Question #e7339

Question

Question #e7339

1 Answer

Impact of this question

1791 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-17T17:00:17

It doesn't.

$cos ({sin}^{- 1} (\frac{2}{3})) = \frac{\sqrt{5}}{3}$ $cos(sin^-1(2/3))=sqrt5/3$

Explanation:

Let's break this down:

$cos ({sin}^{- 1} (\frac{2}{3}))$ $color(red)(cos(color(blue)(sin^-1(2/3)))$

The $cos$ $color(red)(cos)$ function is taking the cosine of the angle ${sin}^{- 1} (\frac{2}{3})$ $color(blue)(sin^-1(2/3))$ .

Let's say that $θ = {sin}^{- 1} (\frac{2}{3})$ $color(blue)(theta=sin^-1(2/3))$ . Remember that this is an angle.

In the triangle where this is true, we see that $sin (θ) = \frac{2}{3}$ $sin(theta)=2/3$ .

We can draw a triangle where $sin (θ) = \frac{2}{3}$ $sin(theta)=2/3$ : this means that the "opposite" side is $2$ $2$ and the hypotenuse is $3$ $3$ .

Through the Pythagorean Theorem we see that the other leg is $\sqrt{5}$ $sqrt5$ , since we have a right triangle.

google drawings

What we want to find is $cos ({sin}^{- 1} (\frac{2}{3}))$ $color(red)(cos(color(blue)(sin^-1(2/3)))$ , which is really $cos (θ)$ $color(red)(cos(color(blue)(theta))$ .

That is, we just want cosine of theta in this triangle.

Cosine is "adjacent" over hypotenuse, so:

$cos (θ) = cos ({sin}^{- 1} (\frac{2}{3})) = \frac{\sqrt{5}}{3}$ $color(red)(cos(color(blue)(theta)))=color(red)(cos(color(blue)(sin^-1(2/3))))=sqrt5/3$

Science

Math

Humanities

... and beyond

Question #e7339

1 Answer

Explanation:

Related questions

Impact of this question