[Math] Why do you multiply one way and divide the other way with these trig problems

trigonometry

I am practicing finding a side of an angle on Khan Academy. I understand SOH CAH TOA and which sin, cos, tan to choose from. But, I don't understand why they multiply sometimes to find the side and divide other times. I am using a calculator.

Here is a multiply example
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and a divide example
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thanks.

Best Answer

In the 1st problem, I think you understand it as far as the next-to-last line, $$\sin A={BC\over\sqrt{97}}$$ Also, you are given the value $$\sin A={9\sqrt{97}\over97}$$ So you have $${9\sqrt{97}\over97}={BC\over\sqrt{97}}$$ Since you want $BC$, you multiply both sides of this equation by $\sqrt{97}$ --- that gets $BC$ all by itself on one side of the equation, which is what you want.

In the second problem, you understand $$\tan A={10\over AC}$$ You want to isolate $AC$. This can be done by multiplying both sides by $AC$, getting $$(AC)(\tan A)=10$$ and then dividing both sides by $\tan A$ to get $$AC={10\over\tan A}$$

Now you are given $$\tan A={10\over3}$$ so you have $$AC={10\over10/3}$$ which simplifies to $3$.

OK?

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