Continuing with some miscellaneous examples, next we have this. In the last module I told you we can also prove this without using formula of sin inverse, let’s prove it.
Say I call this angle A and this angle B.
Now A plus B equal to theta. We want to prove that this theta is equal to sin inverse of 77 by 85. Apply sin on both sides, so sin A cos B plus cos A sin B equal to sin theta. Now try to see, first of all I would like to express everything in terms of sin because angles are in terms of sin inverse. Now look, A is sin inverse 8 by 17, and B is sin inverse 3 by 5. So sin A will be 8 by 17 and sin B will be 3 by 5. So keeping the values here we get number one, A plus 17 then root of minus 9 by 25 plus 3 by 5 root of 1 minus 64 by 289 equal to sin theta. Simplying these values we get 77 by 85 on left hand side. Now finally we need to check because we directly applied sin so we need to check for extra solutions if it is correct or not.
We called A and B this, now try to see as 1 by root 2 is approximately 0.7, and both of these terms 8 by 17 and 3 by 5 are less than 0.7, so these angles A and B both must be less than pi by 4. So they belong to 0 to pi by 4. So theta which is A plus B lies in 0 to pi by 2. So this thing sin theta equal to 77 by 85 will give me theta equal to sin inverse 77 by 85.
So that’s the way to prove this thing without using formula of sin inverse.
Okay, next we have this. We want to prove this result for x positive, okay, let’s try. Number one, let sin inverse x be theta. As x is positive so this theta must lie in 0 to pi by 2. Now applying sin, x is equal to sin theta, always start with right hand side. So put x is equal to sin theta here in this expression. To get sin inverse 1 minus 2 sin square sin theta. I hope you remember this is just cos 2 theta, so it gives me cos inverse cos 2 theta. As theta is 0 to pi by 2, this 2 theta will lie in 0 to pi that is principle domain of cos and hence this gives me 2 theta, theta was sin inverse x. So it is equal to 2 sin inverse which we wanted to prove.
So with that I conclude this module. See you soon with more examples next module.
God bless take care.
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