Question

Find the unknown angles in triangle ABC for each triangle that exists.C = 41° 20', b = 25.9 m, c = 38.4 m

Accepted Answer

Identify the given elements of triangle ABC: angle $$ C = 41^\circ 20' $$, side $$ b = 25.9  m$$, and side $$ c = 38.4  m. We $$need to find the unknown angles $$ A $$ and $$ B . $$Use the Law of Sines, which states that $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} . $$Since we know $$ b $$, $$ c $$, and $$ C $$, we can find angle $$ B  by $$setting up the ratio $$ \frac{b}{\sin B} = \frac{c}{\sin C} . $$Rearrange the equation to solve for $$ \sin B $$:

$$\sin B = \frac{b \cdot \sin C}{c}$$

Calculate $$ \sin C  by $$converting $$ 41^\circ 20'  to $$decimal degrees first (i.e., $$ 41 + \frac{20}{60} ) $$and then find $$ \sin C . $$Determine angle $$ B  by $$taking the inverse sine (arcsin) of the value found for $$ \sin B . $$Note that there may be two possible angles for $$ B $$ because sine is positive in both the first and second quadrants, so consider both $$ B $$ and $$ 180^\circ - B  if $$they satisfy the triangle inequality. Find angle $$ A $$ using the fact that the sum of angles in a triangle is $$ 180^\circ $$:

$$A = 180^\circ - B - C$$

Check for the existence of one or two triangles based on the possible values of $$ B $$ and ensure all angles are positive and less than $$ 180^\circ .$$

Find the unknown angles in triangle ABC for each triangle that exists.

C = 41° 20', b = 25.9 m, c = 38.4 m

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주요 개념

Law of Cosines

Law of Sines

Ambiguous Case of the Law of Sines (SSA Condition)