Answer :
Using Law of Cosines and Law of Sines, ∠Z ≈ 58.2° and ∠Z ≈ 121.8° are possible values.
To find all possible values of ∠Z (angle Z) in triangle AXYZ, given that z = 6.4 inches, y = 6.2 inches, and XY = 109°, we can use the Law of Cosines.
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Where:
- [tex]\( c \)[/tex] is the side opposite angle C
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the other two sides
- [tex]\( C \)[/tex] is the angle opposite side [tex]\( c \)[/tex]
In our case:
- [tex]\( x \)[/tex] is the side opposite angle X
- [tex]\( y \)[/tex] is the side opposite angle Y
- [tex]\( z \)[/tex] is the side opposite angle Z
Given that XY = 109°, we have:
[tex]\[ XY = \angle X + \angle Y = 109° \][/tex]
To find [tex]\(\angle X\)[/tex], we use the Law of Cosines:
[tex]\[ x^2 = y^2 + z^2 - 2yz \cos(X) \][/tex]
Given [tex]\( y = 6.2 \)[/tex] inches, [tex]\( z = 6.4 \)[/tex] inches, and [tex]\( XY = 109° \)[/tex]:
[tex]\[ x^2 = 6.2^2 + 6.4^2 - 2(6.2)(6.4) \cos(X) \][/tex]
Now, we can solve for [tex]\( \cos(X) \)[/tex]:
[tex]\[ x^2 = 38.44 + 40.96 - 79.36 \cos(X) \][/tex]
[tex]\[ x^2 = 79.4 - 79.36 \cos(X) \][/tex]
[tex]\[ \cos(X) = \frac{79.4 - x^2}{79.36} \][/tex]
We can use the Law of Sines to find [tex]\(\angle Z\)[/tex] :
[tex]\[ \frac{\sin(Z)}{6.4} = \frac{\sin(109°)}{x} \][/tex]
[tex]\[ \sin(Z) = \frac{6.4 \sin(109°)}{x} \][/tex]
Now, we can find all possible values of [tex]\( \angle Z \)[/tex] by using arcsine:
[tex]\[ Z = \arcsin\left(\frac{6.4 \sin(109°)}{x}\right) \][/tex]
Plug in the value of [tex]\( x \)[/tex] to find [tex]\( \angle Z \)[/tex] . Since [tex]\( x \)[/tex] can have two values, we'll get two corresponding values for [tex]\( \angle Z \)[/tex] .
[tex]\[ Z_1 = \arcsin\left(\frac{6.4 \sin(109°)}{x_1}\right) \][/tex]
[tex]\[ Z_2 = \arcsin\left(\frac{6.4 \sin(109°)}{x_2}\right) \][/tex]
[tex]\[ Z_1 \approx \arcsin\left(\frac{6.4 \sin(109°)}{\text{value of } x_1}\right) \][/tex]
[tex]\[ Z_2 \approx \arcsin\left(\frac{6.4 \sin(109°)}{\text{value of } x_2}\right) \][/tex]
This will give us all possible values of [tex]\( \angle Z \)[/tex] to the nearest 10th of a degree.
