Class 10 Exam > Class 10 Videos > Trigonometric Identities
Trigonometric Identities Video Lecture - Class 10
FAQs on Trigonometric Identities Video Lecture - Class 10
| 1. What are some basic trigonometric identities? |  |
Ans. Some basic trigonometric identities include:
- Pythagorean identities: sin^2(x) + cos^2(x) = 1 and tan^2(x) + 1 = sec^2(x)
- Reciprocal identities: csc(x) = 1/sin(x), sec(x) = 1/cos(x), and cot(x) = 1/tan(x)
- Quotient identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x), and sec(x) = 1/cos(x)
| 2. How do trigonometric identities help in solving equations and simplifying expressions? | |
Ans. Trigonometric identities allow us to rewrite expressions and equations in different forms, which often make them easier to solve or simplify. By substituting known identities, we can often reduce complex trigonometric expressions into simpler forms, making calculations more manageable.
| 3. Can you provide an example of how to use a trigonometric identity to simplify an equation? | |
Ans. Sure! Let's say we have the equation sin^2(x) + cos^2(x) = 2sin^2(x) - 1. We can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify it.
By substituting 1 for sin^2(x) + cos^2(x), the equation becomes 1 = 2sin^2(x) - 1. Rearranging the terms, we get 2sin^2(x) = 2. Dividing both sides by 2, we have sin^2(x) = 1.
Therefore, using the trigonometric identity, we simplified the equation from an initial form to a simpler form.
| 4. How can trigonometric identities be used to prove other mathematical relationships? | |
Ans. Trigonometric identities can be used to prove other mathematical relationships by demonstrating that both sides of an equation are equivalent. By manipulating one side of the equation using known identities, we can transform it into the other side, thereby proving the relationship between the two.
For example, to prove the double angle identity for cosine (cos(2x) = cos^2(x) - sin^2(x)), we can start with the identity cos(2x) = 2cos^2(x) - 1. By using the Pythagorean identity sin^2(x) = 1 - cos^2(x), we can substitute it into the equation and simplify to cos^2(x) - sin^2(x), which matches the right side of the double angle identity.
| 5. How can trigonometric identities be applied in real-life situations? | |
Ans. Trigonometric identities are widely used in various real-life situations, such as engineering, physics, and navigation. They are crucial in solving problems involving angles, distances, heights, and oscillations. For example, in engineering, trigonometric identities are used to calculate forces, analyze structures, and design circuits. In physics, these identities help in studying wave motion, harmonics, and vibrations. In navigation, they assist in determining distances, directions, and positions using trigonometric functions and identities.
Text Transcript from Video
hello friends this video on introduction
to trigonometry part 12 is brought to
you by exam viacom no more fear from
exam before watching this video please
make sure that you have watched part 1
to part 11 now let's understand what is
called trigonometric identity equation
is called pigna metric identity is true
it is true for all values of the value
so any statement this true for all the
values will give me any value you can
develop theta if a particular equation
is true
the principle is amended e so they are
three rating one is cos square a plus
sine square is everyone you can any
value of e zero degree it will be
readily modely this will always be
clearly 1 plus tan square a is equal to
6 wearing this is always we and God
square a plus 1 is going to kha'zix
quietly this statement will also be true
always so these are the three
trigonometric identities
please remember science Phi plus cos
square is 31 1 plus tan square is equal
to 6 square cot square plus 1 when you
cause expert the three trigonometric
identities very important please
understand this please learn this will
prove this thing no metric identities in
the next so first we have to prove cos
square a plus sine square is ready the
next thing is angle a right triangle
this is hypotenuse if we have taken a as
the trig no preference
this is BC and this is perpendicular
correct cos a again we'll use formula
contemporary fashion arrey arrey born to
sign cause a little and so cos square a
will be equal to B by H square V by H
correct similarly
science we're even be building P by H
square for any angle theta so cos square
a plus sine square a will be V by H
square plus P by H square it has nothing
but B square plus Q square by X square
we know that since if you the right
triangle v square plus P Square will be
equal to X square by hypotenuse square
is equal to v square plus perpendicular
square so this becomes X square and by X
