Problem 28 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$b=14.6, c=7.8, B=40.4^{\circ}$$

Step-by-Step Solution

Answer

Question: Given a triangle with side b = 14.6 units, side c = 7.8 units, and angle B = 40.4 degrees, solve the triangle by finding the remaining side and angles. Answer: To solve the triangle, first find the length of side a using the Law of Cosines: $$a = \sqrt{14.6^2 + 7.8^2 - 2 * 14.6 * 7.8 * \cos(40.4^{\circ})}$$ Next, find angle A using the Law of Sines: $$A = \arcsin\frac{a * \sin(40.4^{\circ})}{14.6}$$ Finally, find angle C using the fact that all angles in a triangle add up to 180 degrees: $$C = 180^{\circ} - A - B$$ Make sure to include the side lengths of a, b, and c, as well as the angles A, B, and C in degrees in your final solution.

Step 1: Find side a using the Law of Cosines

The Law of Cosines states that in any triangle: $$a^2 = b^2 + c^2 - 2bc * \cos A$$ We have the values for sides b and c, as well as angle B. We can rearrange the Law of Cosines to find side a: $$a = \sqrt{b^2 + c^2 - 2bc * \cos B}$$ Now, we can plug in the given values: $$a = \sqrt{14.6^2 + 7.8^2 - 2 * 14.6 * 7.8 * \cos(40.4^{\circ})}$$ Calculate the value to find the length of side a.

Step 2: Find angle A using the Law of Sines

The Law of Sines states that in any triangle: $$\frac{\sin A}{a} = \frac{\sin B}{b}$$ We have the value for angle B, side a, and side b. We can rearrange this formula and solve for angle A: $$A = \arcsin\frac{a * \sin B}{b}$$ Plug in the values for a, b, and B, and calculate angle A: $$A = \arcsin\frac{a * \sin(40.4^{\circ})}{14.6}$$

Step 3: Find Angle C using the fact that all angles in a triangle add up to 180 degrees

Since the sum of all angles in a triangle is 180 degrees, we can find angle C using the equation: $$C = 180^{\circ} - A - B$$ Substitute the values we found for angles A and B and calculate angle C.

Step 4: Write the final solution

With all sides a, b, and c, and all angles A, B, and C now found, write the final solution for the triangle. Make sure to include the side lengths of a, b, and c, as well as the angles A, B, and C in degrees.

Key Concepts

Triangle Solving

Solving a triangle means finding all its sides and angles when some are given. When working with triangles, there are different approaches depending on the information one starts with. Here, what is available is one angle (angle B) and two sides of a triangle (sides b and c).

Let's break down the process:

• Measurements Provided: Side b = 14.6, Side c = 7.8, and Angle B = 40.4°.
• Objective: Find Side a, Angle A, and Angle C.

To solve for these missing values, we first used the Law of Cosines to find the length of side a. This is necessary when you have the measures of two sides and an included angle. Next, the Law of Sines helps find an unknown angle when two sides are known, followed by the use of angle sum property to find the remaining angle.
It's like piecing together a puzzle; each part helps complete the picture until the entire triangle is clearly understood.

Angle Calculation

Knowing how to calculate angles within a triangle is vital for solving triangles fully. The process involves using trigonometric laws and properties that relate the angles and sides of the triangle.

Angle calculation here happens in a couple of important steps:

• Law of Cosines: Start by using the Law of Cosines to find one side. This is useful when you know two sides and the angle between them: \[a = \sqrt{b^2 + c^2 - 2bc \cdot \cos B}\]
• Law of Sines: Once you find all three sides, use the Law of Sines to find one of the other angles, such as angle A: \[A = \arcsin\left(\frac{a \cdot \sin B}{b}\right)\]
• Sum of Angles in a Triangle: Finally, use the fact that the sum of angles in a triangle is always 180° to find the remaining angle: \[C = 180^{\circ} - A - B\]

Each of these steps involves precise calculator work and a solid understanding of trigonometric functions and properties.

Law of Sines

The Law of Sines is an essential principle in triangle solving. It states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. Formally, it is written as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]The Law of Sines is particularly useful when you know:

• Two angles and one side (AAS or ASA situations).
• Two sides and a non-included angle (SSA situation), which is the case in our original problem.

After applying the Law of Cosines to find the missing side, the Law of Sines can be utilized to determine the unknown angles. This process is critical as it allows us to complete the triangle when not all angles or sides are initially known.
Using the Law of Sines often simplifies the complex relationships within a triangle, making it possible to analyze and solve each part effectively. It is a bridge from known quantities to discovering unknown values efficiently in various triangle-related problems.