Calculating $\log_5(.008)$ and $\sin(75)+\sin(120)-\cos(150)+\cos(165)$ by hand

logarithmstrigonometry

Calculating $\log_5(.008)$ and $\sin(75)+\sin(120)-\cos(150)+\cos(165)$ by hand

Sorry if these seems like two different questions, I figured the theme was simiar enough.

I'm helping somebody prepare for a test on which they are not allowed to use a calculator. The review had questions like the two above.

$log_5(.008)y \iff 5^y=.008$

How would you solve this by hand? I mean, the answer is $y=-3$ which is nice, but in theory $y$ could be any real number. I suppose we could just hope the answer is nice an do long division… $\frac{1}{5^2}$ and then $\frac{1}{5^3}$ etc.

And then for the trig question, I'm not seeing any useful trig identies to relate these angles back to something to one of the commonly known angles i.e. angles with reference angle $0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3}$ or $\frac{\pi}{2}$

Thanks in advance

Best Answer

Convert $0.008$ into a fraction : $\frac{8}{1000} = \frac{1}{125}$. If you can now recognize that $125 = 5^3$ then you're done.

For the trig question, all of the numbers are multiples of $15$, and you know $\sin(45^\circ), \sin(30^\circ), \cos(45^\circ), \cos(30^\circ)$, and then apply the sum and difference formulas.

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