Download MATH19730 Part 1 Section2 Trigonometry

MATH19730 Part 1
Trigonometry
Section2
Trigonometry provides methods to relate angles and lengths but the functions
we define have many other applications in mathematics.
An angle can be measured in degrees or radians. There are 360 degrees or
2π radians in a full revolution. This gives us a formula for converting between
the two measurements.
180
× (angle in radians)
π
π
π
For example 90◦ =
rads (approx 1.57 rads), 30◦ =
rads (approx 0.52
2
6
◦
rads), 245 = 4.28 rads to 2 decimal places.
angle in degrees =
2.1 Trigonometric ratios
Let ABC be a right angled triangle containing the angle θ (exercise: draw and
label a triangle to fit with the information below).
We define
sin(θ) =
AC
opposite
=
hypotenuse
AB
cos(θ) =
adjacent
BC
=
hypotenuse
AB
tan(θ) =
AC
opposite
=
.
adjacent
BC
Note that
tan(θ) =
sin(θ)
.
cos(θ)
These definitions can be extended to include all angles, not just angles between 0 and 90o , using the following rules:
• sin(180◦ − θ) = sin(θ)
• cos(180◦ − θ) = − cos(θ)
• sin(−θ) = − sin(θ)
• cos(−θ) = cos(θ)
• sin(360◦ + θ) = sin(θ)
• cos(360◦ + θ) = cos(θ)
• tan(180◦ + θ) = tan(θ)
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Sine and cosine values lie between -1 and +1. Tangent values can take any value
(and are undefined for certain values of θ).
This is a lot to remember but if you know and remember the shapes of
the graphs of these functions (see Section 2.4), plus just a few facts, such as
sin(0) = 0, cos(0) = 1, then you should be able to reconstruct what you need.
The following are sometimes useful special cases:
√
1
3
◦
◦
◦
◦
sin(60 ) = cos(30 ) =
sin(30 ) = cos(60 ) = ,
2
2
1
sin(45◦ ) = cos(45◦ ) = √ ,
2
tan(45◦ ) = 1
Examples
1. Convert the following from degrees to radians, accurate to 2 decimal
places.
(i)31◦
(ii) 225◦
2. Convert the following from radians to degrees, accurate to 1 decimal place.
3π
(i) 1.76 rads
(ii)
rads
4
3. Find the exact value of the following.
2π
(angle in radians)
(i) sin(135o )
(ii) cos( )
3
π
(angle in radians)
(iii) tan(− )
3
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2.2 Trigonometric identities
As well as the three trigonometric functions defined above we also define
sec(θ) =
1
cos(θ)
cosec(θ) =
cot(θ) =
1
sin(θ)
1
tan(θ)
The following identities are true for all values of θ and are really just expressions
of Pythagoras’ Theorem5 about right-angled triangles.
• sin2 (θ) + cos2 (θ) = 1
(where sin2 (θ) is shorthand for (sin(θ))2 )
• tan2 (θ) + 1 = sec2 (θ)
• cot2 (θ) + 1 = cosec2 (θ).
The following compound angle formulas are true for any values of A and B.
• sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
• cos(A + B) = cos(A) cos(B) − sin(A) sin(B)
• sin(2A) = 2 sin(A) cos(A)
• cos(2A) = cos2 (A) − sin2 (A)
Examples
1. Use the trigonometric identities given above to show that for all x,
(i) (sin(x) − cos(x))(1 + sin(x) cos(x)) = sin3 (x) − cos3 (x)
(ii) tan(2x) =
2 tan(x)
.
1 − tan2 (x)
2. By using the compound angle formula show that
√
1+ 3
◦
√ .
sin(75 ) =
2 2
5 This also arose in many different cultures; for instance, it’s known as the Gougu Gao
Theorem in China.
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2.3 Sine and Cosine rules
Take any triangle (not necessarily right-angled) with angles A, B, C and sides
a, b, c opposite to A, B, C respectively. Given partial information about certain
angles and lengths of sides we can use sine and cosine to determine the missing
angles and lengths.
Sine Rule:
a
b
c
=
=
sin(A)
sin(B)
sin(C)
This rule is useful if we know (i) two angles and a side or (ii) two sides and a
non-included angle.
Cosine Rule:
a2 = b2 + c2 − 2bc cos(A)
This rule is useful if we know (i) three sides or (ii) two sides and an included
angle.
