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Chapter 2 Circular (trigonometric) functions
69
remember
remember
1
1
1
1. cosec x = ----------sec x = -----------cot x = -----------sin x
cos x
tan x
2. Symmetry properties of trigonometric functions and their inverses
(a) First quadrant:
π
π
sin --- − θ = cos θ ⇔ cosec --- − θ = sec θ
2
2
π
π
cos --- − θ = sin θ ⇔ sec --- − θ = cosec θ
2
2
(b) Second quadrant:
sin(π − θ) = sin θ ⇔ cosec(π − θ) = cosec θ
cos(π − θ) = −cos θ ⇔ sec(π − θ) = −sec θ
tan(π − θ) = −tan θ ⇔ cot(π − θ) = −cot θ
π
π
sin --- + θ = cos θ ⇔ cosec --- + θ = sec θ
2
2
π
π
cos --- + θ = −sin θ ⇔ sec --- + θ = −cosec θ
2
2
(c) Third quadrant:
sin(π + θ) = −sin θ ⇔ cosec(π + θ) = −cosec θ
cos(π + θ) = −cos θ ⇔ sec(π + θ) = −sec θ
tan(π + θ) = tan θ ⇔ cot(π + θ) = cot θ
3π
3π
sin ------ − θ = −cos θ ⇔ cosec ------ − θ = −sec θ
2
2
3π
π
cos ------ − θ = −sin θ ⇔ sec --- − θ = −cosec θ
2
2
(d) Fourth quadrant:
sin(−θ) = −sin θ ⇔ cosec(−θ) = −cosec θ
cos(−θ) = cos θ ⇔ sec(−θ) = sec θ
tan(−θ) = −tan θ ⇔ cot(−θ) = −cot θ
3π
3π
sin ------ + θ = −cos θ ⇔ cosec ------ + θ = −sec θ
2
2
3π
3π
cos ------ + θ = sin θ ⇔ sec ------ + θ = cosec θ
2
2
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2A
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Example
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Reciprocal trigonometric
functions
1 Copy and complete the following
table using the right-angled
triangles at the top of page 70.
Give exact values for:
i sin x
ii cos x iii tan x
iv cosec x v sec x vi cot x.
sin x
a
b
c
d
e
f
cos x
tan x
cosec x
sec x
cot x
83
Chapter 2 Circular (trigonometric) functions
Compound and double angle formulas
Compound angle formulas
Consider the right-angled triangles shown in the figure at right.
Let AD = 1, ∠BAC = x and ∠DAC = y.
Then
AC = cos y
CD = sin y
1
∠DCE = x (since ∠BCA = 90° − x and ∠ECB = 180°)
⇒ DE = sin x sin y
⇒ CE = cos x sin y
y
⇒ BC = sin x cos y
x
⇒ AB = cos x cos y
A
D
E
C
F
B
BE = sin(x + y) (as BE = FD)
= BC + CE
E
D
= sin x cos y + sin y cos x
∴
sin(x + y) = sin x cos y + cos x sin y
C
and
AF = cos(x + y)
= AB − BF
1
= AB − DE (since DE = BF)
sin (x + y)
= cos x cos y − sin x sin y
∴
cos(x + y) = cos x cos y − sin x sin y.
y
Using a similar approach, or by replacing y
x
with −y, the following identities can also be
A
F
B
derived:
cos (x + y)
1.
sin(x − y) = sin x cos(−y) + cos x sin(−y)
= sin x cos y − cos x sin y (since cos(−y) = cos y and sin(−y) = −sin y)
2.
cos(x − y) = cos x cos(−y) − sin x sin(−y)
= cos x cos y + sin x sin y.
Furthermore:
sin ( x + y )
tan(x + y) = ------------------------cos ( x + y )
Now
sin x cos y + cos x sin y
= ----------------------------------------------------------cos x cos y – sin x sin y
Dividing the numerator and denominator by cos x cos y, this simplifies to:
tan x + tan y
tan(x + y) = ---------------------------------1 – tan x tan y
Similarly:
tan x – tan y
tan(x − y) = ----------------------------------1 + tan x tan y
Note: These identities can also be derived using a unit circle approach.
