If the two equal sides are of length
, then the hypotenuse,
, can be calculated as:
So, we have:
We can try something similar for
and
. We start with an equilateral triangle and we bisect one angle as shown in
[link] . This gives us the right-angled triangle that we need, with one angle of
and one angle of
.
If the equal sides are of length
, then the base is
and the length of the vertical side,
, can be calculated as:
So, we have:
You do not have to memorise these identities if you know how to work them out.
Two useful triangles to remember
Alternate definition for
We know that
is defined as:
This can be written as:
But, we also know that
is defined as:
and that
is defined as:
Therefore, we can write
can also be defined as:
A trigonometric identity
One of the most useful results of the trigonometric functions is that they are related to each other. We have seen that
can be written in terms of
and
. Similarly, we shall show that:
We shall start by considering
,
We see that:
and
We also know from the Theorem of Pythagoras that:
So we can write:
Simplify using identities:
Prove:
Trigonometric identities
Simplify the following using the fundamental trigonometric identities:
Prove the following:
Reduction formula
Any trigonometric function whose argument is
,
,
and
(hence
) can be written simply in terms of
. For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of
. From this we may expect that
.
Function values of
Investigation : reduction formulae for function values of
Function Values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. If P' is the reflection of P about the
-axis (or the line
), use symmetry to write down the coordinates of P'.
Write down values for
,
and
.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the inversion of P through the origin (reflection about both the
- and
-axes) and lies at an angle of
with the
-axis. Write down the coordinates of P'.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Questions & Answers
I'm interested in biological psychology and cognitive psychology
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
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
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills