Download topic 3: circular functions and trigonometry

59 Given the functions f : x 7! 2x ¡ 1 and g : x 7! 2x3 , find
the function (f ± g)¡1 .
OTHER TRIGONOMETRIC FUNCTIONS
You should be able to graph and use:
60 One zero of x4 + 2x3 + 8x2 + 6x + 15 has form bi where
b 6= 0, b 2 R . Find b and all zeros of the polynomial.
TOPIC 3:
² the cosine function y = a cos(b(x ¡ c)) + d
² the tangent function y = a tan(b(x ¡ c)) + d
CIRCULAR FUNCTIONS AND
TRIGONOMETRY
y = tan bx has period
RECIPROCAL TRIGONOMETRIC FUNCTIONS
1
.
cosec x or csc x =
sin x
1
.
secant x or sec x =
cos x
cos x
1
=
.
cotangent x or cot x =
tan x
sin x
RADIAN MEASURE
There are 360± ´ 2¼ radians in a circle.
To convert from degrees to radians, multiply by
¼
.
180
To convert from radians to degrees, multiply by
180
¼ .
¼
.
b
When graphing csc x, sec x, and cot x, there will be vertical
asymptotes corresponding to the zeros of sin x, cos x, and tan x.
cot x will have zeros corresponding to the vertical asymptotes
of tan x.
APPLICATIONS OF RADIANS
For µ in radians:
² the length of an arc of radius r and angle µ is l = rµ
² the area of a sector of radius r and angle µ is A = 12 µr2
TRIGONOMETRIC IDENTITIES
² the area of a segment of radius r and angle µ is
A = 12 µr2 ¡ 12 r2 sin µ.
cos (µ + 2k¼) = cos µ and sin (µ + 2k¼) = sin µ
k2Z
THE UNIT CIRCLE
NEGATIVE ANGLES
The unit circle is the circle centred at the origin O, with radius
1 unit.
cos (¡µ) = cos µ, sin (¡µ) = ¡ sin µ, and
tan (¡µ) = ¡ tan µ
The coordinates of any point P on the unit circle, where the angle
µ is made by [OP] and the positive x-axis, are (cos µ, sin µ).
COMPLEMENTARY ANGLES
cos
¡¼
2
¢
¡ µ = sin µ and sin
¡¼
2
¢
¡ µ = cos µ
µ is positive when measured in an anticlockwise direction from
the positive x-axis.
PYTHAGOREAN IDENTITIES
sin µ
tan µ is defined as
.
cos µ
cos2 µ + sin2 µ = 1, tan2 x + 1 = sec2 x, and
1 + cot2 x = csc2 x
You should memorise or be able to quickly find the values of
cos µ, sin µ, and tan µ for µ that are multiples of ¼2 , ¼4 , and ¼6 .
DOUBLE ANGLE FORMULAE
sin 2A = 2 sin A cos A
(
NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY
B
B
For the triangle alongside:
c
A
A
cos 2A =
a
b
C
tan 2A =
C
sin (A § B) = sin A cos B § cos A sin B
sin A
sin B
sin C
=
=
a
b
c
tan (A § B) =
If you have the choice of rules to use, use the cosine rule to
avoid the ambiguous case.
To solve trigonometric equations we can either use graphs from
technology, or algebraic methods involving the trigonometric
identities. In either case we must make sure to include all
solutions on the specified domain.
If we begin with y = sin x, we can perform transformations to
produce the general sine function f (x) = a sin(b(x ¡ c)) + d.
We have a vertical stretch with factor jaj, a reflection in the
x-axis if a < 0, then a horizontal stretch with factor 1b , and
We need to use the inverse trigonometric functions to invert sin,
cos, and tan.
³ ´
.
For the general sine function:
² the amplitude is jaj
² the principal axis is y = d
² the period is
2¼
b .
Mathematics HL – Exam Preparation & Practice Guide (3rd edition)
tan A § tan B
1 ¨ tan A tan B
TRIGONOMETRIC EQUATIONS
THE GENERAL SINE FUNCTION
c
d
2 tan A
1 ¡ tan2 A
cos (A § B) = cos A cos B ¨ sin A sin B
Cosine rule a2 = b2 + c2 ¡ 2bc cos A
finally a translation with vector
cos2 A ¡ sin2 A
1 ¡ 2 sin2 A
2 cos2 A ¡ 1
COMPOUND ANGLE FORMULAE
Area formula Area = 12 ab sin C
Sine rule
for all
14
Function
Domain
x 7! arcsin x
[¡1, 1]
x 7! arccos x
[¡1, 1]
x 7! arctan x
]¡1; 1[
£
Range
¡ ¼2 ,
¤
¼
2
[0, ¼]
¡ ¼2 ,
¼
2
¤
£
19 Find the period of:
³x´
a y = cos
3
c y = sin 3x + sin x.
The ranges of these functions are important because our
calculator will only give us the one answer in the range.
