vikasana – bridge course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · old hippie...
TRANSCRIPT
![Page 1: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/1.jpg)
Vikasana – Bridge Course 20121
![Page 2: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/2.jpg)
What is trigonometry ?What is trigonometry ?
Trigonometry is a branch of Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles gbetween these sides.
Vikasana – Bridge Course 20122
![Page 3: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/3.jpg)
What is trigonometry ?What is trigonometry ?The word Trigonometry comes f t G k d from two Greek words,
trigonon = triangle, and trigonon triangle, and
metron = measure.
Vikasana – Bridge Course 20123
![Page 4: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/4.jpg)
Origin of Trigonometry
The method was originated in the second
Origin of Trigonometry
century B.C. by Hipparchus and other early Greek astronomers in their attempts to solve certain spherical triangles. The term trigonometry was not used until the close of the sixteenth century.
Vikasana – Bridge Course 20124
![Page 5: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/5.jpg)
Labeling Right Angled TrianglesThe most important skill you need The most important skill you need right now is the ability to correctly l b l h d f h l d label the sides of a right angled triangle.The names of the sides are:
the hypotenusethe hypotenusethe opposite sidethe adjacent side
Vikasana – Bridge Course 2012the adjacent side
5
![Page 6: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/6.jpg)
Labeling Right Triangles
The hypotenuse is easy to locate because it is side opposite to the right pp gangle.
Since this side is across from the
Here is the right angle
across from the right angle, this must be the h tright angle... hypotenuse.
Vikasana – Bridge Course 20126
![Page 7: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/7.jpg)
Before you label the other two Before you label the other two sides you must have a reference angle selectedangle selected.
It can be either of the two acute anglesangles.
Vikasana – Bridge Course 20127
![Page 8: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/8.jpg)
In the triangle below, let’s pick angle B In the triangle below, let s pick angle B as the reference angle.
B This will be our reference angle...
A C
Vikasana – Bridge Course 20128
![Page 9: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/9.jpg)
Remember, angle B is our reference gangle.The hypotenuse is side BC because it is The hypotenuse is side BC because it is across from the right angle.
B (ref angle)B (ref. angle)
hypotenuse
A CVikasana – Bridge Course 2012
A C
9
![Page 10: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/10.jpg)
Side AC is opposite to reference Side AC is opposite to reference angle B. So it is labeled: opposite.
B (ref. angle)
hypotenuse
A Copposite
Vikasana – Bridge Course 201210
![Page 11: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/11.jpg)
The only side unnamed is side AB. yThis must be the adjacent side.
B (ref. angle)
adjacent h tadjacent hypotenuse
A Copposite
Vikasana – Bridge Course 201211
![Page 12: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/12.jpg)
TRIGONOMETRIC RATIOS:
There are six trigonometric ratios:Sine abbreviation sinSine abbreviation sinCosine abbreviation cosTangent abbreviation tanTangent abbreviation tanSecant abbreviation secCosecant abbreviation cosecCosecant abbreviation cosecCotangent abbreviation cot
Vikasana – Bridge Course 201212
![Page 13: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/13.jpg)
Let us define these trigonometric gratios for right angled triangle:
C
site
AB
op
po
s
AB adjacent
Vikasana – Bridge Course 201213
![Page 14: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/14.jpg)
Vikasana – Bridge Course 201214
![Page 15: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/15.jpg)
Vikasana – Bridge Course 201215
![Page 16: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/16.jpg)
We need a way We need a way to remember all of these ratiosratios…
16
![Page 17: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/17.jpg)
SomeOldHippieHippie
CameAHoppin’
Old Hi iOld Hi i
ppThroughOurOld HippieOld Hippie OurApartment
17
![Page 18: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/18.jpg)
O A OA Sin
Opp
SOH CAH TOAOpp
Hyp
Cos
Adjj
Hyp
T
Old Hi iOld Hi iTanOpp
Old HippieOld Hippie Adj18
![Page 19: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/19.jpg)
Consider the right angled triangle g g gwith sides 3, 4 and 5 units
3 5CHere opp = 3
adj = 4Here the angle A is
denoted as θ. It’ G k l tt
4 ABhyp = 5It’s a Greek letter.
Vikasana – Bridge Course 201219
![Page 20: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/20.jpg)
Keen Eye:As hypotenuse is always yp ygreater than the other sides of the right angled triangle. the right angled triangle. There fore sin and cos are < 1Sec and cosec > 1 for acute Sec and cosec > 1 for acute angle.
