Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of sines or the Law of cosines to solve the triangle. $$B=71^{\circ}, a=21, c=29$$

Step-by-Step Solution

Answer

The triangle can be solved using the law of sines and cosines. First, use law of cosines to find angle C using sides a and c and provided angle B, which gives an equation in terms of angle a and side b. Then, use the law of sines with the found angle C, side c and provided angle B to find side b. Lastly, we found the angle a = 180 - B - C and check whether all the angles add up to 180 and the sides follow the law of sines or cosines. The final values of a, b and C depend on the actual computations.

Step 1: Finding the Angle A using the Law of Cosines

First, use the Law of Cosines which states that \(c^{2} = a^{2} + b^{2} - 2*a*b*\cos(C)\). We can rearrange this formula for cosine C as: \(\cos(C) = (a^{2} + b^{2} - c^{2}) / 2ab\). Applying these values, \(\cos(C) = (a^{2} + b^{2} - c^{2}) / 2ab = (21^{2} + b^{2} - 29^{2}) / 2*21*b\). Let's denote this equation as (1).

Step 2: Finding the Angle b

We know that in a triangle the sum of all angles are 180 degrees. Let \(A = a\) and \(B = 71\), hence \(C = 180 - A - B = 180 - 71 - a = 109 - a\). Substitute \(C\) in equation (1), we can solve for \(b\)

Step 3: Finding the Angle C with the Law of Sines

Use the Law of sines, that says \(\sin(A) / a = \sin(B) / b = \sin(C) / c\). We rearrange the formula to solve for \(C = \arcsin(c*\sin(B)/b)\). Substitute \(B = 71\), \(c = 29\), and \(b\) from Step 2 to find \(C\)

Step 4: Checking the Solution

Once all the values for the angles and sides of the triangle have been found, check that the angles add up to 180 degrees, and that the sides follow the law of sines or cosines

Key Concepts

Law of Sines

The Law of Sines is a fundamental rule in trigonometry that helps solve triangles. Especially useful in non-right triangles, it establishes a relationship between the lengths of sides and the sines of their opposite angles. Here's the core idea:

\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
Where each letter represents the side opposite the corresponding angle. This law is particularly helpful when dealing with cases where two angles and a side are known, or two sides and a non-enclosed angle. It helps maintain consistent ratios, allowing us to find unknown angles or sides.

• Ensure you know at least one side and its opposite angle.
• Use the formula to find unknowns, adjusting the setup for the specific case you're solving.

When applied, this tool simplifies the process of solving complex triangles. It's crucial to remember that sometimes using the inverse sine function might give two possible solutions, especially in ambiguous cases. Always verify by adding angles to ensure they total 180 degrees.

Triangle Solution

Triangular problems in mathematics often involve finding missing sides and angles. In any triangle, the sum of the interior angles is always 180 degrees. This basic rule is crucial when solving for an unknown angle after determining one or two other angles.

When tasked with solving a triangle, you can follow these steps:

• Determine all possible known sides and angles.
• Use either the Law of Sines or Law of Cosines, depending on the given information.
• Calculate the remaining sides and angles using these formulas.
• Confirm that the sum of the angles equates to 180 degrees as a check.

The choice between the Law of Sines and the Law of Cosines depends on what data you start with; one approach may be more direct or efficient, while the other ensures all elements conform to the structural rules of geometry. Try solving some examples yourself to see how these formulas consistently guide you to a solution.

Trigonometry Problem Solving

Trigonometry becomes essential when tackling geometric problems that involve triangles. Beyond the basics, it's a field that aids in understanding relationships within a triangle, whether it's measuring distances, calculating angles, or deducing areas. Here are how some concepts can be applied to efficiently solve problems:

• Start by sketching the triangle if a visual reference helps.
• Identify known values such as side lengths and angles, which are key triggers for the right formula.
• Apply the Law of Sines or Cosines strategically, always considering the simplest path to the unknowns.
• If needed, revert to basics like the triangle angle sum for additional unknown angles.

It's also important to keep in mind the utility of inverse trigonometric functions, like arcsin or arccos, when resolving indeterminate forms. Remember to cross-verify with trigonometric identities if you encounter difficulties. With practice, you'll become adept at seamlessly solving trigonometric problems.