Transcript Trigonometry

1
Trigonometry
Trigonometry is
anything using
sin, cos or tan
Maths revision course by Miriam Hanks
2
Trigonometry - Radians
What is a radian ?
r
r
1c
r
One radian is the angle of a
circle which has an arc
length equal to the radius.
There are 2π radians in 360°
Maths revision course by Miriam Hanks
3
Trig – Radians to degrees
Converting:
radians
180  
degrees
degrees
180  
radians


Maths revision course by Miriam Hanks
4
Trigonometry – Exact values

Your exact values need to be learned:
30°
sin
cos
tan
1
2
3
2
1
3

6
45°

4
60° 
1
2
1
2
3
3
2
1
2
1
3
Maths revision course by Miriam Hanks
5
Trigonometry – Graphs
y = sin x
Maths revision course by Miriam Hanks
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Trigonometry – Graphs
y = cos x
Maths revision course by Miriam Hanks
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Trigonometry – Graphs
y = tan x
Maths revision course by Miriam Hanks
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Trigonometry – CAST
When do you use CAST?
180 - x
x
S
T
180 + x
A
C
360 - x
Use CAST quadrants
whenever you press
sin-1, or cos-1 or tan-1
on your calculator.
Maths revision course by Miriam Hanks
9
Trigonometry – formulae
Remember: sin2 x means (sinx)2
sin 2x = 2 sin x cos x
cos 2x = cos2 x – sin2 x
cos 2x = 2 cos2 x – 1
cos 2x = 1 – 2sin2 x
All these
formulae are
on the
formula sheet
Maths revision course by Miriam Hanks
10
Trigonometry – formulae
sin (p ± q) = sin p cos q ± cos p sin q
cos (p ± q) = cos p cos q + sin p sin q
These are also
on the formula
sheet
Maths revision course by Miriam Hanks
11
Trigonometry – formulae
sin x
tan x 
cos x
sin x  cos x  1
2
2
These 2 are NOT
on the formula
sheet
Maths revision course by Miriam Hanks
12
Solving trig equations

If there are “double angles and single
angles”, get rid of the double angles by
using the formulae on the sheet for sin 2A
or cos 2A.
single angle
eg
double angle
3sinx – 1 = cos 2x
3sinx – 1 = 1 – 2sin2x
Maths revision course by Miriam Hanks
13
Solving trig equations

If there is a squared term, then put
everything onto one side, and factorise.
squared term
3sinx – 1 = 1 – 2sin2x
2sin2x + 3sinx – 2 = 0
(2sinx – 1)(sinx + 2)= 0
sin x = ½ or sinx = -2
(impossible)
Maths revision course by Miriam Hanks
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Solving trig equations

If there is only one of sin cos or tan, then
solve by doing the inverse function:
sin x = ½
give answers 0 < x < 2
x = sin-1(½)
x= 
6
or



6
5
6
√
√
(π – x) S
A
T C
Maths revision course by Miriam Hanks
15
Trigonometry –The wave function
eg
sin x + 5cos x = 6
or
3sin x – 2cos x = -1
or
½cos x + 7sin x – 2= 0
If you have a sin x and a
cos x and a constant, it is
not possible to solve the
equation using the above
techniques, so we use the
wave function to reduce
it to a single trig function
Maths revision course by Miriam Hanks
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Trigonometry –The wave function
eg Express
3 cos x – 4 sin x into the form k cos (x + a)
Expand
k cos (x + a) using the
formula sheet first
Maths revision course by Miriam Hanks
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Trigonometry –The wave function
k cos (x + a) = kcos x cos a - k sin x sin a
Then put this equal to the function in the
question:
3 cos x – 4 sin x = kcos x cos a – k sin x sin a
Maths revision course by Miriam Hanks
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Trigonometry –The wave function
3 cos x – 4 sin x = kcos x cos a – k sin x sin a
Now compare how many cos x ’s you have on
each side of the equation:
3 = k cos a
Now compare how many sin x ’s you have on
each side of the equation:
4 = k sin a
Maths revision course by Miriam Hanks
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Trigonometry –The wave function
3 = k cos a
4 = k sin a
sin x
Use tan x 
to divide these equations:
cos x
tan a = 4/3
a = tan-1 (4/3)
a = 0.927 radians (to 3 dps)
Maths revision course by Miriam Hanks
20
Trigonometry –The wave function
3 = k cos a
4 = k sin a
Use Pythagoras to find k:
k = 32 +42
k=5
So, the answer is 5 cos (x + 0.927 )
Maths revision course by Miriam Hanks
21
Trigonometry –Max & min values
If you are asked for a maximum or minimum
value and the question is only worth one point,
then use the fact that sin of anything and cos of
anything have max & min values of 1 and –1.
eg What are the maximum and minimum values
of y when
y = 2 + 3cos(x – 40).
Answer: Since cos(x – 40) has max value 1,
then the max value of y is 5. What is the min?
Maths revision course by Miriam Hanks
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Trigonometry in real life
When is trigonometry
used in real life?
Tidal experts are
meteorologists use
sine waves to predict
tides. These are
invaluable to sailors
and kayakers when
planning a trip.
Maths revision course by Miriam Hanks