Note that sin(cos-1 x) or sin(arc cosx) are algebraic functions, not trigonometric functions. The value of sin(cos-1 x) or sin(arc cosx) is equal to $\sqrt{1-x^2}$. In this post, we will simplify sin(cos^{-1} x).
.png "Simplify sin(cos^-1 x)")
The formula of sin(cos inverse x) is given below:
sin(cos-1 x) = root(1-x2).
Simplify sin(cos-1(x))
We all know that $\sin(\cos^{-1}(x))= \sin(\text{arc} \cos x)$. To simplify this function, we will follow the below steps.
Step 1: In the first step, we let that $\alpha=\cos^{-1} x$ $\cdots (I)$
As $\alpha=\cos^{-1} x$ and we have to simplify $\sin(\cos^{-1}x)$, we need to find out the value of $\sin \alpha$. From the equation $(I)$, we obtain that
$\cos \alpha=x$ $\cdots (II)$
Step 2: Now, we use the Pythagorean trigonometric identity $\sin^2 \alpha+\cos^2 \alpha =1$. From this identity, we have that
$\sin^2\alpha = 1-\cos^2 \alpha$
$\Rightarrow \sin \alpha = \sqrt{1-\cos^2 \alpha}$
$=\sqrt{1-x^2}$ [ putting the value of $\cos \alpha$ from $(II)$ ]
$\therefore \sin \alpha =\sqrt{1-x^2}$
Step 3: Next, we will put the value of $\alpha$ from $(I)$. By doing so we obtain that
$\sin(\cos^{-1}x)=\sqrt{1-x^2}$.
Conclusion: Thus, the formula of $\sin(\cos^{-1}x)$ is equal to $\sqrt{1-x^2}$.
Also Read:
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Question 1: Find the value of sine of cosine inverse 0, that is, find $\sin(\cos^{-1}0)$.
Answer:
Using the above formula, we have that $\sin(\cos^{-1}0)=\sqrt{1-0^2}=1$.
Question 2: Find the value of sine of cosine inverse 1, that is, find $\sin(\cos^{-1}1)$.
Answer:
Using the above formula, we have that $\sin(\cos^{-1} 0)=\sqrt{1-1^2}=0$.
FAQs
Q1: What is sin(cos^-1(x))?
Answer: The value of sin(cos-1x) is equal to $\sqrt{1-x^2}$.