trigonometry
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Introduction• In this chapter you will learn about secant,
cosecant and cotangent, based on cosine, sine and tan
• We will also look at the inverse functions of sine, cosine and tan, known as arcsin, arccos and arctan
• We will build on the Trigonometric Equation solving from C2
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
All 3 are undefined if cosθ, sinθ or
tanθ = 0
1 1xx
1 1coscos
You should remember the index law:
It is NOT written like this in Trigonometry
1cos sec
sintancos
coscotsin
so
Something which will be VERY useful later in the chapter…
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
Example QuestionsWill cosec200 be positive or negative?1cosec200
sin 200
90 180
270 360y = Sinθ
As sin200 is negative, cosec200 will be as well!
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
Example QuestionsFind the value
of:
to 2dpsec 280
1sec 280cos 280
sec 280 5.76
Just use your calculator!
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
Example QuestionsFind the value
of:
to 2dpcot115
1cot115tan115
cot115 0.47
Just use your calculator!
-60
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
Example QuestionsWork out the exact value of:
(you may need to use surds…)
sec 210
1sec 210cos 210
y = Cosθ90 18
0270 360
210
30-60
By symmetry, we will get the same value for cos210 at cos30 (but with the reversed sign)
1sec 210cos30
1sec 2103
2
2sec 2103
2 3or 3
Cos30 = √3/2
Flip the denominator
TrigonometryYou need to know the
functions secantθ, cosecantθ and cotangentθ
6A
1seccos
1cosecsin
1cottan
Example QuestionsWork out the exact value of:
(you may need to use surds…)
3cosec4
3 1cosec34 sin4
3π/4
Sin(3π/4) = Sin(π/4)
Sin(π/4) = Sin45 1/√2
π/2 π 3π/2 2πy = Sinθ
π/4
3 1cosec4 sin
4
3 1cosec 142
3cosec 24
Flip the denominator
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
90 180
270 360y = Sinθ
1
0-1
y = Cosecθ
1cosecsin
At 90°, Sinθ = 1 Cosecθ = 1
At 180°, Sinθ = 0 Cosecθ =
undefined We get an asymptote wherever
Sinθ = 0
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
90 180
270 360
y = Cosθ
1
0-1
y = Secθ
1seccos
At 0°, Cosθ = 1 Secθ = 1
At 90°, Cosθ = 0 Secθ =
undefined We get asymptotes wherever Cosθ = 0
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
1cottan
90 180
270 360
y = Tanθ
y = Cotθ
At 45°, tanθ = 1 Cotθ =
1 At 90°, tanθ = undefined Cotθ =
0
At 180°, tanθ = 0 Cotθ =
undefined
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
90 180
270 360y = Sinθ
10
-1
90 180
270 360
1
0-1
y = Cosecθ
Maxima/Minima at (90,1) and (270,-1)
(and every 180 from then)
Asymptotes at 0, 180, 360
(and every 180° from then)
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
Maxima/Minima at (0,1) (180,-1) and (360,1)(and every 180 from
then)
Asymptotes at 90 and 270
(and every 180° from then)
90 180
270 360
y = Cosθ
1
0-1
90 180
270 360
1
0-1
y = Secθ
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
6B
1seccos
1cosecsin
1cottan
Asymptotes at 0, 180 and 360
(and every 180° from then)
90 180
270 360
y = Tanθ
90 180
270 360
y = Cotθ
-1
TrigonometryYou need to know the
graphs of secθ, cosecθ and cotθ
Sketch, in the interval 0 ≤ θ ≤ 360, the graph of:
6B
1seccos
1cosecsin
1cottan
1 sec 2y
90 180
270 360
1
0-1
y = Secθ
secy
sec 2y
1 sec 2y
Horizontal stretch, scale factor 1/2
Vertical translation, 1 unit
up
90 180
270 360
1
0
y = Sec2θ
y = 1 + Sec2θ
2
TrigonometryYou need to be able to simplify expressions, prove identities and
solve equations involving secθ, cosecθ
and cotθ
This is similar to the work covered in C2, but there are
now more possibilities
As in C2, you must practice as much as possible in order to get
a ‘feel’ for what to do and when…
6C
1seccos
1cosecsin
1cottan
Example QuestionsSimplify
…sin cot sec
sin cos sin
1
cos
sin cos sin cos
1
Remember how we can rewrite cotθ from earlier?
