- Thread starter KatieJM
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Can you expandHi there,

I need some help with this answer ASAP, please and thank you!

Simplify the expression by using a double-angle formula:

View attachment 13240

(I am attaching a screen shot of the formula because I can't find how to type the symbols).

Thanks!

Katie

cos(2x)

in terms of cos(x)?

Left-hand side: \(\cos(16^{\circ}) \approx 0.96126169 \implies 2 \cos^2(16^{\circ}) - 1 \approx 2 \cdot 0.96136169^2 - 1 \approx 0.84804807\)

Right-hand side: \(\cos(32^{\circ}) \approx 0.84804809\)

What say you? You may notice there's a very tiny discrepancy - a difference in the eighth decimal place. Why did this difference appear? What does it mean?

- Joined
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It is aThank you for that explanation!

I am stumped as to why the 8th decimal is different. What am I missing?

\(\displaystyle \frac{1}{3} \approx 0.3333\) and \(\displaystyle \frac{1}{6} \approx 0.1666\)

\(\displaystyle 0.3333 \cdot 0.1666 = 0.0555{\color{red}2778}\)

\(\displaystyle \frac{1}{18} = 0.0555{\color{red}5555}\)

Could we make the approximation tighter? What happens then?

\(\displaystyle \frac{1}{3} \approx 0.33333333\) and \(\displaystyle \frac{1}{6} \approx 0.16666666\)

\(\displaystyle 0.33333333 \cdot 0.16666666 = 0.05555555{\color{red}27777778}\)

\(\displaystyle \frac{1}{18} = 0.05555555{\color{red}55555555}\)

What happens if we use the real, exact value?

\(\displaystyle \frac{1}{3} \cdot \frac{1}{6} = \frac{1}{18}\)

Thank you!It is adifference in rounding, that is all.

Got it, I was overthinking it, thank you!approximationsinstead of the real value?

\(\displaystyle \frac{1}{3} \approx 0.3333\) and \(\displaystyle \frac{1}{6} \approx 0.1666\)

\(\displaystyle 0.3333 \cdot 0.1666 = 0.0555{\color{red}2778}\)

\(\displaystyle \frac{1}{18} = 0.0555{\color{red}5555}\)

Could we make the approximation tighter? What happens then?

\(\displaystyle \frac{1}{3} \approx 0.33333333\) and \(\displaystyle \frac{1}{6} \approx 0.16666666\)

\(\displaystyle 0.33333333 \cdot 0.16666666 = 0.05555555{\color{red}27777778}\)

\(\displaystyle \frac{1}{18} = 0.05555555{\color{red}55555555}\)

What happens if we use the real, exact value?

\(\displaystyle \frac{1}{3} \cdot \frac{1}{6} = \frac{1}{18}\)