Precalculus
will mark brainliest
-see attachment-

.

Step-by-step explanation:

We want to evaluate cos(arcsin(1/4)) probably without a calculator.

If you did want a calculator answer, that would be 0.968245837.

Alright so anyways, this is the way I begin these trig(arctrig( )) types of problems when the trig parts are different.

Let u=arcsin(1/4).

If u=arcsin(1/4) then sin(u)=1/4.

So we want to find cos(u) given sin(u)=1/4.  (I just replace arcsin(1/4) in cos(arcsin(1/4)) with u.)

Let's use a Pythagorean Identity:

.

Let's plug in 1/4 for sin(u):

Simplify a bit:

Subtract 1/16 on both sides:

Simplify the right hand side:

Take the square root of both sides:

Separate the square thing to the numerator and denominator:

Replace with 4 since :

Now how do we determine if the cosine should be positive or negative.

arcsin(1/4) is an angle that is going to be between -pi/2 and pi/2 due to restrictions upon the sine curve to be one to one.

cosine of an angle between -pi/2 and pi/2 is going to be positive because these are the 1st and 4th quadrant where the x-coordinate is positive (the cosine value is positive)

So recall u=arcsin(1/4):

.

For fun, put .  If you don't get  0.968245837 then you made a mistake in the above reasoning. We do get that so the results of the calculator and our trigonometry/algebra confirm each other.

d. bc = 12 and ef = 16

step-by-step explanation:

we are given that δabc : δdef such that .

also, it is given that

so, from the options, we need that .

a.

b.

c.  

d.

as, we can see that only option d gives the same ratio as i.e. .

hence, the possible lengths are bc = 12 and ef = 16.