Problem 96 - Algebra and Trigonometry

Question

Solve triangle \(A B C\) with \(A=39^{\circ}, b=5,\) and \(c=7\) Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Step-by-Step Solution

Answer

The triangle \( ABC \) has side lengths \( a \approx 4.3 \), \( b=5 \), and \( c=7 \) and angles \( A = 39^\circ \), \( B \approx 28^\circ \), and \( C = 113^\circ \)

Step 1: Finding angle B using the Law of Sines

Using the Law of Sines, which states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), and knowing \( A \), \( b \) and \( c \), we can express this as \( B = \sin^{-1} (\frac{b \cdot \sin(A)}{c}) = \sin^{-1} (\frac{5 \cdot \sin(39^\circ)}{7}) \)

Step 2: Computing the value of B

By performing the calculation as suggested in step 1, we find that \( B \approx 28^\circ \)

Step 3: Finding angle C

Since a triangle has \( 180^\circ \), we subtract the measure of \( A \) and \( B \) from \( 180^\circ \) to get \( C = 180^\circ - A - B = 180^\circ - 39^\circ - 28^\circ = 113^\circ \)

Step 4: Finding side a

We can express side \( a \) in terms of known quantities using the Law of Sines as above, \( a = c \cdot \frac{\sin(A)}{\sin(C)} = 7 \cdot \frac{\sin(39^\circ)}{\sin(113^\circ)} \)

Step 5: Computing the value of a

By performing the calculation as suggested in step 4, we find that \( a \approx 4.3 \)

Key Concepts

Solving Triangles

Understanding the process of solving triangles is crucial in trigonometry. When solving triangles, particularly non-right-angled triangles, we commonly use the Law of Sines, the Law of Cosines, or trigonometric ratios.

When provided with some measurements of a triangle such as a couple of sides and an angle or two angles and a side (which are not always right angles), these laws enable us to find the unknown measurements. The Law of Sines, which is applicable in situations where we know either two angles and one side (AAS or ASA) or two sides and a non-enclosed angle (SSA), is especially useful. However, SSA could also result in the ambiguous case, which may lead to zero, one, or two solutions.

Advanced problem-solving might involve the use of both the Law of Sines and the Law of Cosines to solve for unknown sides or angles, especially in cases where the triangle does not include a right angle. This is fundamental as it applies to various real-world scenarios such as navigation, construction, and physics.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The basic functions of trigonometry are sine (sin), cosine (cos), and tangent (tan), each relating a specific angle to a ratio of two sides of a triangle.

These functions are essential in solving triangles, which is a common problem in mathematics and related fields. Trigonometry is not only about solving for missing sides and angles; it also involves understanding the periodic nature of the trigonometric functions, their graphs, and applications in different areas such as sound waves, light waves, and other periodic phenomena.

The ability to navigate trigonometric functions and apply them correctly is paramount in fields that require precise calculations of distances and angles, such as in engineering, astronomy, and computer graphics.

Angle Measurement

Angle measurement, in the context of trigonometry, is essential for solving triangles. Angles can be measured in degrees (\(^\text{o}\)), radians, or grads, with degrees being the most common in trigonometry problems.

There are several vital concepts related to angle measurement such as complementary angles (two angles that add up to 90 degrees), supplementary angles (two angles that add up to 180 degrees), and the fact that the sum of angles in any triangle will always equal 180 degrees. This last concept is especially useful when you're trying to find an unknown angle in a triangle, given the other two angles.

Understanding how to measure and calculate angles is key to successfully applying the Law of Sines and other trigonometric principles to solve triangles. Beyond pure mathematics, angle measurement is also significant in fields like astronomy, where it helps determine the position of celestial bodies, and navigation, where it enables mariners and pilots to chart their courses.