Question

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

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

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.

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: \(Z = cos^{-1}\left(\frac{41^2 + 29^2 - 52^2}{2 \cdot 41 \cdot 29}\right)\).

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: \(Y = sin^{-1}\left(\frac{YZ \cdot sin(Z)}{XZ}\right)\).

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: \(X = 180^\circ - Y - Z\).

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.

Key concepts

Law of Sines

When solving a triangle, the Law of Sines can be a powerful tool. This law helps find unknown angles when you already know one angle and its opposite side.
The Law of Sines states:

• For any triangle with angles A, B, and C, and opposite sides a, b, and c, the Law describes a relationship as \( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \).
• This means each ratio of the sin of an angle to its opposite side is equal to the others.

This relationship comes in handy when two angles or two sides of a triangle are known, allowing us to find the other values easily.
In triangle XYZ, once angle Z was determined using other methods, the Law of Sines allowed calculation of angle Y with the known side lengths. Thus providing a complete set of angle data crucial for solving any triangle accurately.

Triangle Angles

Understanding triangle angles is foundational in trigonometry. In any triangle, the angles are intimately connected with the Law of Sines and the Law of Cosines.
Triangles consist of three angles, and there's an immutable rule: their sum is always \( 180^\circ \).
For example, if you're at a point where two angles of a triangle are known, it's straightforward to find the third:

• Simply subtract the sum of the known angles from \( 180^\circ \).

Solving for angle X in triangle XYZ, knowing Z and Y, becomes as easy as:
\( X = 180^\circ - Y - Z \).
So, understanding this triangle property greatly simplifies your task in determining missing angles.

Angle Calculation

Calculating angles in triangles often involves using trigonometric formulas. Key players in these computations are the Law of Cosines and the Law of Sines.
First, the Law of Cosines helps find an angle when all three sides are known:

• The formula used: \( \cos(Z) = \frac{XZ^2 + YZ^2 - XY^2}{2 \cdot XZ \cdot YZ} \) gives the angle opposite the desired side by calculating cosine first then finding the angle with the inverse cosine function.

After that, with one angle available, like angle Z in triangle XYZ, the more straightforward Law of Sines helps derive the other angles by:

• Using the formula: \( \frac{\sin(Y)}{YZ} = \frac{\sin(Z)}{XZ} \), and finding Y using the inverse sine function.

In essence, angle calculation combines these laws to provide an exact and complete way to determine every angle in a triangle, no matter the known values.