Small approximation- hence approx given value

So, I believe I am correct in my approach for part b. Specifically for 3tantheta. I believe it’s simply 3theta and this is how I found the answer to be correct.

Part b is a little confusing though.

I just presumed replacing the value of theta and finding the answer 0.34 was correct but I see that’s not the solution.

I also considered 1+0.4-(0.4)² and same with value for tan. Also, incorrect.

The examples offered don’t offer a way to do this. Logically, as it says hence, I assume the previous answer should help but I’m not seeing it clearly.




Edit: I also approached it from expanding the function to find this

Youre overcomplicating it. I agree with the series in a). Simply matching terms in part b) what should theta be? Both/either the sin() and tan() terms give it to you.

Edit when you did 0.4, its the ~right idea but not correct.

Look at part a. You have sin 2 theta and tan theta. Look at part b. The sin value is double the tan value. If you notice that pattern, you should figure out that theta = 0.2, and substitute it in for your answer in part a.

Also please note that - sin(0.4) * 3 tan(0.2) = -3 sin(0.4) tan(0.2)

I also expanded my last term incorrectly I tried on the function given.

I should have
1+3 tan 0.2-sin 0.4-3 sin 0.4 tan 0.2.

which finds the correct answer 0.98 if doing it this way.

I also forgot to include the 6 when subbing in 0.4 lol! But as you pointed out is wrong lol

Youve derived a series expansion in a). Simply sub theta=0.2 into it.

I am trying to conclude why theta =0.2? I’m sure it’s obvious but potentially something from previous material that I’ve forgotten.

I am trying to conclude why theta =0.2? I’m sure it’s obvious but potentially something from previous material that I’ve forgotten.

Look at the expressions in parts a) and b). What value of theta are they equal?

I think I figured it out. In part a) it states sin 2 theta, where as in part b) the value is sin 0.4 hence,

2 theta=0.4
tells me theta=0.2.

Yes. And/or
tan(theta) = tan(0.2)
So ...

That’s visibly clear but I can’t assume they have them both in common, can I?

This unit is new to me, so I’m not 100 clear on how it works.

Mmm, I think Is it due to their identity approximates being the same and that their graphs are similar in y=x and that meaning they must share the same theta value, other wise it doesn’t work.

In theory, they obviously could be differerent, but as its a "hence" question part, you'd fully expect that the first expression is valid for a particular value of theta. Therefore you'd expect that the theta and 2theta values match to 0.2 and 0.4 and they obviously do.

Oh, okay. Something else I have to train my self to look out for

Thank you