Remember that other solutions may also be possible. For
example, when using arcsin our calculator will always give
us an acute angle answer, but the obtuse angle with the same
sine may also be valid.
b y = tan(5x)
20 Find the largest angle of the triangle with sides 11 cm, 9 cm,
and 7 cm.
An equation of the form a sin x = b cos x can always be solved
as tan x = ab .
21 Find the equations of the vertical asymptotes on [¡2¼, 2¼]
for:
a f (x) = csc(x)
SKILL BUILDER QUESTIONS
c g : x 7! cot
1 Convert:
a
2¼
9
2 Find the exact value of:
a sin
¡ 5¼ ¢
b cos
3
¡ 3¼ ¢
¡
24 Suppose sin x ¡ 2 cos x = A sin(x + ®) where A > 0 and
0 < ® < 2¼. Find A and ®.
3 A sector of a circle of radius 10 cm has a perimeter of 40 cm.
Find the area of the sector.
25 2 sin2 x ¡ cos x = 1 for x 2 [0, 2¼]. Find the exact value(s)
of x.
4 What consecutive transformations map the graph of y = sin x
onto:
³x´
¡
¢
a y = 2 sin
b y = sin x + ¼3 ¡ 4?
3
26 Find x if arcsin(2x ¡ 3) = ¡ ¼6 .
27 In triangle ABC, AB = 15 cm, AC = 12 cm and angle ABC
measures 30± . Find the size of the angle ACB.
5 Find the amplitude, principal axis, and period of the following
functions:
³ ´
x
a f (x) = sin 4x
b f (x) = ¡2 sin
¡ 1.
2
28 On the same set of axes, sketch the graphs of f (x) = sin x
and g(x) = ¡1 + 2f (2x + ¼2 ) for ¡¼ 6 x 6 ¼.
29 Find the exact value of arcsin(¡ 12 )+arctan(1)+arccos(¡ 12 ).
6 Sketch the graph of y = csc(x) for x 2 [0, 3¼].
7 Simplify sin
¡ 3¼
2
¢
30 µ is obtuse and sin µ = 23 . Find the exact value of sin 2µ.
¡ Á tan(Á + ¼).
31 Solve for x: sin x + cos x = 1 where 0 6 x 6 ¼.
8 If cos(2x) = 58 , find the exact value of sin x.
32
9 Solve sin 2x = sin x for x 2 [¡¼, ¼], giving exact answers.
11 Find the period of:
b y = 2 sin
³ ´
x
2
µ
8 cm
a Find cos µ.
+1
34 Solve the equation cot µ + tan µ = 2 for µ 2 ]¡ ¼2 ,
12 Sketch the graph of y = arccos x, clearly showing the axes
intercepts and endpoints.
35 If 2µ 2 [¼,
tan µ.
2
sin µ
.
1 + cos µ
¡ 5¼ ¢
37 Show that
.
cos µ 6= 0.
16 If cos 2® = sin2 ®, find the exact value of cot ®.
3
2
1
-1
1
= ¡(sec µ + tan µ) provided
tan µ ¡ sec µ
38 Solve for x where x 2 [¡¼, 3¼], giving exact answers:
¡ ¢
p
p
3 tan x2 = ¡1
b
3 + 2 sin(2x) = 0.
a
17 A chord of a circle has length 6 cm. If the radius of the circle is
5 cm, find the area of the minor segment cut off by the chord.
18
and tan(2µ) = 2, find the exact value of
PQ = 60± . Find the length of [PQ], giving your
Rb
answer in radical form.
15 Show that csc(2x) ¡ cot(2x) = tan x and hence find the
12
3¼
2 ]
¼
[.
2
36 In triangle PQR, PR = 12 cm, RQ = 11 cm, and
14 If tan µ = 2, find the exact values of tan 2µ and tan 3µ.
exact value of tan
b Find the area of the triangle.
33 Find the exact period of g(x) = tan 2x + tan 3x.
c y = sin2 x + 5.
13 Simplify 1 ¡
2µ
5 cm
10 A sector of a circle has an arc length of 6 cm and an area of
20 cm2 . Find the angle of the sector.
a y = ¡ sin(3x)
.
23 Given that tan 2A = sin A where sin A 6= 0, find cos A
in simplest radical form.
¢
¼
c tan ¡ 3
4
2
22 Find the exact value of cos 79± cos 71± ¡ sin 79± sin 71± .
b 140± to radians.
radians to degrees
b f : x 7! sec(2x)
³x´
¡
y
39 If sin x = 2 sin x ¡
("r , 3)
Q
¢
, find the exact value of tan x.
40 In a busy harbour, the time difference between successive high
tides is about 12:3 hours. The water level varies by 2:4 metres
between high and low tide. Tomorrow, the first high tide will
be at 1 am, and the water level will be 4:7 metres at this time.
(¼, 1)
P
¼
6
x
(Ef" , -1)
a Find a sine model for the height of the tide H in terms of
time t tomorrow.
b Sketch a graph of the water level in the harbour tomorrow.
For the illustrated sine function, find the coordinates of the
points P and Q.
15
Mathematics HL – Exam Preparation & Practice Guide (3rd edition)