Select the correct answer from each drop-down menu. A student begins a proof of the law of cosines. His work is shown.

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The next step in the proof is to write an expression for the length of side AZ in terms of _
Then the student should write an expression for the length of side AY in terms of_
Next he can use the _
to relate x, y, z, and _
(_) = blank, not known

1. blank choices
y, z and x
y, z and y
x, z and z

2. blank choices
z and Y
x and Z
z and X

3. blank choices
Pythagorean Theorem
Law of sines
tangent function

4. blank choices
X
Z
Y

Thanks for your time, have a nice day!

The answers required to complete the proof are:

1. The expression for length of side AZ would be in terms of x, y and z.

2. An expression for the length of side AY would be in terms of z and X.

3. He can use the trigonometric functions to relate x, y, z and X.

NB: Only the trigonometric functions can be used to relate the three sides of a given triangle with any of its angles.

4. The required angle is <X.

From the given question, the following can be derived:

1. To determine AZ applying the required trigonometric function to ΔXYZ, to have;

Cos(X) =

y =

But, Cos (X) =

AX = zCos(X)

Thus,

y = AX + AZ

So that,

AZ = y - AX

     = y - zCos(X)

AZ  = y - zCos(X)

The length of side AZ can be expressed in terms of x, y and z.

2. From Δ XYA, apply the Pythagoras theorem to have;

= +

= +

But;

zCos(x) = AX

So that;

= +

This implies that;

AY =

The student should write the length of AY in terms of z and X.

3. The student can use the trigonometric functions to relate x, y, z and X.

NB: Only the trigonometric functions can be used to relate the three sides of a given triangle with any of its angles.

4. The reference angle to the three sides of Δ XYZ would be X.

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