ASIN

Returns the inverse sine (the arcsine) of a number.

Syntax:

ASIN(number)

returns the inverse trigonometric sine of number, in other words the angle (in radians) whose sine is number. The angle returned is between -PI/2 and +PI/2.To return the angle in degrees, use the DEGREES function.

Example:

ASIN(0)

returns 0.

ASIN(1)

returns 1.5707963267949 (PI/2 radians).

DEGREES(ASIN(0.5))

returns 30. The sine of 30 degrees is 0.5.

Application:

An application of an ASIN (arcsine) function is in calculating the angle of elevation of a projectile. Imagine a simple physics problem where a ball is thrown, and you want to determine the initial angle at which it was launched.

Let's say you have the following information:

Initial velocity (v0): 20 m/s
Maximum height (hmax): 5.1 m

The formula relating these variables is:

hmax= $\frac{v _{0}^{2} s i n ^{2} ( θ )}{2 g}$

where g is the acceleration due to gravity, approximately 9.8 m/s$^2$.

To find the launch angle θ, you need to rearrange the formula to isolate sin(θ):

sin2(θ)= $\frac{2 g h _{ma x}}{v _{0}^{2}}$

sin(θ)= $\frac{2 g h _{ma x}}{v _{0}^{2}}$

Now, you can use the ASIN function to find the angle θ:

θ= $arcsin (\frac{2 g h _{ma x}}{v _{0}^{2}})$

Let's plug in the numbers:

$θ = arcsin (\frac{2 \times 9.8 \times 5.1}{2 0 ^{2}})$

$θ = arcsin (\frac{100}{400})$

$θ = arcsin (0.25)$

$θ = arcsin (0.5)$

Using a calculator, arcsin(0.5)=30∘. So, the launch angle was 30 degrees.

Here's a table that illustrates this relationship with different maximum heights, assuming the same initial velocity of 20 m/s:

	Maximum Height (hmax)	$\frac{2 g h _{ma x}}{v _{0}^{2}}$	sin(θ)	θ=arcsin(sin(θ))
	A	B	C	D
1	5.1 m	0.5	0.5	30°
2	10.2 m	0.707	0.707	45°
3	15.3 m	0.866	0.866	60°
4	20.4 m	1	1	90°

This table shows how the ASIN function is used to reverse the sine calculation, allowing you to find the angle from the ratio of the sides of a right triangle (which is essentially what sin(θ) represents in this context).