Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\gamma=115^{\circ} 10^{\prime}, \quad a=1.10, \quad b=2.10$$

Step-by-Step Solution

Answer

Use Law of Sines and triangle property to find \( c \) and remaining angles.

Step 1: Identify the Given Information

We are given triangle \( \triangle ABC \) with one angle and two sides: \( \gamma = 115^{\circ} 10' \), \( a = 1.10 \), and \( b = 2.10 \). Our task is to find the unknowns \( \alpha \), \( \beta \), and \( c \).

Step 2: Use the Law of Sines

Since we have one angle and two sides, we can use the Law of Sines: \[\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\] We start by finding \( c \), substituting the values we know:\[\frac{c}{\sin 115^{\circ} 10'} = \frac{1.10}{\sin \alpha}\] Since we need \( c \), continue by finding \( \sin \gamma \):\( \sin 115^{\circ} 10' \approx 0.906 \\) hence:\[ c = \frac{2.10 \times 0.906}{\sin \gamma} \approx 2.10 \\] already calculated for initial purposes.

Step 3: Find Angle α Using Property of Triangle

We know that the sum of angles in a triangle is \( 180^{\circ} \). Thus, if \( \gamma = 115^{\circ} 10' \), then:\[ \alpha + \beta = 180^{\circ} - 115^{\circ} 10' = 64^{\circ} 50' \\] Use additional calculations to find the specific angles if needed using known values.

Step 4: Calculate Angle α

Substitute back into the equation for both sides with known values optimizing values from original input. Compute: \( \alpha = \text{computed value from step 3} \\) This gives remaining expected values.

Step 5: Solve for Remaining Sides If Needed

Continue calculations to determine remaining sides \( c \) using the above law if any inaccuracies in previous calculation are found. Additional trigonometric functions or vectors may assist especially for rounding edges or vaguely determined points.

Key Concepts

Solving Triangles

To solve a triangle means finding all the unknown angles and sides, given some of them. In \(\triangle ABC\), we aim to determine \(\alpha\), \(\beta\), and \(c\) given \(\gamma=115^{\circ} 10'\), \(a=1.10\), and \(b=2.10\).
The Law of Sines is a helpful tool when solving such problems, especially when you know two sides and one angle, or two angles and one side.
For this triangle, an essential step is using the relationships between angles and sides, as stated in the Law of Sines:

• \(rac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\).
• These equivalencies help us to find unknown side lengths and angle measures not directly given in the problem.
• In this case, we start by solving for the side \(c\).

By systematically applying trigonometric principles, one can determine the elements of any triangle.

Angle Calculations

Angle calculations are crucial when solving triangles. In any triangle, the sum of the angles is always \(180^{\circ}\). Applying this to our triangle, after knowing \(\gamma = 115^{\circ} 10'\), we can find \(\alpha + \beta\) by calculating:
\[ \alpha + \beta = 180^{\circ} - 115^{\circ} 10' = 64^{\circ} 50' \]
Knowing the sum of two angles, you can find each by either using additional trigonometric identities or by direct measurement if additional angles are provided. Keep in mind:

• These calculations must ensure the angles fit within the triangle's total 180-degree constraint.
• Being accurate in calculating and checking the sum of angles can help to confirm the sides calculated are correct.

Mastering this consistent check is essential in making sure everything adds up.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in solving triangles. For this specific problem, the Law of Sines leverages the sine function to relate angles to their opposite sides.
Using the sine function, for the angle \(\gamma\) in our given triangle, is necessary to determine side \(c\). Here, we found:

• \(\sin 115^{\circ} 10' \approx 0.906\)
• This allows for framing the equation \(\frac{c}{0.906} = \frac{b}{\sin \gamma}\)
• And solving further gives \(c\) as \(\approx 2.10\) after calculations.

By understanding how these functions work in tandem with the angle relationships, you can calculate unknown sides and angles efficiently with confidence.
While the calculations can become complex, knowing how to effectively use trigonometric functions often simplifies the problem significantly.