Solution:
Given the ΔABC as shown below:
To evaluate the value of x, y, and z,
Let's begin with z.
Step 1: In the ΔBCD, identify the sides.
Thus, in the triangle BCD,
[tex]\begin{gathered} BC\Rightarrow hypotenuse \\ BD\Rightarrow opposite \\ DC\Rightarrow adjacent \end{gathered}[/tex]
Step 2: Evaluate the value of z, using trigonometric ratios.
From trigonometric ratios,
[tex]\begin{gathered} sin\text{ }\theta=\frac{opposite}{hypotenuse} \\ cos\text{ }\theta=\frac{adjacent}{hypotenuse} \\ tan\text{ }\theta=\frac{opposite}{adjacent} \end{gathered}[/tex]
Thus, we have
[tex]\begin{gathered} cos\text{ }\theta=\frac{adjacent}{hypotenuse} \\ where \\ \theta\Rightarrow\angle C=30 \\ adjacent\Rightarrow DC=z \\ hypotenuse\Rightarrow BC=24 \\ thus, \\ cos\text{ 30=}\frac{z}{24} \\ cross-multiply, \\ z=24\times cos\text{ 30} \\ =24\times\frac{\sqrt{3}}{2} \\ \Rightarrow z=12\sqrt{3} \end{gathered}[/tex]
To evaluate the values of x and y, we need to first evaluate the value of BD.
Thus,
[tex]\begin{gathered} \sin\theta=\frac{opposite}{hypotenuse} \\ where \\ \theta=30 \\ opposite=BD \\ hypotenuse=BC=24 \\ thus, \\ \sin30=\frac{BD}{24} \\ cross-multiply, \\ BD=24\times sin\text{ 30} \\ =24\times\frac{1}{2} \\ \Rightarrow BD=12 \end{gathered}[/tex]
Thus, to evaluate the value of x,
step 1: In the ΔABD, identify the sides of the triangle.
Thus, in the triangle ABD,
[tex]\begin{gathered} hypotenuse\Rightarrow AB \\ opposite\Rightarrow AD \\ adjacent\Rightarrow BD \end{gathered}[/tex]
Step 2: Evaluate the value of x, using trigonometric ratios.
From trigonometric ratios,
[tex]\begin{gathered} cos\text{ }\theta=\frac{adjacent}{hyptenuse} \\ where \\ \theta\Rightarrow\angle B=45 \\ adjacent\Rightarrow BD=12 \\ hypotenuse\Rightarrow AB=x \\ thus, \\ cos\text{ 45 = }\frac{12}{x} \\ cross-multiply, \\ x\times cos\text{ 45 = 12} \\ \Rightarrow x\times\frac{\sqrt{2}}{2}=12 \\ divide\text{ both sides by }\frac{\sqrt{2}}{2}, \\ x=12\times\frac{2}{\sqrt{2}}=\frac{24}{\sqrt{2}} \\ rationalize\text{ the denominator of the surd,} \\ x=\frac{24\sqrt{2}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{24\sqrt{2}}{2} \\ \Rightarrow x=12\sqrt{2} \end{gathered}[/tex]
To evaluate the value of y,
In the triangle ABD, using trigonometric ratios,
[tex]\begin{gathered} \tan\theta=\frac{opposite}{adjacent} \\ where \\ \theta=45 \\ opposite\Rightarrow AD=y \\ adjacent\Rightarrow BD=12 \\ thus, \\ \tan45=\frac{y}{12} \\ cross-multiply, \\ y=12\times\tan45 \\ =12\times1 \\ \Rightarrow y=12 \end{gathered}[/tex]
Hence, the values of x, y and z are
[tex]\begin{gathered} x=12\sqrt{2} \\ y=12 \\ z=12\sqrt{3} \end{gathered}[/tex]