square and that is nothing but so we
have proved that cos square equal sine
square is but do nor it's true 1 plus
tan square a is equal to 6 for a again
under Bhaduri Prashad hooray hooray horn
sine cos tan Kozik SEC and cotton this
is going this is a reference this is H
since is the angle we are talking about
my to your reference this becomes me and
this becomes P so I will get 1 plus tan
a is nothing but P by B e square by B
Square right this is nothing but these
four common so tan is P by B so this is
P equals G square common B Square loves
you sir
all right d square plus P Square is s
square so this becomes X square by B
Square and H by P Square is nothing bad
if you see SEC square is sex
and that is what we as official we
approve that 1 plus tan square a it is P
by B square is equal to SEC square so
this is also true for any value of theta
you take any value of theta this will be
let's see the last idea T cot square a
plus 1 is written cos it square again we
use funding by the refresh our array
everyone so sine she scoffs this is
stamp this is Kasich is SEC and business
card since theta is my angle a this is H
this will be base because the site is
the same and this is perpendicular so
cot cot as nothing but this one be by P
me by P Square plus 1 this is nothing
but v square plus P Square by P Square
and B square plus P spies nothing but X
square and right triangle by Pythagoras
theorem by P square H by P is nothing
but if you see H by P is nothing but cos
it little H right this becomes cos x
square and hence this is true for any
value of theta any LOV sorry we have to
express the trigonometric ratio cause
has sake a tanning also signing what we
can do we know the identity sine square
a plus cos square a is equal to 1 so
with this I can get cos i in terms of
sine is what will get cos square a is
nothing but 1 minus sine square e or I
get coffee is equal to root of 1 minus
sine squared a so cos a is the sick is
nothing but 1 by cos it this is nothing
but 1 by root 1 minus sine square a so I
could write se Cayo so in terms of
signing done Tana is nothing but sign a
by causing
sign a is okay because we have write in
terms of sign a only cos e I applied in
some tombow signing
I know causes were 2 root of 1 minus
sine square
thus evident tan a also in terms of sine
cos st. se que caning everything is not
in terms of signing and that is meinster
seeing all these questions when you are
asked to write say a or the in terms of
sign a or cos a or daddy you have to use
other degree vertical entry didn't see
let's take up for example you have to
write all the trigonometric ratios in
terms of sick so first thing we know is
cos is going to one by sinking that the
easiest thing with correct so cos a is
done so what we have to do is sign cause
and Kozik sake is any means there God
always your play cos is done correct
sign we know is nothing but to talk one
minus cos square e the sine square theta
plus cos square a is 1 cosine is nothing
but dou double - prospering cause I've
already written in terms of second so
alike this is nothing but 1 minus 1 by 6
e square for the sine and cos I have
everything actually see sign in cause I
have in terms of say K Tana is nothing
but sign anyway cos it or tan is also 6
square minus 4/3 formula you can either
say Tana is good to sign away Casa de
sign is in the time of six cyclic cos a
is the lower say K akin to the
or I can say six square a tiny
involution square a mess over here user
deletion cotton will be reverse of
tanning that is won by tanning correct
since Danny have already written in
terms of six for a is nothing but 1 by
root down six query Minister then Kasich
is nothing but 1 by signing and this is
nothing but sign is this one this is
nothing one you solve this further just
comes on to be six and A six will be in
history this is nothing but group of six
squared a minus one by sake so Kasich is
one mess in a will become sec a by group
of six value - so if you see I got all
the trigonometric ratio income how sick
what I've done cos a form a pound sign
in and then I can find anything we'll
try this example sine square 63 plus
sine square 27 by cos square seventeen
by cos square 73 this example where we
convert sign to cos a cos cosine so
let's convert everything to cause law in
this case sine 63 I can write as nothing
burns sine of ninety degree minus
seventy meaning square plus sine square
27 over I can i from the gas ninety
degree minus mrs. toosey by this will
become cos of 1700 square 73 so if you
see this this won't get canceled right
because this becomes cos seventy this
cousin
but this is non-dairy cancer for this
approach one one best thing once again
you see 63 plus 27 is equal to 1917 the
17 is equal to 90 so this vs. is not 90
