 𝑥 3 − 7𝑥 − 6 and express f (x) as the
product of three linear factors.
4
The Remainder Theorem
•
𝒃
When a polynomial f (x) is divided by ax + b, the remainder is 𝒇 (− 𝒂).
25 Find each of the remainder when the polynomial 𝑥 3 + 5𝑥 2 − 3𝑥 − 14 is divided by
(a) x – 2
(b) x + 3
(c) 2x + 1
26 Given that when the polynomial 𝑥 3 + 𝑎𝑥 2 − 4𝑥 − 1 is divided by x + 2, the remainder is 3. Find
the value of the constant a.
27 The polynomial 2𝑥 4 − 𝑎𝑥 3 + 𝑥 2 + 𝑏𝑥 + 1 has a remainder of 6 when it is divided by x + 1 and a
remainder of
3
2
when it is divided by 2x – 1. Find the values of the constant of a and b.
28 Find the remainder when the polynomial 4𝑥4 − 4𝑥3 − 9𝑥2 − 𝑥 + 2 is divided by (x + 1)(x – 2).
The Factor Theorem
•
𝒃
For a polynomial f (x), ax + b is a factor of f (x) if and only if 𝒇 (− 𝒂) = 𝟎.
29 Prove that x + 1 and x – 1 are the factors of the polynomial 𝑥6 − 𝑥5 + 𝑥3 − 1.
30 Show that x – 1 is a factor of polynomial 𝑥3 − 2𝑥2 − 5𝑥 + 6. Hence, factorise the polynomial
completely.
31 Factorise the polynomial 𝑓 (𝑥 ) = 𝑥3 + 𝑥2 + 4𝑥 + 4 completely.
32 Given that 4𝑥 2 − 1 is a factor of the polynomial 4𝑥4 + 𝑎𝑥3 − 9𝑥2 + 𝑥 + 𝑏, find the values of the
constants a and b. Hence, factorise the polynomial completely.
33 Solve the cubic equation 2𝑥3 − 3𝑥2 − 44𝑥 − 60 = 0.
34 Given 𝑃(𝑥 ) = 𝑥3 − 3𝑥2 + 𝑥 + 1
(a) Find P(-1) and P(1) and hence write down a factor of P(x).
(b) Express P(x) in the form (𝑥 + 𝑎)(𝑥 2 + 𝑏𝑥 + 𝑐).
(c) Hence, solve the equation P(x) = 0.
35 Solve the equation 𝑥4 − 2𝑥2 − 3 = 0.
36 Solve the equation 𝑥 − 6√𝑥 + 5 = 0.
37 Solve each of the following equations.
(a) 3√1 − 𝑥 = 1 + √3𝑥 − 2
1
1
𝑥
𝑥2
38 If 𝑢 = 𝑥 + , express 𝑥 2 +
(b) √𝑥 + 3 − √𝑥 − 5 = √𝑥 − 2
1
in terms of u. By using the substitution = 𝑥 + , solve the
equation 𝑥4 + 4𝑥3 + 5𝑥2 + 4𝑥 + 1 = 0
𝑥
5
39 Factorise the polynomial 2𝑥3 − 9𝑥2 + 3𝑥 + 4 completely. Hence, find all the real roots of the
equation 2𝑥6 − 9𝑥4 + 3𝑥2 + 4 = 0 .
