Problem 15 - Algebra and Trigonometry

Question

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ A=65^{\circ}, B=65^{\circ}, c=6 $$

Step-by-Step Solution

Answer

Side 'a' is approximately 7.2 units, side 'b' is approximately 7.2 units and angle 'C' is 50 degrees.

Step 1: Calculating angle C

The sum of the interior angles in a triangle is always 180 degrees. Therefore, the measure of angle C is \[180 - A - B = 180 - 65 - 65 = 50^{\circ}\]

Step 2: Calculating side 'a'

Now, use the law of sines to find the remaining sides. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Therefore, we can set up the equation: \[\frac{a}{\sin A} = \frac{c}{\sin C}\]. Plugging in the known values gives us \[\frac{a}{\sin 65^{\circ}} = \frac{6}{\sin 50^{\circ}}\]. Solving for 'a' yields \(a = \frac{6 \cdot \sin 65^{\circ}}{\sin 50^{\circ}}\). Rounding the answer to the nearest tenth, we get \(a \approx 7.2 \)

Step 3: Calculating side 'b'

Similarly, we use the sine rule again to find the length of side 'b'. Setup the equation as \[\frac{b}{\sin B} = \frac{c}{\sin C}\]. Substitute known values, this gives \[\frac{b}{\sin 65^{\circ}} = \frac{6}{\sin 50^{\circ}}\]. Solving it gives \(b = \frac{6 \cdot \sin 65^{\circ}}{\sin 50^{\circ}}\). Rounding the answer to the nearest tenth, we get \(b \approx 7.2 \)

Key Concepts

Solving Triangles

To "solve a triangle" means determining all the sides and angles of the triangle when some are given. This process can involve a mix of different mathematical tools, including congruence, similarity, and trigonometric identities.
When given certain measurements, like two angles and one side, you can apply the Law of Sines. This is useful in non-right triangles and bridges the known measurements to the unknown ones. For instance, given two angles and a non-included side, as in the exercise, the law produces a seamless lead to computing the rest.

• Use trigonometric identities or laws (Law of Sines, Law of Cosines) to find missing measures.
• Follow a systematic approach to ensure accuracy: compute unknown angles first using the angle sum property, then move to determining unknown sides.

In our exercise, knowing two angles (65° and 65°) and one side (c = 6) of the triangle, the third angle was first calculated using the angle sum property, followed by the remaining sides using the Law of Sines.

Angle Sum Property

The angle sum property is a fundamental rule in geometry. It states that the sum of all interior angles in a triangle is always 180 degrees. This is true for all types of triangles- equilateral, isosceles, or scalene.
In problem-solving, the angle sum property can immediately determine any unknown angle of a triangle if the other two's measures are known. It simplifies solving for angles and provides a quick check for angle accuracy.

• Calculate missing angles: If you know two angles, subtract their sum from 180° to find the third.
• Ensure correctness: The total angle measure should always add to 180° in a triangle.

In the exercise, knowing angles A and B are both 65°, angle C was found immediately by subtracting their sum from 180°, resulting in 50°.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. In the case of solving triangles, it helps link angles to side lengths through specific ratios and laws.
For non-right triangles, the Law of Sines is pivotal. It forms a ratio of the length of a side of the triangle to the sine of the angle opposite that side. This relationship holds true across all sides and angles of the triangle.

• Law of Sines Formula: \ \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \ \)
• Use this law when you know two angles and one side or two sides and a non-included angle.

In our example, the Law of Sines helped us find sides 'a' and 'b' using the given angle measures and side 'c'. Calculations provided results rounding to the nearest tenth, with both approximately equaling 7.2.

Geometry

Geometry mimics the physical world to provide a visual framework for understanding mathematical relationships. In triangles, measuring lengths and angles taps into geometric principles and aids in visualizing how each part of the triangle is related.
Through understanding concepts like congruence, similarity, and triangle types (isometric properties), one can begin to predict potential side lengths and angles before performing calculations.
In this exercise, the isosceles nature of the triangle (two angles measuring the same) suggests that the non-calculated sides are equal. This symmetry played a pivotal role in simplifying the calculations for triangle-solving.

• Geometric intuition: Recognize patterns, such as equal angles indicating equal sides.
• Visual aids: Diagrams can assist in ensuring calculations are on point and mistakes are minimized.

By combining geometry's fundamental ideas with trigonometric formulas, determining all unknown components of a triangle becomes an enjoyable challenge rather than a daunting task.