Can your calculator even compute this?

Geometry Level 4

1 0 1 0 10 sin ⁡ ( 109 1 0 1 0 10 ) − 9 9 9 sin ⁡ ( 101 9 9 9 ) − 8 8 8 sin ⁡ ( 17 8 8 8 ) + 7 7 7 sin ⁡ ( 76 7 7 7 ) + 6 6 6 sin ⁡ ( 113 6 6 6 ) \begin{aligned} && 10^{10^{10}} \sin\left( \frac{109}{10^{10^{10}}} \right) - 9^{9^{9}} \sin\left( \frac{101}{9^{9^{9}}} \right) \\ &&- 8^{8^{8}} \sin\left( \frac{17}{8^{8^{8}}} \right) + 7^{7^{7}} \sin\left( \frac{76}{7^{7^{7}}} \right) + 6^{6^{6}} \sin\left( \frac{113}{6^{6^{6}}} \right) \end{aligned}

Evaluate the expression above to 4 significant figures.

Details and assumptions

  • Angles are measured in degrees.
  • No calculators allowed


The answer is 3.1416.

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Shavindra Jayasekera by Shavindra Jayasekera , 21, United Kingdom

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3 solutions

Chew-Seong Cheong
Aug 14, 2015

lim ⁡ x → 0 sin ⁡ x = x \lim_{x \to 0} \sin{x} = x (in radians), therefore,

1 0 1 0 10 sin ⁡ ( 109 1 0 1 0 10 ) − 9 9 9 sin ⁡ ( 101 9 9 9 ) − 8 8 8 sin ⁡ ( 17 8 8 8 ) + 7 7 7 sin ⁡ ( 76 7 7 7 ) + 6 6 6 sin ⁡ ( 113 6 6 6 ) ≈ 109 π 180 − 101 π 180 − 17 π 180 + 76 π 180 + 113 π 180 = 180 π 180 = π = 3.142 \small 10^{10^{10}} \sin{\left( \frac{109}{10^{10^{10}}} \right)} - 9^{9^9} \sin{\left( \frac{101}{9^{9^9}} \right)} - 8^{8^8} \sin{\left( \frac{17}{8^{8^8}} \right)} + 7^{7^7} \sin{\left( \frac{76}{7^{7^7}} \right)} + 6^{6^6} \sin{\left( \frac{113}{6^{6^6}} \right)} \\ \approx \dfrac{109\pi}{180} - \dfrac{101\pi}{180} - \dfrac{17\pi}{180} + \dfrac{76\pi}{180} + \dfrac{113\pi}{180} = \dfrac{180\pi}{180} = \pi = \boxed{3.142}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

I think this is the most efficient method to solving the problem. I used the equations lim ⁡ x → ∞ x sin ⁡ ( y x ) = y \lim _{ x\rightarrow \infty }{ x\sin { (\frac { y }{ x } } ) } =y and lim ⁡ x → ∞ x tan ⁡ ( y x ) = y \lim _{ x\rightarrow \infty }{ x\tan { (\frac { y }{ x } } ) } =y (in radians) which also works the same as the equations which you have used but are longer. However, I am not sure how you get from lim ⁡ x → ∞ x sin ⁡ ( y x ) = y \lim _{ x\rightarrow \infty }{ x\sin { (\frac { y }{ x } } ) } =y to lim ⁡ x → 0 sin ⁡ ( x ) = x \lim _{ x\rightarrow 0 }{ \sin { (x } ) } =x . It would be nice if you could explain how to get from one equation to the other.

Shavindra Jayasekera - 5 years, 9 months ago 
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We have to evaluate this first:

limit of u*sin( a / u ) as u approaches infinity where a = k*pi/180
Obviously the result will be indeterminate so we have to tweak it

sin( a/u ) / (1/u)
Apply L'Hospital's rule: 
cos(a/u) * a * (-1/u**2) / (-1/u**2)
a cos( a/u )

as u approaches infinity:
a cos(a/u) --> a

109 - 101 - 17 + 76 + 113 = 180
180*pi/180 = pi

pi = 3.1416 <-- Answer

2 Helpful 0 Interesting 0 Brilliant 0 Confused

The denominators are extremely large, so that the approximation sin ⁡ x ≈ x \sin x \approx x works well here--but x x must be given in radians. Each term reduces to N sin ⁡ ( a ∘ N ) ≈ N ⋅ a N ⋅ C = a ⋅ C , N \sin\left(\frac {a^\circ} N\right) \approx N\cdot \frac a N \cdot C = a\cdot C, where C C is the conversion factor from degrees to radians. Thus the entire expression is approximately equal to ( 109 − 101 − 17 + 76 + 113 ) ⋅ C = 180 ⋅ C , (109 - 101 - 17 + 76 + 113) \cdot C = 180\cdot C, and we know that 18 0 ∘ 180^\circ corresponds to π \pi radians. Therefore the answer is ≈ π = 3.1416 \approx \pi = 3.1416 .

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