2
1
3
6
40 Find the set of values of x which satisfy the inequality 2 - 2x < 𝑥 + .
41 Find the solution set of the inequality 2𝑥2 − 5𝑥 − 3 > 0 .
42 Find the solution set of the inequality 𝑥2 − 2𝑥 − 15 ≤ 0 .
43 Find the solution set of the inequality (𝑥 + 1)(2𝑥 − 3)(4 − 𝑥) ≥ 0
44 Find the solution set of the inequality (2𝑥 + 1)(𝑥 − 2)2 < 0.
|𝒇(𝒙)| < 𝑘 ↔ −𝑘 < 𝑓 (𝒙) < 𝑘
|𝒇(𝒙)| > 𝑘 ↔ 𝒇(𝒙) > 𝑘 𝑜𝑟 𝑓 (𝒙) < −𝑘
45 Find the set of values of x which satisfy each of the following inequalities.
(a) |𝑥 − 3| < 4
(b) |2 − 5𝑥| ≥ 3
(c) |2𝑥 + 3| ≤ 3𝑥 − 1
(d) |2𝑥 − 7| > 4 − 𝑥
[|𝒇(𝒙)|]𝟐 > 𝑘 , 𝑘 > 0 ↔ |𝒇(𝒙)| > √𝒌
[|𝒇(𝒙)|]𝟐 < 𝑘 , 𝑘 > 0 ↔ |𝒇(𝒙)| < √𝒌
46 Find the solution set for each of the following inequalities.
(a) 𝑥 2 < 9
(b) 𝑥2 − 5 ≥ 0
(c) (𝑥 − 1)2 ≤ 8 (d) (𝑥 + 2)2 − 5 > 0 (e) (𝑥 + 1)2 + 2 > 0
47 Find the solution set for each of the following inequalities.
(a) 𝑥2 − 2𝑥 − 4 ≤ 0
(b) 2𝑥2 + 𝑥 ≥ 4
(e) (2𝑥 − 1)(𝑥 2 − 2𝑥 − 1) ≤ 0
(c) 𝑥2 + 2𝑥 + 5 > 0 (d) (𝑥 + 1)(𝑥 2 + 𝑥 + 1) < 0
(f) (𝑥 − 1)(3 − 𝑥)(𝑥 2 + 𝑥 + 2) > 0
48 Find the solution set for each of the following inequalities
(b) |𝑥2 + 1| < |𝑥2 − 9|
(a) |𝑥 − 4| > |2𝑥 − 1|
3𝑥−2
(c) | 𝑥+2 | < 2
49 Find the solution set for each of the following inequalities
(a)
(e)
13−4𝑥
𝑥+2
<0
(𝑥−1)2
<0
(2𝑥+1)(2−𝑥)
(b)
(𝑥+8)
(𝑥+2)(𝑥−3)
(f)
>0
𝑥2 +2𝑥+2
4𝑥−3
(c)
4
𝑥+3
<2−𝑥
(d)
𝑥−1
𝑥+1
<
𝑥
6
<0
50 Sketch, on the same coordinate axes, the graphs of 𝑦 = |1 + 3𝑥| and 𝑦 = 6𝑥. Hence,
solve the inequality |1 + 3𝑥| < 6𝑥.
51 Sketch, on the same coordinate axes, the graphs of 𝑦 = |𝑥 + 1| and 𝑦 = 4 − |𝑥 + 2|.
Hence, solve the inequality |𝑥 + 1| + |𝑥 + 2| > 4.
6
Partial Fractions
Rule 1 : Corresponding to each linear factor 𝑎𝑥 + 𝑏 in the denominator of a proper rational function,
𝐴
there exists a partial fraction of the form 𝑎𝑥+𝑏 , where a, b and A are constants.
2𝑥+3
52 Express (𝑥+3)(𝑥−6) in partial fractions.
53 Express
3𝑥2 + 𝑥 + 2
(𝑥+4)(𝑥2−𝑥−6)
in partial fractions.
Rule 2 : Corresponding to each quadratic factor 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, which is irreducible, in the
denominator of a proper rational function, there exists a partial fraction of the form
𝐴𝑥+𝐵
𝑎𝑥 2+𝑏𝑥+𝑐
, where
a, b, A and B are constants.
54 Express
55 Express
3𝑥2 +2𝑥
(𝑥+2)(𝑥2+3)
3
1−𝑥3
in partial fractions.
in partial fractions.
Rule 3 : Corresponding to each linear factor 𝑎𝑥 + 𝑏 which is repeated r times in the denominator of a
𝐴
𝐴
1
2
proper rational function, there exists a partial fraction of the form 𝑎𝑥+𝑏
, (𝑎𝑥+𝑏)
2 ,…. ,
𝐴𝑟
(𝑎𝑥+𝑏)𝑟
,
where a, b , 𝐴1, 𝐴2, … ., 𝐴𝑟 and A are constants.
