Question

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!

Answer 1

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) = $\frac{z}{y}$

y = $\frac{z}{Cos(x) }$

But, Cos (X) = $\frac{AX}{z}$

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;

$/Hyp/^{2}$ = $/Adj 1/^{2}$ + $/Adj 2/^{2}$

$z^{2}$ = $(AX)^{2}$ + $(AY)^{2}$

But;

zCos(x) = AX

So that;

$z^{2}$ = $(zCos(X))^{2}$ + $(AY)^{2}$

This implies that;

AY = $\sqrt{z^{2} - (zCos(X))^{2} }$

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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Answer 2

Answer:

1. y, z, and X

2. x and Z

3. Pythagorean theorem

4. X

Step-by-step explanation:

hope this helps