Solve The Triangle. Round Decimal Answers To The Nearest Tenth

Solving a triangle means finding all unknown sides and angles using the given information. This process is essential in trigonometry and practical applications such as navigation, engineering, and physics. A triangle can be solved when enough information is provided, typically through combinations of sides and angles known as cases: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle, which may yield zero, one, or two solutions).

To begin solving a triangle, it is important to identify which case applies. For SSS, the Law of Cosines is used to find the angles. For SAS, the Law of Cosines finds the third side, and then the Law of Sines can be used for the remaining angles. In ASA and AAS cases, the Law of Sines is the primary tool. The SSA case, known as the ambiguous case, requires careful analysis to determine the number of possible triangles.

The Law of Sines states that for any triangle with sides a, b, c and opposite angles A, B, C: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ] The Law of Cosines relates the sides and one angle: [ c^2 = a^2 + b^2 - 2ab \cos C ] When solving, it is helpful to start by finding the largest angle (opposite the longest side) to avoid ambiguity in the Law of Sines. After finding one angle, the others can be determined by subtraction from 180 degrees, since the sum of angles in a triangle is always 180 degrees.

For example, suppose a triangle has sides a = 7, b = 5, and c = 10. To find angle C (opposite side c), use the Law of Cosines: [ \cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{7^2 + 5^2 - 10^2}{2 \times 7 \times 5} = \frac{49 + 25 - 100}{70} = \frac{-26}{70} \approx -0.3714 ] So, angle C is approximately ( \arccos(-0.3714) \approx 111.8^\circ ).

Next, use the Law of Sines to find angle A: [ \frac{a}{\sin A} = \frac{c}{\sin C} \implies \sin A = \frac{a \sin C}{c} = \frac{7 \times \sin(111.8^\circ)}{10} \approx \frac{7 \times 0.927}{10} \approx 0.649 ] Thus, angle A is approximately ( \arcsin(0.649) \approx 40.5^\circ ).

Finally, angle B is ( 180^\circ - 111.8^\circ - 40.5^\circ = 27.7^\circ ).

Rounding to the nearest tenth, the triangle's angles are: A = 40.5°, B = 27.7°, C = 111.8°.

In another scenario, if two angles and a side are known (AAS or ASA), first find the third angle by subtraction, then use the Law of Sines to find the unknown sides. For SSA, check for the ambiguous case by comparing the given side opposite the known angle to the other given side and the altitude from the known angle. This can yield zero, one, or two valid triangles.

Accuracy is crucial. Always carry extra decimal places during calculations and round only at the final step to the nearest tenth, as specified. Using a scientific calculator in degree mode ensures correct trigonometric values.

In summary, solving triangles involves identifying the case, selecting the appropriate law (Sines or Cosines), performing calculations with attention to precision, and rounding final answers to the nearest tenth. Mastery of these steps allows for confident resolution of any triangle problem encountered in academic or real-world contexts.