Without a calculator, how can I find the value of sin67º?

Answer 1

or with a protractor and ruler draw a right triangle with a 67º angle and measure the hypotenuse and leg.

You can get very close: you can get sin of a 22 1/2 degree angle with the half angle formula, and then add that to a 45 degree angle to get the sin of a 67 1/2 degree angle

Answer 2

Clearly, just looking it up in a set of pre-published trigonometric tables would violate the implied conditions in this problem. A question saying "without a calculator" generally means "do it by purely trigonometric means."

But here's another question:

Are you SURE that you weren't asked for sin 67 1/2 degrees? That would really simplify the problem, since 67 1/2 degrees is EXACTLY
(45 + 22 1/2) degrees, or (90 - 22 1/2) degrees.

How does that help? Well, sin and cos (45 deg.) are obviously (1/sqrt [2]), and 22 1/2 deg. is half of 45 deg. There are standard "half-angle formulae," so that from this sine and cosine of 45 deg. you can work out sin and cos 22 1/2 degrees. Finally, there is a standard angle addition or subtraction formula by which you can work out either sin ([45 + 22 1/2] deg.), OR sin ([90 - 22 1/2] deg.)

I was very glad to see that you asked how YOU could do it, rather than ask for a spoonfed answer. I hope this helps. Good luck!

Live long and prosper.

POSTSCRIPT:

If the question really did ask for sin 67 deg. the best way would probably be to first work out sin 67 1/2 degrees as above. That would be an EXACT answer (although probably expressed with surds etc.) Then consider sin ([67 1/2 - 1/2] deg.). Expressed in radians, the power series (NOT approximations!) for sin and cos (1/2 deg.) will converge EXTREMELY QUICKLY: 1/2 deg. is about 1/120th of a radian. So, that would quickly enable you to calculate sin (67 deg.) to as many sig. figs. as is reasonable to expect.)

Answer 3

Alright, my intuition was correct, but it's not pretty.

There are these things called Taylor Series and Taylor Polynomials. If you don't know what those are it doesn't matter. The point is they give us a way of approximating sin(67)

The approximation for sin(x) = x - x^3/3! + x^5/5! - x^7/7! +x^9/9!....

Oh, by the way, x needs to be in radians. To convert degrees to radians you do degrees/180*Pi. So 67 degrees = 1.16937 radians.

The more places you go out, the better of an approximation you'll get.

The exclamation marks mean factorial. Here's an example of a factorial. 3! = 3*2*1. 10! = 10*9*8*7*6*5*4*3*2*1

So, to test it to a few places....
sin(1.16937) = 1.16937 - 1.16937^3 / 6 + 1.16937^5 / 120 = 0.921087

The actual value for sin(67) = 0.920505. See how close they are?

Answer 4

Perhaps the most accurate way is a combination of the above. You can find an exact value for 67.5º because that is one half of 135º. Begin by finding both the sine and cosine. The half angle formulas are:

cos(x) = √((1 + cos(2x))/2)
sin(x) = √((1 - cos(2x))/2)

Since cos(135º) = -1/√2

So:

cos(67.5º) = √((1 + (-1/√2))/2) = (1/2) (√(2 - √2))
sin(67.5º) =√((1 - (-1/√2))/2) = (1/2) (√(2 + √2))

Now that you are so close, use the multiple angle formula:

sin(67º) = sin(67.5º)cos(0.5º) - sin(0.5º)cos(67.5º)

Use small angle approximations for the 0.5º parts:

sin(0.5º) = sin(π/360 radians) = π/360
cos(0.5º) = cos(π/360 radians) = 1 - (1/2)(π/360)^2

So: sin(67º) = (1/2) (√(2 + √2))(1 - (1/2)(π/360)^2) - (π/360)((1/2) (√(2 - √2)))

Even though this uses approximations, the result is correct to 7 significant digits. To evalutate this by hand does require the ability to do square roots. One further simplfication is possible, if you assume that cos(0.5º) = 1 then the answer is still correct to 4 places.

Answer 5

Doesn't look very easy. Certain angles have easy to determine values and then one can use various identities to get a value for another angle.

If you really meant 66 2/3 deg then that's (198+2)/3 = 200/3

(what I'm trying to do is see if this angle is some convenient fraction of pi)

200/3 deg * (2pi/360) = 400pi/1080 = 40/108pi= 20/54pi

Doesn't look like I'm getting anywhere!

Answer 6

Use a Trigonometric Chart.
It's about 0.9205

Answer 7

sin60=root(3)/2 = 1.732/2

sin67 would be a little greater.

Is that close enough?

Answer 8

you can ask on yahoo! answers. Try that.