56 Express
57 Express
𝑥2
(𝑥−3)3
in partial fractions.
2𝑥−7
(𝑥2+4)(𝑥−1)2
in partial fractions.
Rule 4 : Improper rational functions
𝑓(𝑥)
Consider an improper rational function 𝑔(𝑥) , where 𝑓(𝑥) is a polynomial of degree m and 𝑔(𝑥) is a
polynomial of degree n with 𝑚 ≥ 𝑛. As a result of polynomial division, we have
𝑓(𝑥)
𝑔(𝑥)
Remainder
= Quotient +
𝑔(𝑥)
,
Remainder
𝑔(𝑥)
is a proper rational function which can be expressed in
partial fractions by using Rule 1, Rule 2 or Rule 3.
•
•
If 𝑚 = 𝑛, the quotient is a constant .
If 𝑚 > 𝑛, the quotient is a polynomial with degree m – n.
58 Express
59 Express
𝑥2 +3𝑥
𝑥 2−4
in partial fractions.
𝑥4 +1
𝑥 3 +2𝑥
in partial fractions.
7
1.3 Exponential and Logarithmic Functions
60 Sketch the graph for each of the following exponential functions and determine its range.
(a) 𝑓: 𝑥 ⟼ 1 + 𝑒 𝑥 , 𝑥 ∈ ℝ
(b) 𝑔: 𝑥 ⟼ 2𝑒 −𝑥 − 3, 𝑥 ∈ ℝ
61 Sketch the graph for each of the following logarithmic functions and determine its domain
and range. (a) 𝑓: 𝑥 ⟼ ln (x − 2)
(b) 𝑔: 𝑥 ⟼ ln( 𝑥 2 − 9)
62 Without using a calculator, evaluate each of the following.
(a) log10 1000
(b) log 2 128
(c) log5 0.2
63 Express each of the following as a single logarithm.
(b) log 24 – log 6 – log 2
(a) log 2 + log 3 + log 5
1
(b) 2 log 5 + log 4 - 2 log 100
(d) 3 log x – 2 log (xy) + log y
64 Express each of the following in terms of log a, log b and log c.
𝑏
𝑎𝑏
(a) log (𝑎2 )
(b) log ( 2𝑐 )
𝑎
(c) log √𝑏𝑐 2
65 Evaluate each of the following.
(a) log 3 63 – log 9 49
(b) (log 3 m)(log m 81)
66 Solve the following equations.
(a) log x 8 = 1.5
(b)
6𝑥
2 +1
= 18
(c) log 3 (x +2) = log 9 (6x + 4)
(d) 2 log 2 x – log x 2 – 1 = 0
67 Find the set of values of
3
(a) 4𝑥 > 150
(b) 𝑥 2 < 32
1.4 Trigonometric Functions
68 Prove that
(a) sin A cot A = cos A
(c)
1+cos 𝐴
1−cos 𝐴
−
1−cos 𝐴
1+cos 𝐴
(b) (1- cos 2 A)(1 + tan 2 A) = tan 2 A
= 4 cot 𝐴 cosec 𝐴
69 Prove that
(a) (1 + tan A + tan 2 A)(1 – cot A + cot 2A) = tan 2 A + cot 2 A + 1
(b) cosec 4 A(1 – cos 4 A) - 2 cot 2 A = 1
2 𝑥
(c) ( ) < 0.02
3
8
(c)
