Find the largest angle for which sine and tangent agree to within two significant figures.

Please explain ^_^

The answer is indefinite, since both tangent and sin are cyclic functions.

If we look for the largest angle between 0 and 2π, then we set
sin(x)-tan(x) = 0.005 (for two significant digits)
Solving for x, we get
x=1.931688π=6.06858 radians
Check: sin(x) = -0.2129628
tan(x) = -0.2179628
Thus sin(x) and tan(x) agree within 2 significant figures.

so we want

tanx = sinx
sinx/cosx = sinx
cosx = 1
x = 0º, 360º, 720º, ...
As you can see there is no such "largest" angle, they agree (are equal ?) at every multiple of 360º

BTW, then value of the "agreement" would be 0,
i.e. sin360=0, tan360=0
sin 720 = 0, tan 720 = 0 etc.

If you had to put the answer in radians what would it be?

mmmhhh, looks like MathMate and I took a different interpretation.

I think I read is as if there had been a comma after 'agree', ie

Find the largest angle for which sine and tangent agree, to within two significant figures.

I think that is how the question is suppose to be interpreted Reiny. Very confusing wording.

I guess we should use the first derivative to see the max

To find the largest angle for which sine and tangent agree to within two significant figures, we need to identify an angle where the values of the sine and tangent functions are equal up to two decimal places.

Since the sine of an angle is equal to the opposite side divided by the hypotenuse in a right triangle, we can create a table of angles and their corresponding sine values to compare with the tangent values. Here's an example:

Angle | Sine Value
----------------------
10° | 0.174
20° | 0.342
30° | 0.500
40° | 0.643
50° | 0.766
60° | 0.866
70° | 0.939
80° | 0.985
90° | 1.000

Now, let's calculate the tangent values for the same angles:

Angle | Tangent Value
-------------------------
10° | 0.176
20° | 0.364
30° | 0.577
40° | 0.839
50° | 1.191
60° | 1.732
70° | 3.077
80° | 6.313
90° | Not Defined

We can see that at 40°, the sine and tangent values agree up to two decimal places (0.64 and 0.84). Therefore, the largest angle for which sine and tangent agree to within two significant figures is 40°.