Vikasana – Bridge Course 201220
![Page 21: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/21.jpg)
In the ∆ABC, , AB=12cm and BC=5cm then find the values of all trigonometric ratios of the angle C and A.g g
A
By Pythagoras theorem12 13
By Pythagoras theorem
B C5Vikasana – Bridge Course 2012
5
21
![Page 22: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/22.jpg)
In the ∆ABC, , AB=12cm and BC=5cm then find the values of all trigonometric ratios of the angle C and A.g g
AThen AB is opposite side;
12 13C is the ref. angleBC is adjacent side and AC is hypotenuse.
B C5Vikasana – Bridge Course 2012
5
22
![Page 23: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/23.jpg)
AA
12 13
B C5
Vikasana – Bridge Course 201223
![Page 24: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/24.jpg)
ALet us consider A as ALet us consider A as reference angle.Then
12 13Then opposite side=BC=5;adjacent side=AB=12;
B C5
adjacent side AB 12;hypotenuse = AC = 13.
Vikasana – Bridge Course 201224
![Page 25: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/25.jpg)
AA
12 13
B C5
Vikasana – Bridge Course 201225
![Page 26: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/26.jpg)
If tanA=4/3, then find all other trigonometric ratios of A.
Given tanA=4/3. There fore opp=4
4 5and adj = 3
A3
Vikasana – Bridge Course 201226
![Page 27: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/27.jpg)
Vikasana – Bridge Course 201227
![Page 28: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/28.jpg)
If 15cotA=8,find sinA and secAIf 15cotA 8,find sinA and secA
Given cotA=8/15
A
15 17Opp=15; adj=8h 17 A8hyp=17
Vikasana – Bridge Course 201228
![Page 29: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/29.jpg)
If then find the value of If , then find the value of
Clearly, opp=8, adj=7 and hyp= 8yp
7 Vikasana – Bridge Course 2012
7
29
![Page 30: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/30.jpg)
Vikasana – Bridge Course 201230
![Page 31: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/31.jpg)
KEEN EYEKEEN EYE
sinA is one symbol and sinAT i t i l ti d d l sinA is one symbol and sinAdoes not mean sin X A. Similar, is In short, trigonometric ratios are
written as t-ratiosEvery t-ratio is a real number.
Trigonometrical ratio depends only upon the magnitude of the angle
means (sinA)(sinA). It is
l itt the case for cosA and tanA. written as t-ratios.and upon the length of sides.also written as
Vikasana – Bridge Course 201231
![Page 32: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/32.jpg)
Alternative definition of T-ratios
P(x y)P(x,y)yr
MxO
Vikasana – Bridge Course 201232
![Page 33: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/33.jpg)
It is used in satellite systems and astronomy, aviation, engineering, astronomy, aviation, engineering, land surveying, geography and many other fieldsmany other fields.
Vikasana – Bridge Course 201233
![Page 34: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/34.jpg)
Vikasana – Bridge Course 201234
![Page 35: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/35.jpg)
Vikasana – Bridge Course 201235
![Page 36: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/36.jpg)
Vikasana – Bridge Course 201236
![Page 37: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/37.jpg)
Vikasana – Bridge Course 201237
![Page 38: Vikasana – Bridge Course 2012kea.kar.nic.in/vikasana/bridge/maths/chap_17_ppt.pdf · Old Hippie Came A Hoppin’ Old Hi i pp ... Vikasana – Bridge Course 2012 31. Alternative](https://reader033.vdocuments.us/reader033/viewer/2022053022/604efb1908f25533e60c59a4/html5/thumbnails/38.jpg)
Vikasana – Bridge Course 201238