cosec 𝐴 +cot 𝐴
sec 𝐴+ tan 𝐴
=
sec 𝐴−tan 𝐴
cosec 𝐴−cot 𝐴
2 tan 3𝐴
70 Prove that 1+tan 23A = 2 sin 3𝐴 cos 3𝐴 .
sin (A±𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵
tan 𝐴 ± tan 𝐵
tan (A±𝐵) = 1 ∓ tan 𝐴 tan 𝐵
cos (A±𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵
sin 2A= 2 sin 𝐴 cos 𝐴
cos 2A= cos 2 𝐴 − sin2 𝐴
2 tan 𝐴
tan 2A = 1 ∓ tan2 𝐴
71 Prove that following identities :
(a) sin 3A = 3 sin A – 4 sin 3 A
72 (a) Evaluate
cos3 𝐴 − cos 3𝐴
cos 𝐴
+
(b) tan 3A =
3 tan 𝐴−tan3 A
1−3 tan2 𝐴
(c)
2 tan 𝐴−sin 2𝐴
2 cot 𝐴−sin 𝐴
= tan4 𝐴
sin3 𝐴 + sin 3𝐴
sin 𝐴
(b) Find the maximum and minimum value of the expression sin 𝜃 + 2 cos 𝜃
73 Solve the equation sin 𝜃 − √3 cos 𝜃 = 1 in the interval [0, 2𝜋].
74 Solve the inequality sin 2𝜃 < √2 sin 𝜃 in the interval (0, 2𝜋).
75 Solve the inequality √3 cos 𝑥 > sin 𝑥 in the interval (0, 2𝜋).
76 Sketch the graph of 𝑦 =
2𝑥 − 3
𝑥 +1
, stating the equations of the asymptotes. Hence, sketch the graph
2𝑥 − 3
of 𝑦 = | 𝑥 +1 | .
77 Sketch the graph of 𝑦 = sin 𝑥 for 0° ≤ 𝑥 ≤ 360°. Hence, sketch the graph of
(a) 𝑦 = 1 + sin 𝑥
(b) 𝑦 = sin(𝑥 + 30°)
(c) 𝑦 = sin 2𝑥
1
78 Sketch the graph of 𝑦 = 𝑥 . Sketch on separate diagrams, the graphs of
1
(a) 𝑔(𝑥) = 𝑥 + 2
(b) ℎ(𝑥) =
𝑥+1
𝑥
(c) 𝑔(𝑥) = |
𝑥+1
𝑥
|
9
Sum and Difference of sines and cosines
𝑃+𝑄
•
sin 𝑃 + sin 𝑄 ≡ 2 sin (
•
sin 𝑃 − sin 𝑄 ≡ 2 cos (
•
cos 𝑃 + cos 𝑄 ≡ 2 cos (
•
cos 𝑃 − cos 𝑄 ≡ −2 sin (
) cos (
2
𝑃+𝑄
) sin (
2
𝑃+𝑄
2
𝑃−𝑄
) cos (
2
𝑃+𝑄
2
𝑃−𝑄
)
)
2
𝑃−𝑄
) sin (
)
2
𝑃−𝑄
2
)
1 State each of the following as the product of two trigonometric ratios.
(a) sin 3x + sin x
2 Prove that
(b) sin 6x - sin 2x
sin 6𝐴 +sin 4𝐴 + sin 2𝐴
cos 6𝐴 +cos 4𝐴 +cos 2𝐴
(c) cos 8x + cos 2x
(d) cos x - cos 5x
≡ tan 4𝐴.
3 Solve the equation cos 2x = 0.4630 for 0° ≤ 𝑥 ≤ 360° .
4 Solve each of the following equations for 0° ≤ 𝑥 ≤ 360°
(a) cos (x + 50° ) = 0.8517
(b) sin (x - 80° ) = - 0.1124
5 Find all the angles from 0° to 360° that satisfy the equation 2 sec 2 𝑥 + tan2 𝑥 = 2 sec 𝑥.
6 Solve the equation 2 cos 2𝑥 − cos 𝑥 − 1 = 0 for 0° ≤ 𝑥 ≤ 360°.
𝜋
7 Solve the equation cos (𝑥 + 3 ) = sin 𝑥 for - 𝜋 < 𝑥 < 𝜋 .
8 Solve each of the following equations for 0° ≤ 𝑥 ≤ 360°
(a) sin 3𝑥 − sin 2𝑥 + sin 𝑥 = 0
(b) cos 5𝑥 − cos 𝑥 = − sin 3𝑥
10