sintancos
coscotsin
Group up as a single
fraction
Numerator and
denominator are equal
TrigonometryYou need to be able to simplify expressions, prove identities and
solve equations involving secθ, cosecθ
and cotθ
This is similar to the work covered in C2, but there are
now more possibilities
As in C2, you must practice as much as possible in order to get
a ‘feel’ for what to do and when…
6C
1seccos
1cosecsin
1cottan
Example QuestionsSimplify
… sin cos sec cosec
Rewrite the part in brackets
sintancos
coscotsin
1 1sin coscos sin
sin cossin cossin cos sin cos
sin cossin cossin cos
sin cos sin cossin cos
sin cos
Multiply each fraction by the opposite’s
denominator
Group up since the denominators are now
the same
Multiply the part on top by the part
outside the bracket
Cancel the common factor to
the top and bottom
Trigonometry
6C
1seccos
1cosecsin
1cottan
Show that:
32 2
cot cosec cossec cosec
Rewrite both
sintancos
coscotsin
2 2
cot cosecsec cosec
Left side
cot cosec Numerat
orDenominat
or
cossin
1 sin
2
cossin
Group up
2 2sec cosec
2 2
1 1cos sin
2 2
2 2 2 2
sin coscos sin cos sin
2 2
2 2
sin coscos sin
Rewrite both
Group up
Multiply by the
opposite’s denominat
or
2 2
cot cosecsec cosec
2 2
1cos sin
From C2 sin2θ+ cos2θ = 1
2
2 2
cossin
1cos sin
2
cossin
2 2
1 cos sin
2 2cos sin 1
2
cossin
Putting them together
Replace numerator
and denominat
or
This is just a division
Change to a multiplicatio
n
3 2
2
cos sinsin
3cos
Group up
Simplify
TrigonometryYou need to be able to simplify expressions, prove identities and
solve equations involving secθ, cosecθ
and cotθ
You can solve equations by rearranging them in terms of
sin, cos or tan, then using their respective graphs
6C
1seccos
1cosecsin
1cottan
Example Question
sintancos
coscotsin
Solve the equation:
sec 2.5 In the range:
0 360
1 2.5cos
sec 2.5
1 cos2.5
cos 0.4
1cos 0.4
90 180
270 360
y = Cosθ
10
-1
113.6
246.4
Rewrite using cos
Rearrange
Work out the fraction
Inverse cos
Work out the first answer. Add 360 if not in the range we
want…Subtract from 360 (to find the equivalent value in the range
TrigonometryYou need to be able to simplify expressions, prove identities and
solve equations involving secθ, cosecθ
and cotθ
You can solve equations by rearranging them in terms of
sin, cos or tan, then using their respective graphs
6C
1seccos
1cosecsin
1cottan
Example Question
sintancos
coscotsin
Solve the equation:
cot 2 0.6 In the range:
0 360
cot 2 0.6 Rewrite
using tan 1tan 20.6
0 360
0 2 720
Remember to adjust the acceptable
range for 2θ
1 12 tan0.6
Inverse tan
2 59.04
Work out the first value, and others in
the original range (0-360)
y = Tanθ90 18
0270 360
, 239.04You can add 180 to these as the period of tan is
180 2 419.04 , 599.04
29.5, 120, 210, 300 Divide all by 2 (answers to
3sf)
TrigonometryYou need to be able to simplify expressions, prove identities and
