Trigonometry Question: What is sin(-12π)?

Question

I think my calculator (TI 84)has an error. -12π is the same angle as 2π so the sine of the angle should be the same, zero ,but my calculator thinks it should be -4*10^-13. Another weird thing is I used the degree equivalent and it gave me an answer of zero. Is there some weird math principle that I'm missing or is my calculator just being stupid?

Answer 1

The TI-84 Plus can only hold the first 14 digits of pi, so of course there will be slightly off approximations such as the one you detailed here. In these cases, consider these answers as zero.

Answer 2

You're absolutely correct that sin(-12π) = sin(2π) or any multiple of 2π including 0.



That sort of thing happens on all calculators I've used. There's just a little rounding error. Any time you see a number like that, you pretty much know the calculator is trying to say "0".



I think it's because they use Taylor polynomials to calculator trigonometric functions, and the polynomial approximation uses another approximation of pi, so the final approximation isn't long enough to be rounded down to 0. That's just my guess though.

Answer 3

-4*10^-13 is an extremely small number almost 0.

The error is due to the calculator rounding because it doesn't have the exact value of pi, only an approximation. If you used 3.14 instead of pi you would also not get exactly 0 as an answer.

Answer 4

Remember that π = 180°.



By trigonometric identity sin(-x) = sin(x), sin(-12π) = sin(12π). For every value that has the factor of π or 180°, the sine value gives zero. Hence, sin(-12π) = 0.



I hope this helps! Oh yes! When graphing sin(-12π), make sure that you set the mode to "radians".

Answer 5

as said below



the computer cannot store the PRECISE value of pi, thus the error.



some calculators know you have typed 12pi and just set the ans to 0

Answer 6

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SHORTCUT METHOD

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Given: csc x = -3 where x is in quadrant III



Remember that csc x = 1 / sin x.



Since x is in quadrant III, sin x is negative.

csc x = 1 / sin x = -3 ===> sin x = -1 / 3



Remember your unit circle, if you go 12π from where you are, you end up at the same point and, therefore, the same value.



sin(x + 12π) = sin x = -1 / 3



ANSWER: -1 / 3



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LONGER METHOD

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Given: csc x = -3 where x is in quadrant III



Remember that csc x = 1 / sin x.



Since x is in quadrant III, sin x is negative.

csc x = 1 / sin x = -3 ===> sin x = -1 / 3



Remember the Pythagorean Theorem:

a² + b² = c²



Applied to trig, this becomes:

(opposite)² + (adjacent)² = (hypotenuse)²



Remember that sin = opposite / hypotenuse.



Given: sin x = -1 / 3

Means: opposite = -1, hypotenuse = 3



Plug them into the formula to find adjacent.

(opposite)² + (adjacent)² = (hypotenuse)²

(-1)² + (adjacent)² = (3)²

1 + (adjacent)² = 9

(adjacent)² = 8

√(adjacent)² = √8

adjacent = ± √8

adjacent = ± √(4 * 2)

adjacent = ± 2√2



Remember that cos = adjacent / hypotenuse.

cos x = ± 2√2 / 3



Since x is in quadrant III, we know that cos x in quadrant III is negative.

cos x = - 2√2 / 3



sin(x + 12π) = ?



Remember sin (A + B) = sin A cos B + cos A sin B.

sin(x + 12π) =

sin x cos(12π) + cos x sin(12π) =



Plug in your values.

sin x cos(12π) + cos x sin(12π) =

(-1 / 3)cos(12π) + (-2√2 / 3)sin(12π) =

(-1 / 3)cos(12π) - (2√2 / 3)sin(12π) =



Remember your unit circle.

cos(12π) = 1

sin(12π) = 0



Apply this to what you have.

(-1 / 3)cos(12π) - (2√2 / 3)sin(12π) =

(-1 / 3)(1) - (2√2 / 3)(0) =

(-1 / 3) - 0 =

(-1 / 3)



ANSWER: -1 / 3

Answer 7

make sure you're in radian mode when you try it. if it still doesn't work, then reset the calculator and try again. 2nd>+>7>1>2 to reset (i have the same calculator, lol)

Answer 8

sin (-12pi) = --sin(12pi) =--sin(5 (2pi) +2pi} =--sin2pi =0