Sin(Sin^-1 x)=x for -1<=x<=1. True or false?

Mathematics

1 answer:

Umnica [9.8K]3 years ago

5 0

Answer:

True

Step-by-step explanation:

In general, when you compose a function $f(x)$ with its inverse $f^{-1}(x)$ , you always get the identity function:

$f(f^{-1}(x))=x$

The limitation $-1\leq x \leq 1$ is necessary because the arcsin function wouldn't be defined otherwise.

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The line with equation −a+3b=0 coincides with the terminal side of an angle θ in standard position and sinθ<0 .

Anna11 [10]

If $a$ is the variable of the horizontal axis, then you can solve for $b$ to get the equation of the line in slope-intercept form in the $a,b$ plane:

$-a+3b=0\implies b=\dfrac a3$

i.e. a line with slope $\dfrac13$ through the origin, which means it is contained in the first and third quadrants. Since the terminal side of $\theta$ has a negative sine, the angle must lie in the third quadrant.

Because the slope of the line is $\dfrac13$ , you can choose any length along the line to make up the hypotenuse of a right triangle with reference angle $\theta$ . Any such right triangle will have $\tan\theta=\dfrac13$ , regardless of whether the angle is the first or third quadrant. But since $\theta$ is known to lie in the third quadrant, and so $\sin\theta$ and $\cos\theta$ are both negative, you have

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac13\implies \dfrac{\sin\theta}{\cos\theta}=\dfrac{-1}{-3}\implies \cos\theta=-3$