Problem 29 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$b=12.2, c=20, A=65^{\circ}$$

Step-by-Step Solution

Answer

Question: Given a triangle with side b = 12.2, side c = 20, and angle A = $$65^{\circ}$$, find the values of side a, angle B, and angle C. Answer: In the given triangle, side a is approximately 13.43, angle B is approximately $$54.23^{\circ}$$, and angle C is approximately $$60.77^{\circ}$$.

Step 1: Use the Law of Cosines to find side a

The Law of Cosines states that $$a^{2}=b^{2}+c^{2}-2bc\cos{A}$$. Use the given values to solve for side a: $$a^{2} = 12.2^{2} + 20^{2} - 2(12.2)(20)\cos{65^{\circ}}$$ Evaluate the equation to find $$a \approx 13.43$$.

Step 2: Use the Law of Sines to find angle B

The Law of Sines states that $$\frac{a}{\sin{A}} = \frac{b}{\sin{B}}$$. Use the found value of side a and the given angle A to solve for angle B: $$\frac{13.43}{\sin{65^{\circ}}} = \frac{12.2}{\sin{B}}$$ Evaluate the equation to find $$\sin{B} \approx 0.81114$$, so $$B \approx 54.23^{\circ}$$.

Step 3: Find angle C using the triangle angle-sum theorem

The sum of the angles in a triangle is equal to $$180^{\circ}$$. Using the values of angle A and angle B, find angle C: $$C = 180^{\circ} - A - B = 180^{\circ} - 65^{\circ} - 54.23^{\circ} \approx 60.77^{\circ}$$ Now we have all sides and angles: side a is approximately 13.43, angle B is approximately $$54.23^{\circ}$$, and angle C is approximately $$60.77^{\circ}$$. The triangle is solved.

Key Concepts

Law of Sines

The Law of Sines is a key formula in trigonometry that helps us solve triangles, particularly when we have knowledge of angles and sides scattered across the triangle. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is consistent throughout the triangle. In other words, if you have a triangle with sides labeled as \(a\), \(b\), and \(c\), and corresponding opposite angles \(A\), \(B\), and \(C\), the law can be expressed as:

• \(\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}\)

This means if you know any two pairs of side-lengths and their opposite angles, you can find the third.
In our example, after using the Law of Cosines to find side \(a\), we leveraged the Law of Sines to determine angle \(B\). This was done by setting up the proportion \(\frac{13.43}{\sin{65^{\circ}}} = \frac{12.2}{\sin{B}}\), and solving for \(\sin{B}\) to find angle \(B\).
Using precise calculations is crucial here to ensure accuracy, especially in scientific contexts where slight deviations can impact results.

Triangle solving

Solving a triangle typically involves finding unknown angles and side lengths, and may require a variety of trigonometric laws and theorems. In our given exercise, the approach was systematic:

• Using the Law of Cosines to solve for one side when we knew another two sides and an angle.
• Applying the Law of Sines to uncover an unknown angle once one side and angle pair was known.
• Using the Angle Sum Theorem to tie everything together by finding the final unknown angle.

These steps together form a comprehensive approach to solving triangles. Moreover, they emphasize the relationship between angles and sides, highlighting how rules in geometry and trigonometry are interconnected.
Depending on what is given and what needs to be found, different combinations of these theorems and laws may be employed. This makes mastering them quite valuable for anyone interested in mathematics or related fields.

Angle sum theorem

The Angle Sum Theorem is a basic yet powerful theorem in geometry, forming the foundation for solving angles in a triangle. It states that the sum of the interior angles of a triangle is always \(180^{\circ}\). In practical terms, if you know any two angles of a triangle, you can easily determine the third.
For our exercise, once angles \(A\) and \(B\) were calculated, we immediately applied the Angle Sum Theorem to find angle \(C\). The calculation went as follows:

• \(C = 180^{\circ} - A - B\)
• \(C = 180^{\circ} - 65^{\circ} - 54.23^{\circ} = 60.77^{\circ}\)

This process highlights how straightforward this principle is, making it a handy tool whenever tackling problems involving triangles.
Understanding the Angle Sum Theorem provides a stepping stone to exploring more complex geometric concepts, laying the groundwork for virtually all problems involving triangular shapes.