Examples
Calculate all the missing angles and sides in the following triangles ABC.
1. A = 40◦ , B = 30◦ , a = 10
2. a = 3, b = 7, c = 5
3. a = 4, b = 6, B = 25◦
4. b = 5, c = 8, A = 100◦
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2.4 Graphs of Sine, Cosine and Tangent
So far we have thought of the trigonometric functions as acting on angles or,
more generally, rotations (so we distinguish between, say, a rotation anticlockwise of π/4 and a rotation clockwise by 9π/4 even though those two rotations
leave us pointing in the same direction). An angle measured in radians can be
thought of as just a real number, so then we can think of sin(x), cos(x) and
tan(x) as being defined for all real numbers x. In this case we always measure
x in radians.
The graphs of these three functions are given below.
We can see that each one is periodic, meaning that the functions repeat the
same values over a fixed period. Sine and cosine have period 2π and tangent
has period π.
You might have noticed that the angle interpretation of the trig functions is
a bit strange when the angle is ±π/2 (or any integer multiple of π/2) because
then the “triangle” gets squeezed to a line. Sine and cosine still can be made
sense of in that case but not the tangent function which is, therefore, undefined
at ±π/2, ±2π/2, ±3π/2, . . . (and we really do mean undefined, not “infinity”).
These functions have many applications in maths and science apart from
simply dealing with angles.
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Examples
1. Sketch the graphs of the functions sin(2x) and cos(x −
differ from sin(x) and cos(x)?
π
). How do they
3
2. Sketch the graphs of cosec(x), sec(x) and cot(x).
3. Find all the solutions of the following equations for x in radians.
1
(ii) cos(x) = 0
(iii) tan(x) = −1
(i) sin(x) =
2
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2.5 Inverse Trigonometric Functions
From the graphs of sin(x), cos(x) and tan(x) we can see that there are infinitely
many values of x that give the same value of the function. This causes problems
when we want to go back from the value of the function to the value of the
angle that it came from - because there are infinitely many angles that could
have given that value of the function. For example how do we solve
sin(x) = 0?
There are infinitely many solutions to this equation, x = 0, ±π, ±2π, ...
We can define an inverse operation to taking sine if we choose a standard interval
of width π and restrict the values that this inverse can take to that interval.
We’ll choose our standard interval for the inverse sine function6 arcsin to run
from − π2 to π2 (excluding − π2 itself to avoid a repetition). That is, define
π
π
arcsin(x) to be the unique value y with − < y ≤ and sin(y) = x.
2
2
So arcsin(0.5) =
π
rads, arcsin(−0.8) = −0.93 rads.
6
Similarly we can define inverse functions for cosine and tangent.
The inverse cosine, arccos(x), of x is the unique value y with 0 ≤ y ≤ π and
cos(y) = x. (Note we chose a different standard interval for this but, if you
think about the shape of the graph of cos, it does make sense to do so.)
.
6 The notation sin−1 (x) is also used in place of arcsin(x) but we avoid this, and corresponding notations for the other inverse trig functions, here because that notation might (wrongly!)
suggest the meaning 1/ sin(x).
18
The inverse tangent, arctan(x), of x is the unique value y with −
and tan(y) = x.
π
π
<y <
2
2
When we solve equations involving trigonometric functions we should consider
all possible solutions initially, not just those one our chosen interval. Some of
these may be later discarded due to physical restrictions on the range of possible
values (e.g. we may need only positive solutions).
For example consider the equation
cos(5y) = −0.5
(∗)
2π
. From the graph of cos(x) we see
3
that we get the same value of cos(x) = −0.5 for
Using a calculator we get arccos(−0.5) =
x=
2π
4π
± 2nπ and x =
± 2nπ for any n = 0, 1, 2, . . .
3
3
So the solution to (∗) is
y=
4π 2nπ
2π 2nπ
±
and y =
±
for any n = 0, 1, 2, . . .
15
5
15
5
This gives values y = . . . , −
4π 2π 2π 4π 8π
,− ,
,
,
,...
15
15 15 15 15
Examples
1. Calculate in radians: (i) arcsin(0.2)
(ii) arccos(−0.75)
2. Find all solutions to the following equations.
1
(ii) tan(3x) = −1
(i) sin(4x) = √
2
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(iii) arctan(−30).