Cones β Part 2 Slideshow 48, Mathematics Mr Richard Sasaki Room 307.
An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.
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Transcript of An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.
![Page 1: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/1.jpg)
An Introduction to Trigonometry
Slideshow 44, MathematicsMr. Richard Sasaki, Room 307
![Page 2: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/2.jpg)
Objectivesβ’ Learn and recall some components of the
right-angled triangleβ’ Understand the meaning of sine, cosine and
tangent in terms of trianglesβ’ Understand the graphs
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The Right-Angled TriangleTrigonometry is like Pythagoras but includes angles. When we have a specified angle, the vocabulary is different.
πAngle
(Theta)
HypotenuseOpposite
Adjacent
Simple trigonometry involves 2 edges and an angle. If one thing is missing, how do we find it?
![Page 4: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/4.jpg)
Special Case TrianglesWe saw two basic cases, the 30-60-90 triangle and the 45-45-90 triangle.
30π1 2
β345π
1 β2
1We need to think about the relationship between the edge lengths and the angles. Any ideas?The relationship lies with the three main trigonometric functions, , and .sine cosine tangent
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Sine, Cosine and TangentSine, cosine and tangent are the relationships between edge lengths and angles.In calculation, sine, cosine and tangent are shown as , and respectively. We still usually refer to them by their full names though.Each refer to two of the edges.
π
Hypotenuse
Opp
osite
Adjacent
SineCosineTangentS
O HC
A HT
O AYou will need to remember these links.
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π
Sine, Cosine and TangentIn fact, sine, cosine and tangent are functions on angles which equates to the ratio of the corresponding two edges.
Hypotenuse
Opp
osite
Adjacent
π ππ (π )=ΒΏπππππ ππ‘π
π»π¦πππ‘πππ’π π
πππ (π )=ΒΏπ΄πππππππ‘
π»π¦πππ‘πππ’π π
π‘ππ (π )=ΒΏπππππ ππ‘ππ΄πππππππ‘
![Page 7: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/7.jpg)
hypotenuse
adjacent
opposite
0 90 0101 0β
As , neither are the hypotenuse so there is no restriction between their sizes.
12 β3
2
β33β2
2β221
![Page 8: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/8.jpg)
Trigonometric FunctionsUsing the same methods, we can calculate trigonometric values for .
30π60π1 2
β3
π ππ (60 )=ΒΏπππππ ππ‘π
π»π¦πππ‘πππ’π π=ΒΏβ32
πππ (60 )=ΒΏπ΄πππππππ‘
π»π¦πππ‘πππ’π π=ΒΏ12
π‘ππ (60 )=ΒΏπππππ ππ‘ππ΄πππππππ‘=ΒΏβ3With a little more imagination, we can do the
same for and .
![Page 9: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/9.jpg)
BoundariesWe do not have time to explore further but after looking at many values inserted in the trigonometric functions, we would have the following boundaries:
β€ π ππ (π )β€β1 1β€πππ (π )β€β1 1ΒΏ π‘ππ (π )<ΒΏββ β
This means the β axes corresponding to these graphs must obey these rules. What do these graphs look like?
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We know the graph must satisfy .We saw that , and . This isnβt enough data to draw it, but it looks like this:
12
β22
β32
1
β1
π (π₯)
180 360 540 720 90090
Note: We are using degrees, not radians.
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Again, the graph satisfies .We saw that , and . Itβs the opposite, right? Basically itβs like but shifted back :
β32
β22
12
1
β1
π (π₯)
180 360 540 720 90090
![Page 12: An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307.](https://reader036.fdocuments.net/reader036/viewer/2022062601/5a4d1bfd7f8b9ab0599ed11e/html5/thumbnails/12.jpg)
This graph has no boundaries about .What is happening? , and .
β33 1
β3
4
β4
π (π₯)
180 360 540 720 90090
The sizes are increasing.There is a cycle howeverβ¦like this:
Note: tends to infinity.