Evaluate Inverse Sine function

In this section of ask-math, we will discuss how to evaluate inverse sine function.
The inverse sine function is represented as $f(x)=sin^{-1}x$. Here the -1 is not an exponent, it represents the inverse.The another name for $sin^{-1}x$ is arcsinx.
ask-math has already explains you about the sine graph. The domain for sine is $\frac{-\prod }{2}\leq x \leq \frac{\prod}{2}$. The sine inverse function is not one-to-one function so it is restricted sine function.
The another name for inverse sine function is :
$sin^{-1}(x)$ = arcsin (x)
To evaluate inverse sine functions remember that the following statement is equivalent.
$\Theta = sin^{-1}(x) \Leftrightarrow x = sin(\Theta)$

In other words, when we evaluate sine inverse function we are asking what angle,θ, did we plug into the sine function (regular, not inverse!) to get x.


In sine function, we know that $sin \Theta =\frac{opposite side }{hypotenuse}$
The inverse sine function $sin{-1}$ takes the ratio opposite/hypotenuse and gives the angle θ.

∴ θ = $sin^{-1}(\frac{opposite side }{hypotenuse})$

For example, $sin 30^{0} =\frac{1}{2}$

∴ θ = $sin^{-1}(\frac{1}{2})$

⇒ θ = $30^{0}$

Evaluate inverse sine function

1) Evaluate f(x) = $sin^{-1}(\frac{\sqrt{2}}{2})$

Solution : f(x) = $sin^{-1}(\frac{\sqrt{2}}{2})$

y = $sin^{-1}(\frac{\sqrt{2}}{2})$

⇒ y = $45^{0}$ ( For the degree you can use a unit circle or calculator)

2) Evaluate f(x) = $sin^{-1}(\frac{\sqrt{3}}{2})$

Solution : f(x) = $sin^{-1}(\frac{\sqrt{3}}{2})$

y = $sin^{-1}(\frac{\sqrt{3}}{2})$

⇒ y = $60^{0}$

3) Find y, when sin(y) = 0.2384.
Solution : sin(y) = 0.2384.
∴ y = $sin^{-1}(0.2384) $
⇒ y = $13.798^{0}
⇒ y = $13.8^{0}

4) Evaluate : f(x) = $sin^{-1}(sin(\frac{\prod }{4}))$

Solution : f(x) = $sin^{-1}(sin(\frac{\prod }{4}))$

y = $sin^{-1}(sin(\frac{\prod }{4}))$

The value of $sin(\frac{\prod }{4}) = (\frac{\sqrt{2}}{2})$

∴ y = $sin^{-1}(\frac{\sqrt{2}}{2})$

∴ y = $45^{0}$

5) Evaluate f(x) = $sin^{-1}(1)$

Solution : f(x) = $sin^{-1}(1)$

y = $sin^{-1}(1)$

⇒ y = $90^{0}$


11th grade math

precalculus

Home

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.