Question

How do you evaluate `cos(arcsin (1/4))`?

Answer 1

`sqrt(15)/4 approx 0.968`

Explanation:

By the fact that `cos^{2}(theta)+sin^{2}(theta)=1` for all `theta`, when we know the value of `sin(theta)`, there are two possible corresponding values for `cos(theta)`, namely `cos(theta)=pm sqrt{1-sin^{2}(theta)}`.

Next note that `arcsin(x)` always gives an answer between `-pi/2` and `pi/2` radians, where the cosine function is positive. Hence, `cos(arcsin(1/4)) geq 0`.

Thus, `cos(arcsin(1/4))=sqrt(1-sin^{2}(arcsin(1/4)))`

`=sqrt(1-(1/4)^2)=sqrt(1-1/16)=sqrt(15/16)=sqrt(15)/4 approx 0.968.`

It's also possible to solve this problem by drawing a right triangle, labeling one of the non-right angles as `arcsin(1/4)`, using the Pythagorean Theorem and SOH, CAH, TOA to label and find possible side lengths and ultimately the final answer.

Answer 2

Less formal style of solution

`sqrt(15)/4~~0.9682` to 4 decimal places

Explanation:

arcsin of some value gives you the angle that was used derive the that value of the sine

`color(brown)("sin, cos and tangent are just another way of defining ratios")`

The value given can be used in conjunction with the properties of sine to determine a related triangle.

From this and using Pythagoras we can determine the length of the adjacent.

`x^2+1^2=4^2`

`=>x=sqrt(15)`

so `cos(theta)= x/4=sqrt(15)/4 ~~ 0.9682` to 4 decimal places