solve equations involving secθ, cosecθ
and cotθ
You can solve equations by rearranging them in terms of
sin, cos or tan, then using their respective graphs
6C
1seccos
1cosecsin
1cottan
Example Question
sintancos
coscotsin
Solve the equation:In the
range:0 360
Rewrite each side
Cross multiply
Divide by Cosθ
Divide by 2
Rewrite the right-hand
side
Trigonometry
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
5tan12
A
Given that:
and A is obtuse, find the exact value of secA
Example Question
θ
tan OppAdj
5
12
13
cos AdjHyp
Ignore the negative, and use Pythagoras to work out the missing
side…
12cos13
90 180
270 360
y = Cosθ
10
-1
12cos13
13sec12
Replace A and H from the triangle…
A is obtuse (in the 2nd quadrant)
Cos is negative in this range
Flip the fraction to get Secθ
Trigonometry
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
5tan12
A
Given that:
and A is obtuse, find the exact value of
cosecA
Example Question
θ
tan OppAdj
5
12
13
sin OppHyp
Ignore the negative, and use Pythagoras to work out the missing
side…
5sin13
90 180
270 360
10
-1
5sin13
13cosec5
Replace A and H from the triangle…
A is obtuse (in the 2nd quadrant)
Sin is positive in this range
Flip the fraction to get Secθ
y = Sinθ
TrigonometryYou need to know and be able to use
the following identities
You might be asked to show where these come from…
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
2 21 tan sec 2 21 cot cosec
2 2sin cos 1
2 2
2 2 2
sin cos 1cos cos cos
2tan 1 2 sec
Divide all by cos2θ
Simplify each part
TrigonometryYou need to know and be able to use
the following identities
You might be asked to show where these come from…
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
2 21 tan sec 2 21 cot cosec
2 2sin cos 1
2 2
2 2 2
sin cos 1sin sin sin
1 2 cot 2 cosec
Divide all by sin2θ
Simplify each part
Trigonometry
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
2 21 tan sec 2 21 cot cosec
Prove that:Example Question
24 4
2
1 coscosec cot1 cos
4 4cosec cot
2 2 2 2cosec cot cosec cot
2 2 2 2cosec cot 1 cot cot
2 2cosec cot
2
1sin
2
2
cossin
2
2
1 cossin
2 2sin cos 1
2
2
1 cos1 cos
Factorise into a double bracket
Replace cosec2θ
1The second bracket =
1
Left hand side
Rewrite
Group up into 1
fraction
Rearrange the bottom (as in C2)
Trigonometry
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
2 21 tan sec 2 21 cot cosec
Prove that:Example Question
2 2 2 2sec cos sin (1 sec )
2 2sin cos 1
2 2sin (1 sec )
2 2 2sin sin sec
2 22
1sin sincos
22
2
sinsincos
2 2sin tan
21 cos 2 sec 1
2 2sec cos
Multiply out the bracket
Right hand side
Replace sec2θ
Rewrite the second term
Replace the fraction
Rewrite both terms based on the inequalities
The 1s cancel out…
This requires a lot of practice and will be slow to begin with. The more questions
you do, the faster you will get!
