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$$ \begin{array}{l}{\text { (III) For small angles } \theta \text { , the numerical value of } \sin \theta \text { is }} \\ {\text { approximately the same as the numerical value of tan } \theta \text { . }} \\ {\text { Find the largest angle for which sine and tangent agree to }} \\ {\text { within two significant figures. }}\end{array}$$

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0.24 radians.

Physics 101 Mechanics

Chapter 1

Introduction, Measurement, Estimating

Physics Basics

Cornell University

Rutgers, The State University of New Jersey

Hope College

University of Sheffield

Lectures

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

09:56

In mathematics, algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

0:00

For small angles $\theta$,…

01:03

Find the smallest positive…

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01:35

00:35

00:54

So here we know that the small angle approximation lets us know that sign of Fada is approximately equal to tangent of data for very small angles. So we can actually set up in a fit of the chart here. Pie and radiant. Azari, Fada and Radiance. My apologies. Oh, and then it will be sign of data and tangent to fatal. And we want to find the maximum angle measure such that it is accurate to two significant digits. So let's just start with zero. Then we'll be Goto 00.10 Then we'll go to 0.12 We can go to 0.20 and we can stop there and see what happens. So sign of zero would, of course, be zero chance and observe. Of course, be zero side of 00.10 point 10 Radiance would be again 100.10 Tangents of 0.10 would be 0.10 Ah, sign of 0.12 Radiance would have again be 0.12 tangent of 0.12 Radiance would be 0.12 uh, sign of 0.20 radiance again. 0.20 Tangent of 0.20 Radiance 0.20 So we know that we're still accurate. Let's go to the next one. Let's go to point 24 Let's try 240.24 0.24 is again 0.24 And for a tangent of 0.24 radiance is again 0.24 So we need to go toe again. Further, Let's go to 0.25 Radiance sign of 0.25 Radiance is 0.25 and then tangent of 0.25 Radiance is rounded toe 0.26 and this is where it deviates So we can say that angle rather fatal Max would be equal toe 0.25 radiance um, for small angle approximation, uh, in degrees approximation in degrees to approximately 12 degrees. So anything larger than 12 degrees, you won't be able. Thio, use the small angle approximation. And again, this is ah, 222 significant figures. So it'll still be accurate. Um however, uh, anything greater than 12 degrees, then we're starting to deviate toothy tens to the hundreds place. So essentially Ah, we can only really go to a fate of Max of 12 degrees. In order for this approximation to be accurate. That is the end of the solution. Thank you for watching

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