Question

match the following each letter may be used more than once.a. 12/15b. 15/12c. 9/15d. 9/12e. 12/9f. 15/9

Answer 1

We have a right triangle and we have to write some of the trigonometric ratios.

A trigonometric ratio relates a trigonometric function of an angle of the tiangle with a quotient of two of the sides of the triangle.

The basic trigonometric ratios are:

`\begin{gathered} \sin (\alpha)=\frac{\text{Opposite}}{\text{Hypotenuse}} \\ \cos (\alpha)=\frac{\text{Adyacent}}{\text{Hypotenuse}} \end{gathered}`

We can also write the trigonometric ratio for the tangent:

`\tan (\alpha)=(\sin (\alpha))/(\cos (\alpha))=\frac{\text{Opposite}}{\text{Adyacent}}`

Now, we can write sin(x):

`\sin (X)=\frac{\text{Opposite}}{\text{Adyacent}}=(YZ)/(XZ)=(12)/(15)`

The opposite side to X is YZ and the hypotenuse is XZ, so sin(X) = YZ/XZ = 12/15.

In the same way, we can calculate cos(x):

`\cos (X)=\frac{\text{Adyacent}}{\text{Hypotenuse}}=(XY)/(XZ)=(9)/(15)`

The tan(x) can be calculated as:

`\tan (X)=\frac{\text{Opposite}}{\text{Adyacent}}=(YZ)/(XY)=(12)/(9)`

For Z, the opposite and adyacent angles are different than for X, so we can write:

`\tan (Z)=\frac{\text{Opposite}}{\text{Adyacent}}=(XY)/(YZ)=(9)/(12)`

Answer:

sin(X) = 12/15

cos(X) = 9/15

tan(X) = 12/9

tan(Z) = 9/12