Trigonometry
6D
1seccos
1cosecsin
1cottan
sintancos
coscotsin
2 21 tan sec 2 21 cot cosec
Solve the Equation:
Example Question 24cosec 9 cot
2 2sin cos 1
24cosec 9 cot
24 1 cot 9 cot
24 4cot 9 cot
24cot cot 5 0
(4cot 5)(cot 1) 0
5cot4
cot 1 or
4tan5
tan 1 or
in the interval: 0 360
y = Tanθ90 18
0270 360
38.7, 219 135, 315
Replace cosec2θ
4/5
-1
A general strategy is to replace terms until they are all of the same type (eg
cosθ, cotθ etc…)
Multiply out the bracket
Group terms on the left
sideFactorise
Solve
Invert so we can use the tan graph
Use a calculator for the first answer
Be sure to check for others in the given range
Trigonometry
6E
Undefined√311/√3 or √3/30Tanθ
00.51/√2 or √2/2√3/21Cosθ
1√3/21/√2 or √2/20.50Sinθ
90°60°45°30°0°
Copy and complete, using surds where appropriate…
Trigonometry
6E
Undefined√311/√3 or √3/30Tanθ
00.51/√2 or √2/2√3/21Cosθ
1√3/21/√2 or √2/20.50Sinθ
π/2π/3
π/4π/60
The same values apply in radians as well…
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
These are the inverse functions of sin, cos and tan respectively
However, an inverse function can only be drawn for a one-to-one
function
(when reflected in y = x, a many-to-one function would become
one-to many, hence not a function)
6E
π/2-π/2
-1
1
1
π/2
-π/2
-1
y = sinx
y = arcsinx
y = sinxDomain: -π/2 ≤ x ≤ π/2Range: -1 ≤ sinx ≤
1
y = arcsinxDomain: -1 ≤ x ≤ 1Range: -π/2 ≤ arcsinx ≤
π/2
y = x
Remember that from a function to its inverse, the domain and range
swap round (as do all co-ordinates)
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
These are the inverse functions of sin, cos and tan respectively
However, an inverse function can only be drawn for a one-to-one
function
(when reflected in y = x, a many-to-one function would become
one-to many, hence not a function)
6E
π/2-1
1
π
-1y = cosx
y = arccosx
y = cosxDomain: 0 ≤ x ≤ π
Range: -1 ≤ cosx ≤ 1
y = arccosxDomain: -1 ≤ x
≤ 1Range: 0 ≤ arccosx ≤ π
y = x
Remember that from a function to its inverse, the domain and range
swap round (as do all co-ordinates)
π/2
π
1
We can’t use –π/2 ≤ x ≤ π/2 as the domain for cos,
since it is many-to-one…
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
These are the inverse functions of sin, cos and tan respectively
However, an inverse function can only be drawn for a one-to-one
function
(when reflected in y = x, a many-to-one function would become
one-to many, hence not a function)
6E
y = tanxDomain: -π/2 < x < π/2 Range: x ε
R
y = arctanxDomain: x ε R
Range: -π/2 < arctanx < π/2
y = tanx
y = arctanx
π/2-π/2
π/2
-π/2
Subtle differences… The domain for tanx cannot equal
π/2 or –π/2
The range can be any real number!
Trigonometry
6E
1
-π/2
-1
y = arcsinx -1
y = arccosx
π/2
π
1
y = arctanx
π/2
-π/2
π/2
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
Work out, in radians, the value of:
6E
arcsin(0.5)
1sin (0.5)
arcsin(0.5)
6
30
Arctan just means inverse
sin…Remember the
exact values from earlier…
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
Work out, in radians, the value of:
6E
arctan( 3)
1tan ( 3)
arctan( 3)
3
60
Arctan just means inverse
tan…Remember the
exact values from earlier…
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
Work out, in radians, the value of:
6E
2arcsin2
Arcsin just means inverse
sin…
Ignore the negative for now, and remember the
values from earlier…2arcsin2
1 2sin2
1 2sin2
4
1 2sin2
4
45
Sin(-θ) = -Sinθ(or imagine the Sine
graph…)
π/2-π/2
-1
1 y = sinx
You need to be able to use the inverse trigonometric
functions, arcsinx, arccosx and arctanx
Work out, in radians, the value of:
√2/2
-√2/2
π/4
-π/4
TrigonometryYou need to be able to use the
inverse trigonometric functions, arcsinx, arccosx
and arctanx
Work out, in radians, the value of:
6E
Arcsin just means inverse
sin…
cos arcsin 1
You need to be able to use the inverse trigonometric
functions, arcsinx, arccosx and arctanx
Work out, in radians, the value of:
cos arcsin 1
1cos sin 1
cos2
cos2
0
Think about what value you need
for x to get Sin x = –1
π/2-π/2
-1
1 y = sinx
Cos(-θ) = Cos(θ)
π/2-π/2
-1
1Remember it, or read from the graph…
y = cosx
Summary
6E
• We have learnt about 3 new functions, based on sin, cos and tan
• We have seen some new identities we can use in solving equations and proof
• We have also looked at the inverse functions, arc sin/cos/tanx
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