Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\gamma=73.01^{\circ}, \quad a=17.31, \quad c=20.24$$

Step-by-Step Solution

Answer

\( b \approx 22.45 \), \( \alpha \approx 55.48^{\circ} \), \( \beta \approx 51.51^{\circ} \)

Step 1: Identify the Given Data and Conditions

We have a triangle \( \triangle ABC \) with side \( a = 17.31 \), side \( c = 20.24 \), and angle \( \gamma = 73.01^{\circ} \). We need to find the missing side \( b \) and angles \( \alpha \) and \( \beta \).

Step 2: Use the Law of Cosines to Find Side 'b'

Since we know sides \( a \) and \( c \) and the included angle \( \gamma \), we use the Law of Cosines:\[b^2 = a^2 + c^2 - 2ac \cos(\gamma)\]Substitute the known values:\[b^2 = 17.31^2 + 20.24^2 - 2 \times 17.31 \times 20.24 \times \cos(73.01^{\circ})\]Calculate \( \cos(73.01^{\circ}) \), then continue with the calculations to find \( b \).

Step 3: Compute Side 'b'

Calculate:\[b^2 = 299.8561 + 409.6576 - 702.7308 \times 0.2924 \approx 299.8561 + 409.6576 - 205.456 \approx 503.81\]Take the square root to find \( b \):\[b \approx 22.45\]

Step 4: Use the Law of Sines to Find Angle 'α'

With side \( a = 17.31 \), side \( b \approx 22.45 \), and angle \( \gamma = 73.01^{\circ} \), apply the Law of Sines:\[\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\]Use \( \frac{c}{\sin(\gamma)} = \frac{20.24}{\sin(73.01^{\circ})} \) to find \( \sin(\alpha) \) using\[\sin(\alpha) = \frac{a \cdot \sin(73.01^{\circ})}{20.24}\]

Step 5: Calculate Angle 'α'

Calculate:\[\sin(73.01^{\circ}) \approx 0.9563\]\[\sin(\alpha) = \frac{17.31 \times 0.9563}{20.24} \approx \frac{16.54}{20.24} \approx 0.8178\]\( \alpha \approx \arcsin(0.8178) \approx 55.48^{\circ} \).

Step 6: Compute Angle 'β' Using the Angle Sum Property

Use the sum of angles in a triangle which is \( 180^{\circ} \):\[\beta = 180^{\circ} - \alpha - \gamma = 180^{\circ} - 55.48^{\circ} - 73.01^{\circ}\]Calculate \( \beta \):\[\beta \approx 51.51^{\circ}\]

Step 7: Summary of the Solution

The calculated sides and angles for \( \triangle ABC \) are:- \( b \approx 22.45 \)- \( \alpha \approx 55.48^{\circ} \)- \( \beta \approx 51.51^{\circ} \)

Key Concepts

Law of Cosines

The Law of Cosines is a powerful tool in trigonometry, used when you know two sides and the included angle of a triangle. It's especially useful when dealing with non-right angled triangles. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is given by:\[ b^2 = a^2 + c^2 - 2ac \cos(\gamma) \] For the triangle \(\triangle ABC\) in this exercise, this means if you know sides \(a\) and \(c\), and angle \(\gamma\) between them, you can find the missing side \(b\).

• Start by squaring the sides \(a\) and \(c\).
• Multiply the product of \(a\) and \(c\) with the cosine of angle \(\gamma\).
• Subtract this product from the sum of the squares of \(a\) and \(c\).
• Finally, take the square root to compute the length of side \(b\).

Using the given values \(a = 17.31\), \(c = 20.24\), and \(\gamma = 73.01^{\circ}\), we substitute these into the formula and calculate \(b\), ultimately finding it to be approximately \(22.45\). This provides the crucial third side needed to solve the rest of the triangle's properties.

Law of Sines

Once a side length and its opposite angle are known, the Law of Sines helps find the remaining angles and sides. This law is especially useful for solving an ambiguous case, where multiple triangles can form with the given data. It is expressed as:\[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] The principle behind the Law of Sines is that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.In our triangle \(\triangle ABC\), with known \(a = 17.31\), \(b \approx 22.45\), and \(\gamma = 73.01^{\circ}\), once we find \(\gamma\)'s sine using a calculator, we can solve for \(\alpha\) as follows:

• Calculate \(\sin(\gamma)\) to be approximately \(0.9563\).
• Substitute \(a\) and \(\sin(\gamma)\) into the Law of Sines to find \(\sin(\alpha)\).
• Solve for \(\alpha\) by taking the inverse sine (\(\arcsin\)).

This approach yields \(\alpha \approx 55.48^{\circ}\), providing another crucial angle in simplifying our understanding of \(\triangle ABC\).

Triangle Angle Sum Property

The triangle angle sum property is a fundamental rule in geometry that states that the sum of the angles in any triangle is always \(180^{\circ}\). This simple, yet effective property is universally true and vital in finding unknown angles once two angles are identified.By utilizing this property, we can easily determine \(\beta\) in our given triangle \(\triangle ABC\).Given:

• \(\alpha \approx 55.48^{\circ}\)
• \(\gamma = 73.01^{\circ}\)

You can find \(\beta\) as follows:

• Subtract \(\alpha\) and \(\gamma\) from \(180^{\circ}\).
• \(\beta \approx 180^{\circ} - 55.48^{\circ} - 73.01^{\circ}\)

After calculating, \(\beta\) is found to be approximately \(51.51^{\circ}\). This property ensures we verify the consistency and completion of the triangle's angle measures and is an elegant testament to the balance inherent in all triangles.