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Chapter 10: Problem 18

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{XZ}=85 \mathrm{~cm}, \mathrm{YZ}=70 \mathrm{~cm}\) and \(Z=59^{\circ}\).

Short Answer

Expert verified
Question: Determine the other angles and sides of a non-right triangle XYZ, given side XZ = 85 cm, side YZ = 70 cm, and angle Z = 59°. Answer: In triangle XYZ, the other angles and sides are approximately X ≈ 40.78°, Y ≈ 80.22°, and XY ≈ 54.19 cm.

Step by step solution

01

Use the Law of Cosines to find side XY

In triangle \(\Delta XYZ\), applying the Law of Cosines, we have: \[XY^2 = XZ^2 + YZ^2 - 2(XZ)(YZ) \cos{Z}\] \[XY^2 = 85^2 + 70^2 - 2(85)(70) \cos(59^\circ)\] Now, use a calculator to determine the value of \(XY^2\) and take the square root to find the length of side \(XY\). \[XY \approx 54.19 \,\text{cm}\]
02

Use the Law of Sines to find angle X

Applying the Law of Sines to find angle \(X\), we get: \[\frac{\sin{X}}{XY} = \frac{\sin{Z}}{YZ}\] \[\sin{X} = \frac{XY \sin{Z}}{YZ}\] \[\sin{X} = \frac{54.19 \sin(59^\circ)}{70}\] Now, use a calculator to determine the value of \(\sin{X}\) and take the inverse sine to find angle \(X\). \[X \approx 40.78^\circ\]
03

Find angle Y

To find angle \(Y\), we use the fact that the sum of angles in a triangle is equal to \(180^\circ\). \[Y = 180^\circ - (X + Z)\] \[Y = 180^\circ - (40.78^\circ + 59^\circ )\] Calculate the value of angle \(Y\). \[Y \approx 80.22^\circ\] The solved triangle \(\Delta XYZ\) has sides \(XY \approx 54.19\) cm, \(XZ = 85\) cm, \(YZ = 70\) cm, and angles \(X \approx 40.78^\circ\), \(Y \approx 80.22^\circ\), \(Z = 59^\circ\).

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