Verify the equality! arcsin (4/5) + arcsin (5/13) + arcsin(16/65)=pi/2

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To start off, arcsin (4/5), arcsin (5/13), and arcsin (16/65) are angles, but we don't know the value of them. These being angles, it means that we can apply trigonometric functions. In this case, arcsin is the inverse function of trigonometric function sine. But there is a problem: we can calculate the sine of only a two angles addition, not three, like in this case. So, in order to proceed with the addition, we have to move any term from the left side to the right side. We've chosen to move the last term from the left side: arcsin (16/65).

arcsin (4/5) + arcsin (5/13) = pi/2 - arcsin (16/65)

sin [ arcsin (4/5) + arcsin (5/13)]=sin [pi/2 - arcsin (16/65)]

We'll use the formula sin(a+b)=sin(a)cos(b)+sin(b)cos(a) and instead of a and b angles, we'll have arcsin (4/5), arcsin (5/13), arcsin (16/65).

sin [arcsin (4/5)]*cos[arcsin (5/13)] + sin[arcsin (5/13)]*cos[arcsin (4/5)] = sin(pi/2)*cos[arcsin (16/65)] - sin[arcsin (16/65)]*cos(pi/2)

But sin arcsin a = a and cos arcsin a=[1-(arcsin a)^2]^1/2

sin [arcsin (4/5)] = 4/5

cos[arcsin (5/13)]=(1 - 25/169)^1/2=12/13

sin[arcsin (5/13)] = 5/13

cos[arcsin (4/5)]=(1 - 16/25)^1/2=3/5

sin(pi/2) = 1

cos[arcsin (16/65)]= (1-256/4225)=63/65

sin[arcsin (16/65)] = 16/65

cos(pi/2)=0

Now, we'll substitute the above values:

4/5*12/13 + 5/13*3/5 = 1*63/65 - 16/65*0

[(4*12) + (5*3)]/5*13=63/65

(48+15)/65 = 63/65

63/65 = 63/65

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neela | High School Teacher | (Level 3) Valedictorian

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If arcsinx=A, then x=sinA. Smilarly, arcsiny = B=> y=sinB.

Therefore we can write,

sin(A+B)=sinAsinB+coAcosB=xsqrt(1-y^2)+sqrt(1-x^2)y. Taking inverse on both sides, we get:

A+B=Arcsin{xsqrt(1-y^2)+sqrt(1-x^2)y}.We use this on left side of the given epression to simplify and see whether we arrive at what is on the right side.

Taking the first two terms on the left,

arcsin(4/5)+arc(5/13) =Arcsin{(4/5)sqrt[1-(5/13)^2]+{sqrt(1-(4/5)^2(5/13) = arcsin[(4/5)(12/13)+(3/5)(5/13)}=arcsin(63/65).

The third term will now be added  to this and simplified similarly :

arcsin(63/65)+arcsin(16/65)=arcsin{(63/65)[sqrt(1-(16/65)^2+(16/65)sqrt(1-(63/65)^2(16/65)}

=arcsin{(63/65)(63/65)+(16/6)(16/65)}

=arcsin{(63^2+16^2/65}= arcsin{65/65}=arcsin(1)=Pi/2 the right side.

Verified the equality,arcsin (4/5)+arcsin (5/13) + arcsin(16/65)=pi/2,and it is true.

 

 

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