Problem 29 - Precalculus with Limits

Question

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. \(A\ =\ 58^{\circ}\), \(a\ =\ 11.4\), \(b\ =\ 12.8\)

Step-by-Step Solution

Answer

The two possible solutions to this problem are: \n1) Angles in degrees are approximately A=58°, B=67.83°, C=54.17° and sides in length are a=11.4, b=12.8, c=13.87.\n2) Angles in degrees are approximately A=58°, B=112.17°, C=9.83° and sides in length are a=11.4, b=12.8, c=2.08.

Step 1: Calculate the respective sin ratios

For angle A and side a: \(sinA/a = sin(58)/11.4\).\n For side b and unknown angle B, it will be: \(sinB/b = sinB/12.8\).

Step 2: Use Law of Sines to solve for sinB

Since we know that \(sinA/a = sinB/b\), we can set sinB equal to \((b/a) * sinA\), which gives sinB = \((12.8/11.4) * sin(58)\).

Step 3: Calculate angle B

For first solution, angle B = \(arcsin((12.8/11.4)*sin(58))\) (which lies in first quadrant).\n For the second solution, angle B = \(180 - arcsin((12.8/11.4)*sin(58))\) (which lies in second quadrant).

Step 4: Check validity of solutions

Ensure that the calculated angles don't violate the property that sum of all angles in a triangle should be 180 degrees. If no violation occurs, multiple solutions exist.

Step 5: Calculate for angle C

Once valid angle B values are found, calculate angle C using \(C = 180 - (A + B)\). Do this for both possible solutions.

Step 6: Calculate for side c

Using Law of Sines again, find side c for both possible triangles. Set \(sinC/c = sinA/a\), and solve for c.

Key Concepts

triangle solving

Solving a triangle means finding the unknown lengths of sides and measurements of angles. To solve a triangle completely, you need to have at least one complete angle-side pair, such as a given angle and its opposite side. This is particularly useful when using tools like the Law of Sines or the Law of Cosines. In the example given, we start with angle \( A = 58^{\circ} \), side \( a = 11.4 \), and side \( b = 12.8 \). We aim to find the other angle (\( B \)), the remaining side (\( c \)), and the last angle (\( C \)).

• First, determine what you know: often two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
• Apply relevant trigonometric rules, like the Law of Sines, which is useful when working with any given pair of angle and opposite side.
• Continue by using known information to solve for other unknowns in the triangle.

By following this approach, solving triangles becomes a structured method, taking one step at a time to find each missing piece.

trigonometry

Trigonometry is a branch of mathematics focusing on relationships between angles and sides of triangles. In triangle solving, trigonometric laws are fundamental. The Law of Sines is particularly useful in our context.The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant. It can be written as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This law becomes particularly powerful when you have one complete angle-side pair, allowing you to find unknown sides or angles. For example, if you know \( a \) and \( A \), you can easily find \( b \) or \( B \) using:\[\sin B = \frac{b}{a} * \sin A\]

• This approach helps to directly relate side lengths with their respective angles.
• It allows for straightforward calculations when two angles and one side are known, or two sides and a non-included angle are given.

Trigonometry thus provides the mathematical tools and formulas needed to understand these interrelationships within a triangle.

angle calculation

Angle calculation is a critical step in solving triangles, often requiring the use of trigonometric functions such as sine. When using the Law of Sines, you can solve for an unknown angle once you have identified a complete angle-side pair in the triangle. To calculate an unknown angle \( B \), knowing \( \sin B \) is a crucial first step. Using the formula \[\sin B = \frac{b}{a} * \sin A\]you can determine \( \sin B \). Then, use the arc-sine function to find angle \( B \):\[B = \arcsin\left(\frac{b}{a} * \sin A\right)\]

• Remember that the \( \arcsin \) function will initially give an angle in the first quadrant.
• For a triangle, there can be a second possible solution for angle \( B \) in the second quadrant: \( 180 - B \).

Complete the triangle by ensuring the sum of angles adds up to \( 180^{\circ} \), then solve for the remaining angle \( C \) as:\[C = 180^{\circ} - (A + B)\]This process ensures all angles are calculated accurately, allowing for a comprehensive analysis of the triangle.