Question

Solve triangle \(A B C\). $$a=11.3 \quad b=15.6 \quad c=12.8$$

Short answer

First, compute angle C using the Law of Cosines, then find angle A using the Law of Sines, and finally calculate angle B knowing the sum of the angles in a triangle is 180 degrees.

Step-by-step solution

Apply the Law of Cosines to find one angle

First, use the Law of Cosines to find one of the angles. We can solve for angle C using the formula: \(c^2 = a^2 + b^2 - 2ab\cos(C)\). Rearrange the formula to solve for \(\cos(C)\) and calculate the value of angle C.

Calculate the cosine of angle C

Plug in the values given for a, b, and c into the rearranged Law of Cosines to find \(\cos(C)\): \(\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\). Substitute a = 11.3, b = 15.6, and c = 12.8 and solve for \(\cos(C)\).

Find the measure of angle C

Use an inverse cosine function (usually on a calculator), to find the measure of angle C from the \(\cos(C)\) value found in Step 2.

Apply the Law of Sines to find another angle

Now, apply the Law of Sines to find another angle, say angle A. The Law of Sines states that \(\frac{\sin(A)}{a} = \frac{\sin(C)}{c}\). Rearrange to solve for \(\sin(A)\) and calculate the value of angle A using the known measures of a and C.

Find the measure of angle A

Solve for angle A by using \(\sin(A) = \frac{a \cdot \sin(C)}{c}\). Then compute A using the value of a and the measure of angle C obtained previously. Use the inverse sine function to find the measure of angle A.

Determine the measure of the remaining angle

The sum of angles in any triangle is 180 degrees. Use this fact to find the remaining angle B. Calculate angle B as \(180^\circ - A - C\).

Key concepts

Law of Cosines

Understanding the Law of Cosines is vital for solving triangles, especially when we are given the lengths of all three sides and need to find the angles. Essentially, it's an extension of the Pythagorean theorem to any triangle, not just right-angled ones. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and the angle opposite to side c being C, the relation is given by:
\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\].
This formula allows us to find the angle C once we know the lengths of the sides. To uncover the angle, we need to isolate \(\cos(C)\) from the formula:
\[\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\].
By substituting the side lengths into this equation, we can calculate the cosine of angle C. From there, using an inverse cosine function—often just a button on your calculator—you can determine the angle measurement itself. It's like a mathematical detective, searching for the hidden angles based on the evidence provided by the side lengths.

Law of Sines

Once we've established one angle using the Law of Cosines, it's time to bring in the Law of Sines for further investigation. The Law of Sines is a proportion that relates the ratios of side lengths to their opposite angles in any triangle. Specifically, the formula is:
\[\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\].
With this relationship, if we know the length of one side of the triangle and the measure of its opposite angle, we can determine another angle. For instance, if we've found angle C using the Law of Cosines, we can use its sine value to find angle A:
\[\sin(A) = \frac{a \cdot \sin(C)}{c}\].
After calculating \(\sin(A)\), we use the inverse sine function to find the measure of angle A. This step continues the process of solving the triangle, and when we have two angles, the third one is straightforward to find since the angles of a triangle always add up to 180 degrees. Each angle sheds light on the next, allowing us to piece together all parts of the triangular puzzle.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the detectives of the triangle world, revealing the angles when we only know the sides. After using the Law of Cosines or the Law of Sines, we often end up with the sine or cosine of an angle. To solve for the actual angle, we use inverse trigonometric functions:
\[\text{angle} = \arccos(\text{value})\ \text{or}\ \arcsin(\text{value})\].
In our exercise, after calculating the cosine and sine of angles, the next step is to find their measures, which is done by using \(\arccos\) or \(\arcsin\) functions. Remember, the outputs of these inverse functions are in radians by default on most calculators, so you might need to convert them to degrees. Moreover, it's crucial to ensure your calculator is set to the correct mode (degree or radian) based on what the problem requires. These functions are the final step in our trigonometric investigation, providing the last piece of information needed to completely solve the triangle.