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Email UsThis session It also introduces the unit circle and how it can be used to derive simple trigonometric identities. Upon mastering the content of this session, you will:
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T1.2 Radians
Understand the unit circle definition of
Understand the unit circle definition of
| Name | Prework | Homework |
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The trigonometric identities provides us with a means of simplifying and modifying trigonometric expressions.
It is very useful to translate angles into their first quadrant equivalent. This allows us to find alternate / simplified trigonometric expressions for angles.
We can use the rules below to translate angles from each quadrant into the first:
Starting Quadrant | Translation Process | Effect on Coordinates and Gradient |
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2nd Quadrant | Reflect across the y axis |
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3rd Quadrant | Draw the opposing parallel radius |
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4th Quadrant | Reflect across the x axis |
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This technique can be used to translate a trigonometric expressions with angles in the 2nd, 3rd or 4th quadrant into an expression with an angle from the 1st quadrant. An example of this can be seen below:
First Quadrant Equivalent Expressions
Imagine we wanted to convert
As we can see, the first quadrant equivalent angle for
Adding a full revolution to any angle brings you back to the same place on the unit circle, this property is known as periodicity. This works in degrees, radians and revolutions.
Since we return to the same place on the unit circle, the output of all the trigonometric functions remain the same. We can therefore conclude that all trigonometric functions are periodic.
Periodic Identities ( Units In Degrees )
Where
is any integer number
An application of trigonometric periodicity is shown below:
Trigonometric Periodicity Example
Imagine we wanted to find the exact value of
We can now use our special ratios and replace
The supplementary identities relate angles that sum to give
These angles are reflections of each other about the y-axis. This reflection negates the
Supplementary Identities ( Units In Degrees )
An application of supplementary angles is shown below:
Supplementary Angle Example
Imagine we wanted to find the exact value of
We can now use the special trigonometric ratios to get:
Similar to the supplementary identities, the complementary identities relate angles that add up to produce an angle of
By applying SOH CAH TOA to this triangle we can get the following identities:
Complementary Identities ( Units In Degrees )
An application of complementary angles is shown below:
Complementary Angle Example
Imagine a scenario where we know the exact value of
Therefore, we can conclude that:
Difficulty
00
Time
SOLUTION
Difficulty
3 Mins
Time
Find the first quadrant equivalent angle for the following:
SOLUTION
For each of these questions, we will draw the angle on the unit circle and then translate it into the first quadrant.
Part 1
Since
Part 2
Since
Part 3
Since
Difficulty
3 Mins
Time
Find the exact value of the following expressions:
SOLUTION
We can find the exact values of these expressions by utilising the supplementary trigonometric identities and the special triangles shown below.
Part 1
We can now utilise the exact values from the special triangles:
Part 2
We can now utilise the exact values from the special triangles:
Difficulty
5 Mins
Time
If
SOLUTION
We will begin answering both of these problems by finding the first quadrant equivalent angle.
Part 1
As we can see above, the first quadrant equivalent angle for
Part 2
As we can see above, the first quadrant equivalent angle for
At this stage we should realise that we can use the complementary identities to convert the sin into a cos.
Difficulty
4 Mins
Time
Simplify the expression below:
SOLUTION
To answer this question we must draw the first quadrant equivalent angle.
As we can see the angle represented by
At this stage, we should realise that this expression can be further simplified using the complementary identities.
Difficulty
7 Mins
Time
Show that the following identity is true:
SOLUTION
To show this property holds, we will simplify the LHS of the expression.
We notice that the inputs of each function is
As we can see, the first quadrant equivalent has the same
Putting these fractions on a common denominator gives:
We can now utilise the pythagorean identity to write this as:
Hence, this identity is true.
Difficulty
4 Mins
Time
Classify
SOLUTION
A function is defined to be even if
Sine Odd / Even
We can start by letting
From our unit circle we can see that the coordinates at
Therefore, sin is an odd function
Cosine Odd / Even
We can start by letting
From our unit circle we can see that the coordinates at
Therefore, cos is an even function
Tan Odd / Even
We can start by letting
From our unit circle we can see that the gradients at
Therefore, tan is an odd function
Difficulty
5 Mins
Time
Find the exact value of the expression below:
SOLUTION
To find the exact value of the entire expression, we must first know the values of each trigonometric expression. These can be found with the aid of the special triangles shown below.
To find the value of the tan expression, we can utilise the supplementary angle:
Finding The Value Of The Net Expression
By definition, two angles are supplementary if they:
Sum to any multiple of
Sum to
Sum to the angle in a semi-circle
Sum to the angle in three quarters of a revolution
Which of the following identities indicate that cosine is periodic?
A supplementary identity looks at points on a unit circle that are:
Rotated by some multiple of
Separated by some small constant angle
Reflected around the origin
Flipped along either the x or y axis (or both)
Note that it is is also true that:
Which indicates that
But since
We can repeat this for sine
By definition, two angles are supplementary if they:
Sum to the angle in a semi-circle
Sum to the angle in three quarters of a revolution
Sum to any multiple of
Sum to
Which of the following identities indicate that cosine is periodic?
A supplementary identity looks at points on a unit circle that are:
Rotated by some multiple of
Flipped along either the x or y axis (or both)
Reflected around the origin
Separated by some small constant angle
Note that it is is also true that:
Which indicates that
But since
We can repeat this for sine
Two angles are complementary if they:
Complete the angle required for a full revolution together
Sum to the angle of a quarter circle
Are reflected across either the x or y axis
Multiply together to produce some multiple of
Given the fact that
If points
It needs to be flipped along a line at
It’s angle needs to be halved
It needs to be rotated by an angle of
It’s angle needs to be doubled
Two angles are complementary if they:
Complete the angle required for a full revolution together
Are reflected across either the x or y axis
Sum to the angle of a quarter circle
Multiply together to produce some multiple of
Given the fact that
If points
It needs to be rotated by an angle of
It needs to be flipped along a line at
It’s angle needs to be halved
It’s angle needs to be doubled
