Question

In questions $1-10$ solve $\mathrm{△}\mathrm{ABC}$ given $\mathrm{BC}=36\mathrm{cm},\mathrm{AC}=92\mathrm{cm},C={51}^{\circ }$

Short answer

Answer: The side lengths of triangle ABC are approximately AB = 120.5 cm, BC = 36 cm, and AC = 92 cm. The angles are approximately A = 38.3°, B = 90.7°, and C = 51°.

Step-by-step solution

Check for the existence of triangle using Law of Sines

Use the Law of Sines to determine if the given information results in a unique triangle: $\frac{\mathrm{BC}}{\sin A}=\frac{\mathrm{AC}}{\sin C}$ Plug in the given side lengths and angle C: $\frac{36}{\sin A}=\frac{92}{\sin {51}^{\circ }}$ Now, solve for $\sin A$: $\sin A=\frac{36\times \sin {51}^{\circ }}{92}\approx 0.6237$ Since $0\le \sin A\le 1$, there exists a unique triangle with the given side lengths and angle.

Find angle A using the inverse sine function

To find angle A, take the inverse sine of the value we found for $\sin A$: $A={\sin }^{-1}(0.6237)\approx {38.3}^{\circ }$

Find angle B using the angle sum property of triangles

Since the sum of the angles in a triangle is always ${180}^{\circ }$, we can find angle B as follows: $B={180}^{\circ }-A-C\approx {180}^{\circ }-{38.3}^{\circ }-{51}^{\circ }\approx {90.7}^{\circ }$

Find side AB using the Law of Sines

Finally, we can find the remaining side length (AB) using the Law of Sines: $\frac{\mathrm{AB}}{\sin B}=\frac{\mathrm{AC}}{\sin C}$ Plug in the values of AC, C, and B: $\mathrm{AB}=\frac{\mathrm{AC}\times \sin B}{\sin C}=\frac{92\times \sin {90.7}^{\circ }}{\sin {51}^{\circ }}\approx 120.5\mathrm{cm}$ Now we have completely solved triangle ABC with side lengths AB $\approx 120.5\mathrm{cm}$, BC $=36\mathrm{cm}$, and AC $=92\mathrm{cm}$, and angles A $\approx {38.3}^{\circ }$, B $\approx {90.7}^{\circ }$, and C $={51}^{\circ }$.

Key concepts

Law of Sines

The Law of Sines is a key concept in trigonometry for solving triangles. It relates the ratios of the sides of a triangle to the sines of its angles. It's especially useful when dealing with non-right triangles. The formula is:

• $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

where $a,b,$ and $c$ are the lengths of the sides, and $A,B,$ and $C$ are the respective opposite angles. In the context of the given problem, the Law of Sines is used to determine if the provided measurements form a valid triangle. By verifying that the calculated sine values fall within the possible range $0\le \sin A\le 1$, we can confirm the triangle's existence. This process highlights how the Law of Sines not only aids in solving triangles but also in verifying them.

Triangle Solving

Solving a triangle means finding the lengths of all its sides and the measures of all its angles when some of these values are given. In our problem, we start with one known angle and two side lengths. Solving involves a couple of key steps:

• Calculate another of the triangle's angles using trigonometric ratios, such as the Law of Sines or Cosines, based on the available data.
• Use the inverse trigonometric functions to find angle measures, as shown by using ${\sin }^{-1}$ to solve for angle $A$.

These steps systematically find missing parts of the triangle by using an appropriate blend of trigonometric identities and inverse operations. Conclusively, solving triangles is about employing trigonometric tools efficiently to deduce unknown aspects of the shape from the known quantities.

Angle Sum Property

The angle sum property is a fundamental principle in geometry, stating that the sum of the interior angles of a triangle is always ${180}^{\circ }$. This property is crucial in trigonometry for determining unidentified angles in a triangle once one or two angles are known.

• For example, in our exercise, we've found angles $A$ and $C$, so to find $B$, we simply subtract these from ${180}^{\circ }$: $B={180}^{\circ }-A-C$.
• This provides a reliable method to compute missing angles without additional information.

The angle sum property is consistently applied in triangle-solving processes. It guarantees that once two angles are known, the third angle can be easily found, completing the triangle's angular description. This concept is vital for the cohesive application of the Law of Sines and other trigonometric techniques.