Question

Which property can be justified using the ratios in triangle XYZ

Answer 1

Option (B) is correct.

`\sin Y= \cos (90^(\circ)-Y)`

Explanation:

Given : A figure showing triangle XYZ with right angle at X and measurement of angles and sides are given.

We have to choose the correct option from the given options.

Consider the given options , we will check each one by one.

A)

`\cos Y =(x)/(z)`

We know, Cosine of an angle gives the relationship between base and hypotenuse.

`\cos\theta =(base)/(hypotenuse)`

Thus, For
`\theta=Y`, we have base is z and hypotenuse = x

Thus,
`\cos Y =(z)/(x)`

So, (A) is false.

B)

`\sin Y= \cos (90^(\circ)-Y)`

We know
`\sin\theta= \cos (90^(\circ)-\theta)`

Thus,
`\sin Y= \cos (90^(\circ)-Y)` is true.

SO, (B) is correct.

C)

`\cos Z= \sin (90^(\circ)-Y)`

We know
`\cos\theta= \sin (90^(\circ)-\theta)`

Thus,
`\cos Z= \sin (90^(\circ)-Z)`

SO, (C) is not correct.

D)

`\tan Z=(\cos Y)/(\sin Z)`

Since,
`\tan Z=(\sin Z)/(\cos Z)`

Thus,
`\tan Z=(\cos Y)/(\sin Z)` is incorrect.

Thus, Option (B) is correct.

Answer 2

The correct answer is B.

We can actually tell this without even doing work as every answer for Cos(α) = Sin(90 - α) due to the way that Sin and Cos wave behave. However, we can also tell this by using some mathematical evaluation.

Sin(Y) = Cos(90 - Y)
Sin(50) = Cos(90 - 50)
.766 = Cos(40)
.766 = .766

This should that they are exactly the same.