Can anyone see an easy way to explain why tan 89.9 is approximately the same as 10x tan 89?

Just messing around on the calculator and they are remarkably close..

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- Jan 6th 2018, 01:56 AM #1

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- Jan 6th 2018, 05:47 AM #2

- Jan 6th 2018, 08:36 AM #3

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## Re: tan 89 or tan 89.9

Consider the complimentary angle to theta, which we'll call alpha and is equal to theta - pi/2. We can use the fact that cos(theta) = sin (alpha), and also the fact that as alpha approaches 0 the value of sin(alpha) is very close to alpha itself. Hence for small values of alpha the value of tan(theta) is very close to 1/alpha. So, since the angle alpha is ten times larger for theta= 89 degrees than for theta = 89.9 degrees, the tangent will be approximately 1/10 as large for theta = 89 degrees compared to theta = 89.9 degrees.

- Jan 7th 2018, 06:11 AM #4

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## Re: tan 89 or tan 89.9

Hence for small values of alpha the value of tan(theta) is very close to 1/alpha.

Can you clarify.?

if tan (theta) = sin (theta)/cos(theta) = sin(theta)/ sin(pi/2-theta) = theta/pi/2-theta for small values of theta???? except the denominator is no longer small?

Sorry i am getting confused?

- Jan 7th 2018, 06:13 AM #5

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- Jan 7th 2018, 06:59 AM #6

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## Re: tan 89 or tan 89.9

Not quite. The angle theta is close to pi/2, so it’s not small. But sin(pi/2 - theta) is small, so appromimately equal to pi/2-theta. So for theta close to pi/2 we have tan(theta) is approximately equal to sin(pi/2) / sin(pi/2 - theta), which is approximately equal to 1/(pi/2-theta). For theta equal to 89 or 89.9 degrees the denominator is pi/180 or pi/1800, respectively (in radians). Hence the tangent is approximately 180/pi or 1800/pi, respectively.

- Jan 7th 2018, 09:19 AM #7

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- Jan 7th 2018, 11:08 AM #8

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## Re: tan 89 or tan 89.9

I think i have worked out a simpler way..

tan 1 is approx equal to 10tan(0.1)

so given tan 89 = 1 /tan1 and tan 89.9 = 1/(tan 0.1) the result follows?

Does that feel ok?

It does rely on tan x = 1/tan(90-x) , which i am not sure how to prove yet..

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