I have a question that asks me khổng lồ find an algebraic expression for sin(arccos(x)). From the lone example in the book I seen they"re doing some multistep thing with the identities, but I"m just not even sure where khổng lồ start here. It"s supposed to lớn be $sqrt1-x^2$



Set $alpha=arccos x$. Since, by definition, $alphain<0,pi>$, you know that $sinalphage0$, so$$sinarccos x=sinalpha=sqrt1-cos^2alpha=sqrt1-(cosarccos x)^2=sqrt1-x^2$$


Let $arccos(x)= heta$

$x=cos( heta)$

You require $y=sin(arccos(x))=sin( heta)$

Identity: $cos^2( heta)+sin^2( heta)=1$



What is $arccos(x)$? it is the angle given by the ratio of sides of length $x$ và $1$. That is, on a triangle with adjacent side length $x$ and hypotenuse length $1$ we will find the angle $arccos(x)$.

(There is a picture for this which I hope you can get from what I describe).

Bạn đang xem: What is sin of arccos of x

Now what is $sin(y)$? It is the ratio of the opposite side to the hypotenuse of a triangle with angle $y$ in the respective part. So if we use $y=arccos(x)$, we have that $sin(y)$ is the ratio of the sides with degree given by a triangle whose adjacent and hypotenuse sides are length $x$ và $1$ respectively.

Since we have the adjacent & hypotenuse lengths, we can calculate the opposite sides length by the Pythagorean theorem. This means, if we use $z$ for the opposite side, that $z^2+x^2=1^2=1$. Solving for $z$ gives $z=sqrt1-x^2$.

Then $sin(y)$ is the ratio of the opposite size, $z=sqrt1-x^2$ & the hypotenuse, $1$. We may now say $sin(arcos(x))=sqrt1-x^2$.

edited Apr 26, năm ngoái at 6:30
answered Apr 26, 2015 at 6:21

5,54011 gold badge1616 silver badges3333 bronze badges
địa chỉ cửa hàng a comment |
Draw a right triangle with one of the angles being $cos^-1 x$, then fill out the lengths of the sides of the triangle using the definition of cosine for two of the sides, & Pythagoras"s theorem for the third side. Then, use the definition of sine lớn find $sin(cos^-1 x)$. For questions lượt thích these, always try drawing a triangle first.

cốt truyện
answered Apr 26, 2015 at 6:15
Kevin HsuKevin Hsu
47533 silver badges1313 bronze badges
add a phản hồi |
$$y=sinarccos x$$$$x=cosarccos x$$

Therefore $x$ và $y$ are the sin and cos of the same angle and


mô tả
answered Apr 26, năm ngoái at 6:21
62.6k22 gold badges5151 silver badges100100 bronze badges
showroom a comment |

Your Answer

Thanks for contributing an answer to magmareport.netematics Stack Exchange!

Please be sure khổng lồ answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to lớn other answers.Making statements based on opinion; back them up with references or personal experience.

Use magmareport.netJax lớn format equations. magmareport.netJax reference.

To learn more, see our tips on writing great answers.

Draft saved
Draft discarded

Sign up or log in

Sign up using Google
Sign up using Facebook
Sign up using e-mail and Password

Post as a guest

thư điện tử Required, but never shown

Post as a guest


Required, but never shown

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged trigonometry or ask your own question.

The Overflow Blog
Featured on Meta
The angle $alpha$ is greater than $90$ and less than $360$ & $cos(alpha)=frac23$. Find the exact value of $ an(alpha)$.

Xem thêm: Lý Thuyết Phép Tịnh Tiến Là Gì ? Nghĩa Của Từ Phép Tịnh Tiến Trong Tiếng Việt

Rewrite the expression in the khung $A sin(x+C)$
Rewriting an expression in the size of $A sin(x + C)$
Easiest way in general khổng lồ find the sin, cos, arcsin, arccos, of "not so easy" angles/values without using a calculator?
Simplify Inverse Trigonometric Expression
An algebraic expression for a trigonometric function
$arcsin x- arccos x= pi/6$, difficult conclusion
Let $g(x)=arccos(1/cosh(5x))$. Find $g′(ln(3))$
Solving $arccos(fracxr) + arccos(fracx cdot dr) = fracpin$
Evaluating $operatornamearccos frac2sqrt5 + operatornamearccos frac3sqrt10$
Write $sin^2(arctan x)$ as an algebraic expression
Hot Network Questions more hot questions

Question feed
Subscribe to lớn RSS
Question feed lớn subscribe to this RSS feed, copy và paste this URL into your RSS reader.

Stack Exchange Network
Site thiết kế / logo sản phẩm © 2022 Stack Exchange Inc; user contributions licensed under cc by-sa. Rev2022.4.21.42004

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.