Question

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

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

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 \(XY\). \[XY \approx 54.19 \,\text{cm}\]

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 \(X\). \[X \approx 40.78^\circ\]

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 \(X \approx 40.78^\circ\), \(Y \approx 80.22^\circ\), \(Z = 59^\circ\).

Key concepts

Law of Cosines

The Law of Cosines is a useful tool in trigonometry for solving triangles when you know two sides and the angle between them. This law is especially handy in non-right triangles, also known as oblique triangles. It states that for a triangle with sides of lengths \(a\), \(b\), and \(c\), opposite the respective angles \(A\), \(B\), and \(C\), the law can be expressed as:
\[c^2 = a^2 + b^2 - 2ab \cos(C)\]
This formula helps find the length of an unknown side when two sides and the included angle are known, as demonstrated in solving for side \(XY\) in triangle \(\Delta XYZ\). By substituting the known values into the formula, you can calculate \(XY\) accurately. The process involves using a calculator to manage the trigonometric calculations efficiently. Remember to take the square root of your result to find the length of side \(XY\), rather than the squared value.

Law of Sines

When you're looking to find unknown angles or sides in a non-right triangle, the Law of Sines can be extremely helpful. The Law of Sines expresses a relationship between the sides and angles of a triangle. It is given by:
\[\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\]
This law can be used when you know either:

• Two angles and one side (AAS or ASA cases)
• Two sides and an opposite angle (SSA case)

In our exercise, once side \(XY\) was found, the law was used to determine angle \(X\). By rearranging the Law of Sines, we solved for \(\sin(X)\) using known values and calculated angle \(X\) using the inverse sine function (\(\sin^{-1}\)). This process let us pin down every angle in the triangle based on side lengths and existing angle measurements.

Triangle Solving

Solving a triangle means finding all the unknown sides and angles when given some initial information. By using trigonometric laws such as the Law of Cosines and the Law of Sines, you can solve essentially any triangle, whether it's right or oblique.
The approach usually involves:

• Identifying what is given - sides, angles, or both.
• Deciding which law or rule is applicable - whether it's the Law of Cosines for missing sides or the Law of Sines for missing angles.
• Performing calculations step-by-step to uncover unknown values.

Once you determine the necessary values, ensure they make sense in the context of a triangle (sides should logically work together, and angles should always sum to 180 degrees). This logical consistency check helps to identify any potential errors in calculation.

Angle Calculation

Calculating angles within a triangle often involves finding the missing angles once some are known. We commonly utilize the angle sum property of triangles, which states that the interior angles of a triangle will always add up to \(180^\circ\).
In triangle \(\Delta XYZ\), after finding side \(XY\) and angle \(X\), we calculated angle \(Y\) using this property, knowing that\(X\), \(Y\), and \(Z\) must sum to \(180^\circ\). To calculate angle \(Y\):

• Add the already known angles \(X\) and \(Z\).
• Subtract the sum from \(180^\circ\) to find \(Y\).

This straightforward calculation ensures precision in finding the last missing angle, allowing for the complete solution of the triangle. Always double-check your results to ensure accuracy across all calculated angles.