- #1
Wild ownz al
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Prove:
a) cos(sin-1x) = √(1-x2)
b) cos-1a+cos-1b = cos-1(ab-√(1-a2)√(1-b2) (edited)
(VERY HARD)
a) cos(sin-1x) = √(1-x2)
b) cos-1a+cos-1b = cos-1(ab-√(1-a2)√(1-b2) (edited)
(VERY HARD)
Last edited:
Olinguito said:Hi Wild ownz al.
Try using the identity $\cos^2\theta+\sin^2\theta=1$.
(a) We have $\sin^2(\sin^{-1}x)=x^2$ and so
$$\cos^2(\sin^{-1}x)\ =\ 1-\sin^2(\sin^{-1}x)\ =\ 1-x^2$$
$\implies\ \cos(\sin^{-1}x)\ =\ \sqrt{1-x^2}$
taking the positive square root because the range of $\sin^{-1}x$ (for $-1\le x\le1$) is $\displaystyle\left[-\frac{\pi}2,\,\frac{\pi}2\right]$ on which the cos function takes non-negative values.
(b) Check your equation. There should be an equals (“=”) sign, which is missing.
Wild ownz al said:Prove:
a) cos(sin-1x) = √(1-x2)
b) cos-1a+cos-1b = cos-1(ab-√(1-a2)√(1-b2) (edited)
(VERY HARD)
Wild ownz al said:I'm a bit confused as to your steps...did you manipulate the left hand side or the right hand side? Also could you start from the given equation? I corrected part b). Thanks.
This equation is a trigonometric identity, which means it is always true for any value of x. It relates the cosine and sine functions through the inverse sine function and shows that the cosine of the inverse sine of x is equal to the square root of 1 minus x squared.
The equation can be derived using the Pythagorean identity for sine and cosine, as well as the definition of the inverse sine function. It involves manipulating the equations to get them in a form where they can be substituted for each other, resulting in the given identity.
This equation is important in trigonometry and calculus, as it allows for simplification of complex trigonometric expressions and can be used to solve various problems involving trigonometric functions. It also helps to establish connections between different trigonometric functions.
Yes, this equation can be used to solve for x in certain cases. However, it is important to note that the inverse sine function has a restricted domain of -1 to 1, so the equation may not be applicable for all values of x.
Yes, this equation has various applications in fields such as physics, engineering, and astronomy. For example, it can be used to calculate the angle of elevation or depression in a right triangle given the lengths of the sides, or to determine the velocity of an object in projectile motion.