# Having trouble setting up this Trig problem.

## Homework Statement

Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution θ (see the figure) can be improved by using a lens with a larger diameter D. The relationship between θ in degrees and D in meters is given by sin θ = 1.22λ/D, where λ is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of λ = 550 × 10−9 meter, its resolution is approximately 0.00003754°. Approximate the diameter of the lens. (Round your answer to two decimal places.)

## Homework Equations

sin(.00003754)=[1.22(550x10^-9)]

## The Attempt at a Solution

When I do it set up this way, I get a decimal, which cannot be the size of largest telescope lens.
If I divide like this: sin(.00003754)/1.22(550x10^-9) = 55.9463m this was not right.

I googled the actual telescope lens diameter and it is about 40 ft. (or 12.192m)

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SteamKing
Staff Emeritus
Homework Helper
Your math skills (and reading comprehension) need work. The lens diameter for the Yerkes telescope is 40 inches (not feet) or 1.02 m

The formula you were given is sin theta = 1.22 lambda / D [Note: the D was omitted from the formula in sec. 2 of the OP]

When you substitute the given data into the given formula, and invert, D = 1.02 m

tiny-tim
Homework Helper
hi jrjack!
The relationship between θ in degrees and D in meters is given by sin θ = 1.22λ/D…

sin(.00003754)=[1.22(550x10^-9)]
you seem to be using sin θ/1.22λ = D (instead of 1/D)

(and are you converting to radians?)

Your math skills (and reading comprehension) need work. The lens diameter for the Yerkes telescope is 40 inches (not feet) or 1.02 m

The formula you were given is sin theta = 1.22 lambda / D [Note: the D was omitted from the formula in sec. 2 of the OP]

When you substitute the given data into the given formula, and invert, D = 1.02 m
I do struggle with both skills, but hopefully more practice will help. sorry I forgot to type the / D
The section we are learning has only dealt with right triangles, and has not mentioned Lambda or wavelength yet, so I am having trouble visuallizing how the lens dia is related to theta.

I tried it in degrees and radians and I'm still not getting the right answer.
sin (00003754)degrees =.000000655

.000000655/ 1.22 (550x10^-9) = .9764479899 in degrees

I know I am missing something or not doing something right.

SteamKing
Staff Emeritus
Homework Helper
The argument of the sine function (0.00003754) is already in degrees.

The sine of an angle (whether the angle is measured in degrees or radians) is a pure number without units.

The formula you use to calculate D of the lens is:

sin (theta) = 1.22 lambda / D, where theta is in degrees and lambda and D are in meters

In order to solve for D, you must apply a little algebra to the given formula:

D = 1.22 lambda / sin (theta) [divide both sides by sin (theta) and multiply both sides by D]

Plug your data into the modified formula above to calculate D.

Thanks.
I'm slowly learning, It's been 14 years since high school, and my algrebra refresher course was lacking.
I now realize when I converted the formula, I ended up with the sin (theta) as the numerator, when it should have been the denominator. Then I had to make sure I put my calculator back in degree mode.
Got it now, thanks.