TutorMe homepage

SIGN IN

Start Free Trial

Don S.

Teacher / Tutor for 12 years

Tutor Satisfaction Guarantee

Geology

TutorMe

Question:

Which mineral (often found in a box in an undergraduate lab) sticks to your tongue if you lick it?

Don S.

Answer:

Kaolinite.

Pre-Calculus

TutorMe

Question:

Simplify $$(\log_8 27)(\log_9 16)$$.

Don S.

Answer:

We proceed by using a change of base of logarithms. Recall $$\log_y x = \frac{\log_b x}{\log_b y}$$. Thus, our problem simplifies as follows $$(\log_8 27)(\log_9 16) = \frac{\log27\, \cdot\, \log16}{\log8 \, \cdot \, \log9}$$ $$= \frac{\log3^3\, \cdot\, \log2^4}{\log2^3\, \cdot\, \log3^2}$$ $$=\frac{3\log3\, \cdot\, 4\log2}{3\log2\, \cdot\, 2\log3}$$ $$= 2 $$

Trigonometry

TutorMe

Question:

Find an algebraic expression for the somewhat intimidating-looking function $$f(x)=\sin(\tan^{-1}x)$$. Note that you are being asked to write down this expression in terms of only $$x$$, and not using any trigonometric functions.

Don S.

Answer:

For some reason, people tend to hate these kinds of questions. I like them; I am odd--I know this. In any case, let's look at this problem in a slightly simpler form. It looks very much like $$\sin(\theta)$$ if we let $$\theta=\tan^{-1}x$$. Enjoying this new name for the argument of the sine function, we see immediately, by taking the tangent of both sides of the equation, that $$x=\tan\theta$$. This is where a picture is truly worth a thousand words, but they aren't seeming to let me draw one in this space :( If it were there, it would be a simple right triangle drawn in the first quadrant, with angle $$\theta$$ in the lower left corner. This is all we need! Since $$\tan\,\theta=x$$ this triangle has vertical side length $$x$$, and horizontal side length 1 (since $$\tan\,\theta = \text{opposite / adjacent}$$). By the Pythagorean Theorem, the hypotenuse of the triangle must be $$\sqrt{1+x^2}$$. We are nearly finished. The original question requested an algebraic expression for $$\sin(\tan^{-1}x)$$, we called this $$\sin\,\theta$$ and by our picture, we see that $$f(x) = \sin(\tan^{-1}x) = \sin\,\theta =\frac{x}{\sqrt{1+x^2}}$$ (since $$\sin\,\theta = \text{opposite / hypotenuse}$$). Now, wasn't that fun?! (Next time we will do this with a double angle identity for extra fun...)

Send a message explaining your

needs and Don will reply soon.

needs and Don will reply soon.

Contact Don

Ready now? Request a lesson.

Start Session

FAQs

What is a lesson?

A lesson is virtual lesson space on our platform where you and a tutor can communicate.
You'll have the option to communicate using video/audio as well as text chat.
You can also upload documents, edit papers in real time and use our cutting-edge virtual whiteboard.

How do I begin a lesson?

If the tutor is currently online, you can click the "Start Session" button above.
If they are offline, you can always send them a message to schedule a lesson.

Who are TutorMe tutors?

Many of our tutors are current college students or recent graduates of top-tier universities
like MIT, Harvard and USC.
TutorMe has thousands of top-quality tutors available to work with you.

Made in California

© 2018 TutorMe.com, Inc.