Introduction to Trigonometry
Core 2 and 3
Trig basics
A
normal
axes…
Radius
Draw
inOP
aset
unit
is of
the
circle
hypotenuse
of
(aacircle
right-angled
of radiustriangle
1)
whose
adjacent
side(1,0)
is given
Now take
the point
and rotate it
by
the x-coordinate
P origin,
anticlockwise
aboutofthe
and
whose
opposite
side is
so that
radius
OP makes
an angle θ
given
by the
y-coordinate
with the
positive
x-axis of P.
+
y = sin θ
(0,1)
cos θ = adj / hyp
and hyp = radius = 1
=> x-coord. of P = adj = cos θ
O
sin θ = opp / hyp
and hyp = radius = 1
=> y-coord. of P = opp = sin θ
Hence no matter how far round the circle P moves,
x = cos θ and y = sin θ
P
θ
-
(Note: sin & cos are still in alphabetical order – or you can think
of the vertical axis as a “sine-post”)
x θ
x = cos
(1,0)
+
-
One more thing: Knowing that tan θ = opp / adj,
how could you express tan θ in terms of sin θ and cos θ?
Sketching trig graphs
Can you predict what the graph of y = sin θ will look like as θ changes?
Think about what is happening at each stage:
y = sin θ
• At the starting point (1,0), what is sin θ?
• What is sin θ when you get to (0,1)?
• What about when θ=45 degrees,
(0,1)
i.e. when P is half way from (1,0) to (0,1)?
P
• At what angle will sin θ be ½?
• What will happen in the other quadrants?
• What will happen if you go round again?
x = cos θ
θ
O
(1,0)
How about the graph of y = cos θ?
What will that look like?
+
-
For more of a challenge,
can you predict how the graph of y = tan θ will look?
Click here to see
the sine curve
Click here to see
the cosine curve
+
Click here to see
the tangent curve
Trig mnemonics
Think about the quadrants where the different trig ratios are positive
(the “Curves compared” slide may help you with this)…
90°y = sin θ
Between 0 and 90°,
• sin θ goes from 0 up to 1
• cos θ goes from 1 down to 0
• tan θ = sin θ / cos θ = +ve / +ve
so sin, cos and tan θ are ALL positive.
Between 90 and 180°,
• sin θ goes from 1 down to 0
• cos θ goes from 0 down to -1
• tan θ = sin θ / cos θ = +ve / -ve
so only sin θ is positive.
+
180°
-
Carry on and fill in the other two quadrants.
The traditional mnemonic for this diagram is
“All Stations To Coventry”
- but you can invent your own if you prefer!
S A
O
T C
x = cos θ
+ 0°360°
270°
Curves compared
Solving trig equations 1
Now let’s look at how this diagram helps us to identify different angles
with the same value of sin θ, cos θ or tan θ.
Imagine that you are asked to solve the
equation sin θ = 0.4 for 0 ≤ θ ≤ 360°.
Nowprincipal
The
use this diagram
solution, to
θ (in
findthis
thecase
23.6°), issolution:
second
the acute angle that your
calculator
The
sine curve
givesisyou
symmetrical
for “sin-10.4”.
about
θ
180°
However,
90°,
so you
the
reflect
symmetry
the line
of showing
the curve means
that there
where
θ is,will
in be
theanother
90° line,solution
i.e. the in the
same period.
vertical
axis.
Look at the sine curve and see if you can
How
work far
outround
wherehave
it willwe
be.come from the 0° line?
90°y = sin θ
+
180°-θ
S A
O
T C
-
θ
x = cos θ
+ 0°360°
So the second solution in the range
270°
specified is 180° - θ, giving 180 - 23.6 = 156.4°. Answer: Add 360° to each of
What would the additional solutions be
the previous solutions.
if the range were 0 ≤ θ ≤ 720°?
Sine curve Cosine curve Tangent curve
Solving trig equations 2
Now try equations
solving cos
= 0.8
Solving
in θtan
θ is easier,
for 0 ≤ θ ≤the
360°.
because
tangent curve simply
Use the itself
sameevery
approach,
repeats
180°. but this time
we havethat
a different
symmetry.
(Check
you canline
seeofthis
from
the tangent graph.)
So all
you need
is extend the
The
cosine
curvetoisdo
symmetrical
line showing
principal
angle,
about
0° and the
about
180°, so
the
straight through
horizontal
axis isthe
ourorigin.
reflection line.
90°y = sin θ
+
Try solving tan θ = 0.8 for 0 ≤ θ ≤ 360°. 180°+θ
How far round have we come
from the 0° line this time?
S A
180°
θ
O
T C
θ
θ
x = cos θ
+ 0°360°
360°-θ
Solution: θ = 38.7°, 218.7°
Solution: θ = 36.9°, 323.1°
270°
What would the additional solutions be for the interval -360 ≤ θ ≤ 360°?
Answer: Subtract 360° from each of
the previous solutions.
Sine curve Cosine curve Tangent curve
Solving trig equations 3
We already know that the first quadrant of the graph
contains the principle value of θ.
90°
Now we also know how to find the second
solution in a given period:
• With sin θ, the second solution
is given by 180°- θ
• With cos θ, it’s 360°- θ
180°
• With tan θ, it’s 180°+ θ
-
+
y = sin θ
180°- θ
θ
S A
O
T
If solutions are required outside
the interval 0 to 360° then
180°+θ
we add 360° to / subtract 360° from
each solution already found, until we have
all the solutions in the interval specified.
270°
Sine curve
x = cos θ
0°
+ 360°
C
360°-θ
Cosine curve
±360°
Tangent curve
Solving trig equations in radians
The same approach can be used for problems
where the angle is in radians.
π/2
π radians are equivalent to 180 degrees so:
+
• With sin θ, the second solution
is given by π - θ
• With cos θ, it’s 2π - θ
• With tan θ, it’s π + θ
π
-
y = sin θ
π-θ
θ
S A
O
T
If solutions are required outside
the interval 0 to 2π then
π+θ
we add 2π to / subtract 2π from
each solution already found, until we have
all the solutions in the interval specified.
3π/2
x = cos θ
0
+ 2π
C
2π-θ
±2π
Calculus with trig (Core 3)
If y = sin x then dy/dx = cos x
If y = cos x then dy/dx = -sin x
It’s essential that you remember
when the sign needs to change,
otherwise you could lose a lot of exam marks!
The same axes we used for “ASTC” can help
you here.
All you have to remember is to go
• clockwise for differentiation
• anticlockwise for integration.
+
y = sin θ
Int.
Diff.
x = cos θ
+
O
So differentiating, we have:
sin x → cos x → -sin x → -cos x → sin x
And integrating, we have:
sin x → -cos x → -sin x → cos x → sin x
-
Remember that calculus can only be used with trig functions
if the angle is in radians!
Sine curve
Cosine curve
Tangent curve
Back to “Sketching trig graphs” slide
Cosine curve
Sine curve
Tangent curve
Back to “Sketching trig graphs” slide
Tangent curve
Cosine curve
Sine curve
Back to “Sketching trig graphs” slide
Curves compared
y = sin x
y = cos x
y = tan x
Back to “Trig mnemonics” slide