Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$A=29.7^{\circ}, b=41.5 \text { feet, } a=27.2 \text { feet }$$

Step-by-Step Solution

Answer

There are two triangles: one with \((A, B, C) = (29.7^\circ, 49.2^\circ, 101.1^\circ)\) and another with \((A, B, C) = (29.7^\circ, 130.8^\circ, 19.5^\circ)\).

Step 1: Use Law of Sines

Use the Law of Sines to find another angle of the triangle: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the given values: \( \frac{27.2}{\sin 29.7^\circ} = \frac{41.5}{\sin B} \). Calculate \( \sin B \approx \frac{41.5 \cdot \sin 29.7^\circ}{27.2} \).

Step 2: Calculate \( \sin B \)

Using a calculator, find \( \sin 29.7^\circ \approx 0.497 \). Substitute into the equation from Step 1, \[ \sin B \approx \frac{41.5 \cdot 0.497}{27.2} \approx 0.757 \].

Step 3: Determine Angle B

Check if \( \sin B \leq 1 \). Since \( 0.757 < 1 \), angle \( B \) exists. Use the inverse sine function: \( B \approx \sin^{-1}(0.757) \approx 49.2^\circ \).

Step 4: Check for Ambiguous Case

Since there is an ambiguous case (angle B is less than 90 degrees), calculate the other possible angle: \( B' = 180^\circ - 49.2^\circ = 130.8^\circ \).

Step 5: Verify Triangle Possibilities

Calculate \( C = 180^\circ - (A + B) \) for both possible angles. For \( B = 49.2^\circ \), \( C = 180^\circ - (29.7^\circ + 49.2^\circ) = 101.1^\circ \). For \( B = 130.8^\circ \), \( C = 180^\circ - (29.7^\circ + 130.8^\circ) = 19.5^\circ \). Both are valid triangles because all angles are positive and sum to 180 degrees.

Step 6: Conclusion

There are two possible triangles: \( A = 29.7^\circ, B = 49.2^\circ, C = 101.1^\circ \) and \( A = 29.7^\circ, B = 130.8^\circ, C = 19.5^\circ \).

Key Concepts

triangle solving

Solving a triangle means finding all its sides and angles. This includes identifying all missing elements given some initial measurements. The process often uses trigonometric principles. For any triangle problem, it's best to start by listing known values:

\[ \begin{align*} A &= 29.7^{\circ}, \ a &= 27.2 \text{ feet}, \ b &= 41.5 \text{ feet} \ \end{align*} \]

With these, the Law of Sines helps find the unknowns. This law relates the lengths of sides of a triangle to the sines of its angles:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

This relation is invaluable in calculating missing angles and sides, ensuring a logical path through the problem.

ambiguous case

The ambiguous case happens in triangle solving when using the Law of Sines. This case arises in an oblique triangle (not a right triangle) under certain conditions. Specifically, in the angle-side-side (ASS) scenario, there can be two possible triangles or sometimes none. It happens because the sine function can yield two possible acute angles for a given positive sine value:

• The calculated angle, from \( \sin^{-1} \), is generally the acute one.
• The supplementary angle, \( 180^{\circ} - \text{angle} \), can also be valid.

This is why checking for an additional possible triangle is crucial whenever you solve triangles with two given sides and a non-included angle. Not doing so might miss a valid solution. Each potential triangle must be examined to ensure it meets the fundamental property: the sum of angles equaling 180 degrees.

angle calculation

Understanding angle calculation when solving triangles is essential for precisely determining all angles. After finding one angle using the initial data and the Law of Sines, other angles can be calculated easily.

1. **Inverse Sine Function:**
After finding \( \sin B \approx 0.757 \), use the inverse sine function to get angle \( B \):

\[ B \approx \sin^{-1}(0.757) \approx 49.2^{\circ} \]

2. **Complementary Angles:**
If the ambiguous case is relevant, calculate the supplementary angle. Its value: \( B' = 180^{\circ} - B \). So here: \( B' = 180^{\circ} - 49.2^{\circ} = 130.8^{\circ} \).

3. **Remaining Angle:**
Find angle \( C \) using the property that the sum of angles in a triangle is always 180 degrees. Compute for both \( B \) and \( B' \):

\[ C = 180^{\circ} - (A + B) \]
Calculate:

\[ C = 180^{\circ} - (29.7^{\circ} + 49.2^{\circ}) = 101.1^{\circ} \]
For the other scenario, \[ C' = 180^{\circ} - (29.7^{\circ} + 130.8^{\circ}) = 19.5^{\circ} \]

This sequence completes the triangle, ensuring all angles are accounted for.