Now that we have graphs for
and
, there is an easy way to visualise the tangent graph. Let us look back at our definitions of
and
for a right-angled triangle.
This is the first of an important set of equations called
trigonometric identities . An identity is an equation, which holds true for any value which is put into it. In this case we have shown that
for any value of
.
So we know that for values of
for which
, we must also have
. Also, if
our value of
is undefined as we cannot divide by 0. The graph is shown in
[link] . The dashed vertical lines are at the values of
where
is not defined.
Functions of the form
In the figure below is an example of a function of the form
.
Functions of the form
:
On the same set of axes, plot the following graphs:
Use your results to deduce the effect of
.
On the same set of axes, plot the following graphs:
Use your results to deduce the effect of
.
You should have found that the value of
affects the steepness of each of the branches. The larger the absolute magnitude of
a , the quicker the branches approach their asymptotes, the values where they are not defined. Negative
values switch the direction of the branches.
You should have also found that the value of
affects the vertical shift as for
and
.
These different properties are summarised in
[link] .
Table summarising general shapes and positions of graphs of functions of the form
.
Domain and range
The domain of
is all the values of
such that
is not equal to 0. We have already seen that when
,
is undefined, as we have division by zero. We know that
for all
, where
is an integer. So the domain of
is all values of
, except the values
.
The range of
is
.
Intercepts
The
-intercept,
, of
is again simply the value of
at
.
Asymptotes
As
approaches
,
approaches infinity. But as
is undefined at
,
can only approach
, but never equal it. Thus the
curve gets closer and closer to the line
, without ever touching it. Thus the line
is an asymptote of
.
also has asymptotes at
, where
is an integer.
Graphs of trigonometric functions
Using your knowldge of the effects of
and
, sketch each of the following graphs, without using a table of values, for
Give the equations of each of the following graphs:
The following presentation summarises what you have learnt in this chapter.
Summary
We can define three trigonometric functions for right angled triangles: sine (sin), cosine (cos) and tangent (tan).
Each of these functions have a reciprocal: cosecant (cosec), secant (sec) and cotangent (cot).
We can use the principles of solving equations and the trigonometric functions to help us solve simple trigonometric equations.
We can solve problems in two dimensions that involve right angled triangles.
For some special angles, we can easily find the values of sin, cos and tan.
We can extend the definitions of the trigonometric functions to any angle.
Trigonometry is used to help us solve problems in 2-dimensions, such as finding the height of a building.
We can draw graphs for sin, cos and tan
End of chapter exercises
Calculate the unknown lengths
In the triangle
,
cm,
cm and
. The perpendicular line from
to
intersects
at
. Calculate
the length
,
the length
, and
the angle
A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder?
A ladder of length 25 m is resting against a wall, the ladder makes an angle
to the wall. Find the distance between the wall and the base of the ladder?
In the following triangle find the angle
In the following triangle find the length of side
and
. Find the angle between the line through A and B and the x-axis.
and
. Find the angle between the line through C and D and the y-axis.
A
ladder is placed
from the wall. What is the angle the ladder makes with the wall?
Given the points: E(5;0), F(6;2) and G(8;-2), find angle
.
An isosceles triangle has sides
and
. Find the size of the smallest angle of the triangle.
A right-angled triangle has hypotenuse
. Find the length of the other two sides if one of the angles of the triangle is
.
One of the angles of a rhombus (
rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter
is
.
Find the sides of the rhombus.
Find the length of both diagonals.
Captain Hook was sailing towards a lighthouse with a height of
.
If the top of the lighthouse is
away, what is the angle of elevation of the boat to the nearest integer?
If the boat moves another
towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
(Tricky) A triangle with angles
and
has a perimeter of
. Find the length of each side of the triangle.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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Jharna
continue..
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Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
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Jharna
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Jharna
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