this belt also not 90 this is not
forming a player but 63 verse 27 is
performing a pair and desperate this is
forming a pair so what we can do so if I
can convert this into cos square so
there is a gong sine square 63 is equal
cause by 63 sine square plus cos Phi is
equal to 1 here his cause if I can
convert this into sine we call square X
the signs for a living one because this
end is formula P so let's in this way
this become sine square 63 plus sine
square 90 minus 27 by cos square
seventeen plus cos square 19 giving
minus 17 degree 90 minus X if we ask why
because this is forming a pair 90
systems 53 a 27 so this comes out to be
sine square 63 plus cos square 16 by cos
square seventeen by sine squared 70 sine
square theta plus cos square theta is
equal to 1 cos square theta plus sine
square theta spending money so one by
one went to 1 and that is mad early we
tried to convert the one the given
earlier to cost but that didn't work out
so then we observe that 63 plus 27 is
going to 1972 the 73 is between nineteen
these are forming a pl so we can read
this sign to cause in this cos to sign
and we got sine square theta plus cos
square theta pattern that became one
click thank you visit exam viacom to
watch free education videos try free
online tests in the best quality so
Dimitri
is study from the best tutors and
mentors and mushroom thanks a serene
to trigonometry part 12 is brought to
you by exam viacom no more fear from
exam before watching this video please
make sure that you have watched part 1
to part 11 now let's understand what is
called trigonometric identity equation
is called pigna metric identity is true
it is true for all values of the value
so any statement this true for all the
values will give me any value you can
develop theta if a particular equation
is true
the principle is amended e so they are
three rating one is cos square a plus
sine square is everyone you can any
value of e zero degree it will be
readily modely this will always be
clearly 1 plus tan square a is equal to
6 wearing this is always we and God
square a plus 1 is going to kha'zix
quietly this statement will also be true
always so these are the three
trigonometric identities
please remember science Phi plus cos
square is 31 1 plus tan square is equal
to 6 square cot square plus 1 when you
cause expert the three trigonometric
identities very important please
understand this please learn this will
prove this thing no metric identities in
the next so first we have to prove cos
square a plus sine square is ready the
next thing is angle a right triangle
this is hypotenuse if we have taken a as
the trig no preference
this is BC and this is perpendicular
correct cos a again we'll use formula
contemporary fashion arrey arrey born to
sign cause a little and so cos square a
will be equal to B by H square V by H
correct similarly
science we're even be building P by H
square for any angle theta so cos square
a plus sine square a will be V by H
square plus P by H square it has nothing
but B square plus Q square by X square
we know that since if you the right
triangle v square plus P Square will be
equal to X square by hypotenuse square
is equal to v square plus perpendicular
square so this becomes X square and by X
square and that is nothing but so we
have proved that cos square equal sine
square is but do nor it's true 1 plus
tan square a is equal to 6 for a again
under Bhaduri Prashad hooray hooray horn
sine cos tan Kozik SEC and cotton this
is going this is a reference this is H
since is the angle we are talking about
my to your reference this becomes me and
this becomes P so I will get 1 plus tan
a is nothing but P by B e square by B
Square right this is nothing but these
four common so tan is P by B so this is
P equals G square common B Square loves
you sir
all right d square plus P Square is s
square so this becomes X square by B
Square and H by P Square is nothing bad
if you see SEC square is sex
and that is what we as official we
approve that 1 plus tan square a it is P
by B square is equal to SEC square so
this is also true for any value of theta
you take any value of theta this will be
let's see the last idea T cot square a
plus 1 is written cos it square again we
use funding by the refresh our array
everyone so sine she scoffs this is
stamp this is Kasich is SEC and business
card since theta is my angle a this is H
this will be base because the site is
the same and this is perpendicular so
cot cot as nothing but this one be by P
me by P Square plus 1 this is nothing
but v square plus P Square by P Square
and B square plus P spies nothing but X
square and right triangle by Pythagoras
theorem by P square H by P is nothing
but if you see H by P is nothing but cos
it little H right this becomes cos x
