Solve The Triangle. Round Decimal Answers To The Nearest Tenth

Solving a triangle involves determining all unknown side lengths and angle measures when enough information is provided. This process, often called solve the triangle, relies on fundamental trigonometric principles such as the Law of Sines and the Law of Cosines. In many educational contexts, after calculating the required values, students are instructed to round decimal answers to the nearest tenth to present results in a concise, easily interpretable format. This article walks you through the essential concepts, systematic steps, and practical examples needed to master triangle solving while adhering to the rounding convention.

Understanding the Basics of Triangle Solving

Key Concepts

A triangle has three sides (commonly labeled a, b, c) and three interior angles (labeled A, B, C). The relationships among these elements are governed by two primary trigonometric laws:

• Law of Sines: a / sin A = b / sin B = c / sin C
• Law of Cosines: c² = a² + b² – 2ab cos C (and its cyclic permutations)

These formulas allow you to transition between known sides and angles. Additionally, the sum of interior angles in any triangle is always 180°, a fact that helps verify the consistency of your calculations.

Step‑by‑Step Guide to Solve the Triangle

When faced with a triangle problem, follow a logical sequence to uncover all missing measurements:

1. Identify the Given Data

   • Note which sides and angles are known. Common scenarios include:
     • SSA (two sides and a non‑included angle)
     • SAS (two sides and the included angle)
     • ASA or AAS (two angles and a side)
     • SSS (all three sides)

2. Select the Appropriate Law

   • For SAS or SSS, employ the Law of Cosines to find the unknown side or angle.
   • For ASA, AAS, or SSA, the Law of Sines is usually the most efficient choice.

3. Compute the Missing Side or Angle

   • Rearrange the chosen formula to isolate the desired variable.
   • Perform algebraic manipulations carefully to avoid sign errors.

4. Determine the Remaining Unknowns

   • Use the angle‑sum property (180°) to find any remaining angle.
   • Apply the appropriate law again if a side is still unknown.

5. Round Decimal Answers to the Nearest Tenth

   • Examine each decimal result.
   • Look at the hundredths place: if it is 5 or greater, increase the tenths digit by one; otherwise, keep the tenths digit unchanged.
   • Record the rounded value with one decimal place.

6. Check for Ambiguities (SSA Case)

   • The SSA configuration can yield two possible triangles (the “ambiguous case”).
   • Verify whether the computed angle is acute or obtuse and whether a second solution satisfies the triangle inequality.

Using the Law of Sines and Cosines

Law of Sines in Practice

When you know two angles and a side (ASA or AAS), you can find the remaining sides by setting up proportions:

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

Example: If A = 42°, B = 58°, and a = 7.5, then

[ \frac{7.5}{\sin 42°} = \frac{b}{\sin 58°} ]

Solving for b yields b ≈ 9.3 after rounding to the nearest tenth.

Law of Cosines for SAS and SSS

For a triangle where two sides and the included angle are known (SAS), the Law of Cosines provides the third side:

[ c = \sqrt{a^{2} + b^{2} - 2ab\cos C} ]

Example: With a = 5.2, b = 8.7, and C = 30°,

[ c = \sqrt{5.2^{2} + 8.7^{2} - 2(5.2)(8.7)\cos 30°} \approx 7.9 ]

If all three sides are known (SSS), the Law of Cosines can be rearranged to find any angle:

[ \cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab} ]

Then compute C and round the result.

Rounding Decimal Answers to the Nearest Tenth

Rounding is a crucial step for presenting clean, interpretable answers. The procedure is straightforward:

• Identify the digit in the tenths place (the first digit after the decimal point).
• Look at the hundredths digit (the second digit after the decimal).
• If the hundredths digit is 5, 6, 7, 8, or 9, increase the tenths digit by one.
• If the hundredths digit is 0–4, leave the tenths digit unchanged.
• Drop all digits beyond the tenths place.

Illustration:

• 12.37 → 12.4 (since 7 ≥ 5)
• 5.84 → 5.8 (since 4 < 5)

Applying this rule consistently ensures that every numeric answer in your triangle‑solving work adheres to the required format.

Worked Example: Solving a Triangle with SAS and Rounding

Suppose you are given a triangle where side a = 6.4, side b = 9.1, and the included angle *C = 55°

Now that the three pieces of data have been laid out, the next move is to determine the length of the side opposite the known angle.
Applying the cosine relationship for the included angle gives

[ c=\sqrt{a^{2}+b^{2}-2ab\cos C} ]

Substituting the numbers:

[ c=\sqrt{6.4^{2}+9.1^{2}-2(6.4)(9.1)\cos 55^{\circ}} ]

Evaluating the cosine (≈ 0.5736) and performing the arithmetic yields

[ c\approx\sqrt{40.96+82.81-2(6.4)(9.1)(0.5736)} =\sqrt{123.77-6.71} =\sqrt{117.06} \approx 10.8 ]

The result is already expressed to one decimal place, so no further rounding is required for this side.

With the third side in hand, the remaining interior angles can be uncovered.
Because the triangle is now known completely (SSS), the Law of Sines offers a quick path to the unknown angles. Using the proportion

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

we isolate (\sin A):

[ \sin A = \frac{a\sin C}{c} = \frac{6.4\sin 55^{\circ}}{10.8} ]

Since (\sin 55^{\circ}\approx0.8192),

[ \sin A \approx \frac{6.4 \times 0.8192}{10.8} \approx \frac{5.2429}{10.8} \approx 0.4855 ]

Taking the inverse sine produces two mathematical possibilities, but only the acute value fits the geometry of an SAS triangle (the obtuse counterpart would force the sum of angles to exceed 180°). Hence

[ A \approx \sin^{-1}(0.4855) \approx 29.0^{\circ} ]

The last angle follows from the angle‑sum rule:

[ B = 180^{\circ} - A - C = 180^{\circ} - 29.0^{\circ} - 55^{\circ} \approx 96.0^{\circ} ]

Both computed angles are reported to the nearest tenth, as instructed.

Verifying Consistency

A quick sanity check confirms that the three angles add to roughly 180°, and the side lengths obey the triangle inequality (each side is shorter than the sum of the other two). No second triangle emerges from this configuration because the given angle is the one enclosed by the two known sides; the ambiguous SSA scenario does not apply here.

Conclusion

The process of solving a triangle when two sides and their included angle are known proceeds in a clear, repeatable sequence:

1. Compute the unknown side with the cosine formula, then round the outcome to the nearest tenth.
2. Derive the remaining angles using the sine relationship, again rounding each result to one decimal place.
3. Confirm that the angles sum to 180° and that the side lengths satisfy the triangle inequality, ensuring the solution is unique and physically plausible.

By adhering to these steps—particularly the disciplined rounding routine—students and practitioners can present triangle solutions that are both mathematically sound and neatly formatted, ready for inclusion in reports, exams, or practical applications.