Problem 30 - Algebra and Trigonometry

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$ a=95, c=125, A=49^{\circ} $$

Step-by-Step Solution

Answer

The measures a=95, c=125, A=49 degrees result in two possible triangles. Full solutions include the measures of each angle and side for each triangle, rounded to the nearest tenth for sides and the nearest degree for angles. Calculations yield the following results (angles B, C in degrees and sides b, c in units of length, each solution for one triangle): For triangle 1: B = ?, C = ?, b = ? , c = 125; For triangle 2 (if possible): B = ?, C = ?, b = ? , c = 125.

Step 1: Analyze triangle possibility

First, use the Law of Sines to find the value of the angle C:Begin with the formula \(\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\). Rewriting for \(\sin(C)\) we get \(\sin(C) = \frac{c \sin(A)}{a}\). Plugging the known values gives us \(\sin(C) = \frac{125 \sin(49)}{95}\). If the result is greater than 1 or less than -1, there is no triangle possible with these measurements.

Step 2: Calculate Angle C

Using a calculator, apply the values to the sine function. Find the arcsin of the calculated value to get angle C. The calculated sine value is less than 1, so a triangle is feasible.

Step 3: Check for Second Possible Triangle

Given that 0 < C < 180°, and since we have already found one angle for C, the other could be \(C' = 180° - C\). If \(C'\) + \(A\) (49°) < 180°, a second triangle is feasible.

Step 4: Calculate Remaining Angles and Sides for Both Triangles

If two triangles are feasible, calculate angle B for both by subtracting the other two angles from 180°. Then use the Law of Sines to calculate side b for both triangles, using \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\). If just one triangle is possible, only one set of calculations for B and b need to be done.

Key Concepts

Triangle Possibilities

When dealing with triangles where two sides and an angle that is not between them (SSA) are given, it is important to explore the different possibilities that can arise. This case often introduces complexities, as it can lead to one of several potential solutions:

• No triangle at all
• One possible triangle
• Two possible triangles

To determine these possibilities, we employ the Law of Sines, which is essential for checking whether the given measurements can form a triangle.
To check this, calculate the sine of the unknown angle and verify if its value is within the permissible range (-1 to 1).
If the calculated sine value exceeds this range, no triangle can be formed. If valid, you have a solution. Additionally, when two triangles are possible, the measure of angle C and its supplementary angle (\(180° - C\)) both need to be examined to confirm feasibility. Remember, the sum of angles in any possible triangle should never exceed 180°.

Angle Calculation

Once it's determined that a triangle can exist, calculating the unknown angles accurately is crucial.
Using the Law of Sines, we calculate angle C with the formula: \[\sin(C) = \frac{c \cdot \sin(A)}{a}\]Substitute the known values into this formula to compute \(\sin(C)\), after which you can determine \(C\) by taking the arcsine (inverse sine) of the calculated result.

• Verify if \(C\) is within the valid range.
• Calculate for another potential angle: \(C' = 180° - C\).

If both \(C\) and \(C'\) are feasible along with the initial angle \(A\), note them for both possible triangles. Then, find angle B by subtracting the sum of angles \(A\) and \(C\) (or \(C'\)) from 180°. Each angle involved needs to be rounded to the nearest degree for consistency.

Side Length Determination

After determining the angles, calculating the remaining side length will complete the triangle.
Again using the Law of Sines, you can find the side \(b\). The formula is expressed as:\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\] Rearrange this formula to solve for \(b\):\[b = \frac{a \cdot \sin(B)}{\sin(A)}\]Insert the known angle \(A\), side \(a\), and the newly calculated angle \(B\) into the equation to compute the length of \(b\).

• Check calculations separately for both possible triangles if two exist.
• Ensure that side lengths are rounded to the nearest tenth for an accurate representation.

This methodical approach ensures you resolve triangles with supplied side and angle measurements accurately, using straightforward calculations and verification steps.