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Chapter 10: Problem 21

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{XZ}=41 \mathrm{~cm}, \mathrm{YZ}=29 \mathrm{~cm}\) and \(\mathrm{XY}=52 \mathrm{~cm}\).

Short Answer

Expert verified
Question: In triangle XYZ, side XZ = 41 cm, side YZ = 29 cm, and side XY = 52 cm. Calculate the angles of triangle XYZ. Answer: Follow the step-by-step solution provided to find the angles of triangle XYZ. Calculate angle Z using the Law of Cosines, then angle Y using the Law of Sines. Finally, use the angle sum property to find angle X. Provide the final solution with all angles and side lengths of triangle XYZ.

Step by step solution

01

Apply the Law of Cosines to find angle Z

The Law of Cosines can be applied to find the angle Z since we have the side lengths XZ, YZ, and XY. The formula for angle Z is: \(cos(Z) = \frac{XZ^2 + YZ^2 - XY^2}{2 \cdot XZ \cdot YZ}\). Substitute the given values and calculate angle Z.
02

Calculate angle Z in degrees

After the substitution, the expression becomes: \(cos(Z) = \frac{41^2 + 29^2 - 52^2}{2 \cdot 41 \cdot 29}\). Solve the equation and calculate the value of angle Z in degrees using the inverse cosine function: \(Z = cos^{-1}\left(\frac{41^2 + 29^2 - 52^2}{2 \cdot 41 \cdot 29}\right)\).
03

Apply the Law of Sines to find angle Y

Now that we have angle Z, we can use the Law of Sines to find the other angles. The formula for angle Y is: \(\frac{sin(Y)}{YZ} = \frac{sin(Z)}{XZ}\). Solve for \(sin(Y)\) and calculate the value of angle Y in degrees using the inverse sine function: \(Y = sin^{-1}\left(\frac{YZ \cdot sin(Z)}{XZ}\right)\).
04

Use the angle sum property to find angle X

Since the sum of the angles in a triangle equals \(180^\circ\), we can use the angle sum property to find angle X: \(X = 180^\circ - Y - Z\).
05

Final solution of \(\Delta XYZ\)

By following the steps mentioned above, you have calculated angles X, Y, and Z. The final solution for \(\Delta XYZ\) can be given as XZ = 41 cm, YZ = 29 cm, XY = 52 cm, and angles X, Y, and Z as calculated in the previous steps.

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Most popular questions from this chapter

Convert the following angles to degree/ minute/second format: (a) \(7.3614^{\circ}\) (b) \(10.0932^{\circ}\) (c) \(14.9610^{\circ}\)

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=21 \mathrm{~cm}, \mathrm{AC}=30 \mathrm{~cm}, A=42^{\circ}\)

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=29 \mathrm{~cm}, \mathrm{BC}=41 \mathrm{~cm}, B=100^{\circ}\)

The angle of elevation to the top, \(\mathrm{B}\), of a vertical tower \(\mathrm{AB}\) is \(19^{\circ} 3^{\prime}\) when measured from a point, \(\mathrm{C}, 27.3 \mathrm{~m}\) from the base of the tower. Calculate (a) the height of the tower (b) the distance BC.

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{YZ}=15 \mathrm{~cm}, X=39^{\circ}\) and \(Y=75^{\circ}\).
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