In summary, the compound angle formulas are:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
tan A + tan B
tan(A + B) = --------------------------------------1 – tan A tan B
sin(A − B) = sin A cos B − cos A sin B
cos(A − B) = cos A cos B + sin A sin B
tan A – tan B
tan(A − B) = ---------------------------------------1 + tan A tan B
90
Maths Quest 12 Specialist Mathematics
π
π
10 If cos x = −0.4, x ∈ ---, π and cosec y = 1.25, y ∈ 0, --- , find the value of each of
2
2
the following, correct to 2 decimal places.
a sin x
b tan(x + y)
c cos y
d cos(x − y)
11 Simplify each of the following expressions using the double angle formulae.
x
2 tan --a sin2x − cos2x
b sin x cos x
2
c -----------------------x
2
1 – tan --2
cos 2x
sin 2 x – sin 4 x
e ------------------------------d ------------------------------f (sin x − cos x)2
sin 2x
sin 4 x – cos 4 x
π
12 If cos x = −0.6 and x ∈ ---, π , find the values of:
2
16
a cos 2x
b sin 2x
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Example
xample
π
13 If tan x = 2 and x ∈ 0, --- , find the exact values of:
2
a tan 2x
b sin 2x
14 If sin A =
a sin 2A
1
------5
c
tan 2x.
c
cos 2x.
π
and x ∈ 0, --- , find the exact values of:
2
b cos 2A
c
sin 3A
A
d sin2 --- .
2
π
15 If cos B = 0.7 and x ∈ 0, --- , then find each of the following, correct to 2 decimal
2
places:
B
B
B
B
a cos --b sin --c tan --d cot --- .
2
2
2
2
Work
ET
SHE
2.1
16 Use the double angle formulas to find the exact values of:
π
π
a sin --b cos --c
8
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More identities
Prove the following identities.
1 cot x sec x = cosec x
2 (1 + cot2x)(1 – cos2x) = 1
1
3 (1 + sin x)(1 – sin x) = -----------sec2 x
4 cosec2x + sec2x = cosec2x sec2x
π
tan --- .
8
Chapter 2 Circular (trigonometric) functions
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Example
xample
19
97
4 Simplify each of the following.
a sin(sin−1 0.5)
b cos(sin−1 0.5)
c tan sin−1 ------2-
π
e sin−1 ⎛ cos--- ⎞
⎝ 6⎠
π
f cos−1 tan ⎛– --- ⎞
⎝ 4⎠
5π
g tan−1 ⎛ 2 cos------ ⎞
⎝
6⎠
2π
h sin−1 ⎛ cos------ ⎞
⎝
3⎠
5π
i cos−1 ⎛ sin ------ ⎞
⎝
4 ⎠
7π
j sin−1 ⎛ tan ------ ⎞
⎝
4 ⎠
k tan−1[cos (−π)]
11 π
l cos−1 ⎛ cos --------- ⎞
⎝
6 ⎠
(
2
)
d cos(tan−1 3 )
5 multiple
ultiple choice
π
Consider the function y = sin−1(x + 1) + --- .
3
a The implied domain of this function is:
A [−2, 0]
B [−1, 1]
C (−1, 1)
D [0, 2]
E (0, 2)
b The range of this function is:
A
π π
– ---, --2 2
B
5π π
– ------, --6 6
5π π
C ⎛– ------, --- ⎞
⎝ 6 6⎠
D
π 5π
– ---, -----6 6
E R
6 multiple
ultiple choice
Consider the function with the rule f (x) = cos−1 (sin 2x).
a The implied domain of f (x) is:
A [−π, π ]
B
π π
– ---, --4 4
π π
– ---, --2 2
C
D R
E [0, π ]
D [0, π]
E [−1, 1]
b The range of f (x) is:
A [−2, 2]
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Example
xample
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B R
[
C − 1--- ,
2
1
--2
]
7 State the i implied domain and ii range of each of the following.
a y = sin−1(1 − x)
b y = cos−1(x + 1)
c
y = tan−1(2x)
d y = sin−1(x2)
e y = cos−1(2x + 3)
f
y = tan−1(4 − x)
g y = cos−1(x2 − 1)
h y = sin−1(cos x)
i
y = tan(2 sin−1x)
k y = cos(2 sin−1x)
l
y = cos−1( 3 tan x)
j
y = tan−1(sin x)
8 Sketch the graph of the following functions and state the i implied domain
and ii range of each.
21
π
b y = 2 sin−1x + --a y = tan−1(x + 1)
2
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Math
y = cos−1(x − 1)
π
e f (x) = 3 tan−1x + --4
g f (x) =
1
--3
π
sin−12x + --3
d y = sin−1(2 − x)
f
f (x) = 2 cos−1 x +
(
1
--2
)
π
h f (x) = cos−1(x + 2) − --2
Graphs of
inverse
trig.
functions
ET
SHE
Work
c
cad
Example
xample
2.2