square and hence this is true for any
value of theta any LOV sorry we have to
express the trigonometric ratio cause
has sake a tanning also signing what we
can do we know the identity sine square
a plus cos square a is equal to 1 so
with this I can get cos i in terms of
sine is what will get cos square a is
nothing but 1 minus sine square e or I
get coffee is equal to root of 1 minus
sine squared a so cos a is the sick is
nothing but 1 by cos it this is nothing
but 1 by root 1 minus sine square a so I
could write se Cayo so in terms of
signing done Tana is nothing but sign a
by causing
sign a is okay because we have write in
terms of sign a only cos e I applied in
some tombow signing
I know causes were 2 root of 1 minus
sine square
thus evident tan a also in terms of sine
cos st. se que caning everything is not
in terms of signing and that is meinster
seeing all these questions when you are
asked to write say a or the in terms of
sign a or cos a or daddy you have to use
other degree vertical entry didn't see
let's take up for example you have to
write all the trigonometric ratios in
terms of sick so first thing we know is
cos is going to one by sinking that the
easiest thing with correct so cos a is
done so what we have to do is sign cause
and Kozik sake is any means there God
always your play cos is done correct
sign we know is nothing but to talk one
minus cos square e the sine square theta
plus cos square a is 1 cosine is nothing
but dou double - prospering cause I've
already written in terms of second so
alike this is nothing but 1 minus 1 by 6
e square for the sine and cos I have
everything actually see sign in cause I
have in terms of say K Tana is nothing
but sign anyway cos it or tan is also 6
square minus 4/3 formula you can either
say Tana is good to sign away Casa de
sign is in the time of six cyclic cos a
is the lower say K akin to the
or I can say six square a tiny
involution square a mess over here user
deletion cotton will be reverse of
tanning that is won by tanning correct
since Danny have already written in
terms of six for a is nothing but 1 by
root down six query Minister then Kasich
is nothing but 1 by signing and this is
nothing but sign is this one this is
nothing one you solve this further just
comes on to be six and A six will be in
history this is nothing but group of six
squared a minus one by sake so Kasich is
one mess in a will become sec a by group
of six value - so if you see I got all
the trigonometric ratio income how sick
what I've done cos a form a pound sign
in and then I can find anything we'll
try this example sine square 63 plus
sine square 27 by cos square seventeen
by cos square 73 this example where we
convert sign to cos a cos cosine so
let's convert everything to cause law in
this case sine 63 I can write as nothing
burns sine of ninety degree minus
seventy meaning square plus sine square
27 over I can i from the gas ninety
degree minus mrs. toosey by this will
become cos of 1700 square 73 so if you
see this this won't get canceled right
because this becomes cos seventy this
cousin
but this is non-dairy cancer for this
approach one one best thing once again
you see 63 plus 27 is equal to 1917 the
17 is equal to 90 so this vs. is not 90
this belt also not 90 this is not
forming a player but 63 verse 27 is
performing a pair and desperate this is
forming a pair so what we can do so if I
can convert this into cos square so
there is a gong sine square 63 is equal
cause by 63 sine square plus cos Phi is
equal to 1 here his cause if I can
convert this into sine we call square X
the signs for a living one because this
end is formula P so let's in this way
this become sine square 63 plus sine
square 90 minus 27 by cos square
seventeen plus cos square 19 giving
minus 17 degree 90 minus X if we ask why
because this is forming a pair 90
systems 53 a 27 so this comes out to be
sine square 63 plus cos square 16 by cos
square seventeen by sine squared 70 sine
square theta plus cos square theta is
equal to 1 cos square theta plus sine
square theta spending money so one by
one went to 1 and that is mad early we
tried to convert the one the given
earlier to cost but that didn't work out
so then we observe that 63 plus 27 is
going to 1972 the 73 is between nineteen
these are forming a pl so we can read
this sign to cause in this cos to sign
and we got sine square theta plus cos
square theta pattern that became one
click thank you visit exam viacom to
watch free education videos try free
online tests in the best quality so
Dimitri
is study from the best tutors and
mentors and mushroom thanks a serene
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Nov 08, 2024 Last updated |
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