Download MAT1A01: Trigonometry (Appendix D)

MAT1A01: Trigonometry (Appendix D)
Dr Craig
6 February 2013
Information
Make sure that you have a MAT1A01 Learning
Guide. Extra copies are available at the department
(C-Ring).
My office: C-Ring 533A (Stats Dept corridor)
Consulting hours: Monday 12pm-1pm, Wednesday
12pm-1pm
Email: [email protected]
Tel: 011 559 3024
Or, just drop in and see if I am there. If I am free, I
will help.
Radian measure
In this course, angles will be measured in radians.
Radians measure the ratio between the arc length
and the radius. When a =arc length, we have:
a
θ=
and a = r θ.
r
Note: these formulas are only valid when θ is
measured in radians.
Example: 90◦ = π/2 radians.
How do we get this?
Consider a circle of radius 1. The circumference of
the circle is given by 2πr = 2π × 1 = 2π. So, the
arc length of 90◦ is 2π/4 = π/2.
∴
90◦ = (π/2)/1 = π/2 radians.
Formulas to convert between degrees and radians:
1 radian =
180
π
1◦ =
Example: convert
(a) 36◦ to radians,
(b) −7π/2 to degrees.
π
radians.
180
30◦ =
π
6
45◦ =
π
4
60◦ =
π
3
90◦ =
π
2
120◦ =
2π
3
135◦ =
3π
4
150◦ =
5π
6
180◦ = π
270◦ =
3π
2
360◦ = 2π
Trigonometric functions
Trig functions take as input an angle (measured in
radians) and output the ratio between two distances.
hypotenuse
opposite
θ
adjacent
sin θ =
opp
hyp
cos θ =
adj
hyp
tan θ =
opp
adj
csc θ =
hyp
opp
sec θ =
hyp
adj
cot θ =
adj
opp
More generally, angles can be measured in a
coordinate system:
θ>0
θ<0
A positive and negative angle drawn in the standard
position.
Trig functions in a coordinate system
P(x, y )
θ
r
sin θ =
y
r
cos θ =
x
r
tan θ =
y
x
csc θ =
r
y
sec θ =
r
x
cot θ =
x
y
Signs of trig functions
sin θ > 0
S
A
All > 0
tan θ > 0
T
C
cos θ > 0
Special angles
π
6
√
π/4
√
2
2
1
π/3
π/4
1
1
Example: calculate all of the trig ratios for
θ = 4π/3.
3
Trig identities
Trig identities are useful relationships between trig
functions. Some of the basic identities are:
1
1
csc θ =
sec θ =
sin θ
cos θ
cot θ =
1
tan θ
cot θ =
cos θ
sin θ
tan θ =
sin θ
cos θ
More trig identities
sin2 θ + cos2 θ = 1
More trig identities
sin2 θ + cos2 θ = 1
Divide both sides by cos2 θ to get
1
sin2 θ cos2 θ
+
=
cos2 θ cos2 θ
cos2 θ
∴
tan2 θ + 1 = sec2 θ
Now divide both sides of the original by sin2 θ:
sin2 θ cos2 θ
1
+
=
sin2 θ sin2 θ
sin2 θ
∴
1 + cot2 θ = csc2 θ
Addition formulas
sin(x + y ) = sin x. cos y + cos x. sin y
cos(x + y ) = cos x. cos y − sin x. sin y
How can we use
sin(x + y ) = sin x. cos y + cos x. sin y
to get a formula for
sin(x − y )?
Addition formulas
sin(x + y ) = sin x. cos y + cos x. sin y
cos(x + y ) = cos x. cos y − sin x. sin y
How can we use
sin(x + y ) = sin x. cos y + cos x. sin y
to get a formula for
sin(x − y )?
Result:
sin(x − y ) = sin x. cos y − cos x. sin y
We can also use the addition formulas to get the
double-angle formulas:
sin(2x) = 2 sin x. cos x
cos(2x) = cos2 x − sin2 x
cos(2x) = 2 cos2 x − 1
cos(2x) = 1 − 2 sin2 x
Example: Find all of the values of x in the interval
[0, 2π] such that sin x = sin 2x.
Graphs of trig functions
Graph of f (x) = sin(x). Note: −1 6 sin x 6 1.
Question: why is sin(π/2) = 1?
Graph of f (x) = cos(x). Again, −1 6 cos x 6 1.
Note: cos 0 = 1.
Also sin(x + 2π) = sin x and cos(x + 2π) = cos x.
Graph of f (x) = tan(x).
Note the vertical asymptotes at π/2 and −π/2.
Graph of f (x) = csc(x).
Note: | csc(x)| > 1, x ∈
/ { z.π | z ∈ Z }.
Graph of f (x) = csc(x) and g (x) = sin(x).
Note: | csc(x)| > 1, x ∈
/ { z.π | z ∈ Z }.
Graph of f (x) = sec(x).
Graph of f (x) = cot(x).
Note: cot(x) = 0 wherever tan(x) has an
asymptote.
Graph of f (x) = cot(x) and g (x) = tan(x).
Note: cot(x) = 0 wherever tan(x) has an
asymptote.