Problem 11 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$a=15, b=12, B=20^{\circ}$$

Step-by-Step Solution

Answer

Question: Given a triangle with sides a = 15, b = 12, and angle B = 20°, find all the sides and angles of the triangle. Answer: The triangle has sides a = 15, b = 12, and c ≈ 14.46, with angles A ≈ 23.20°, B = 20°, and C ≈ 136.80°.

Step 1: Use the Law of Cosines to find the side c.

The Law of Cosines states that for any triangle with sides a, b, and c, and angle B, $$c^2 = a^2 + b^2 - 2ab\cos{B}$$ Given $$a=15, b=12, \text{ and } B=20^{\circ}$$, we can plug in these values to find the side c: $$c^2 = 15^2 + 12^2 - 2(15)(12)\cos{20^{\circ}}$$ $$c^2 = 225 + 144 - 360\cos{20^{\circ}}$$ $$c^2 \approx 209.21$$ $$c \approx 14.46$$

Step 2: Use the Law of Sines to find angle A.

The Law of Sines states that for any triangle, the ratio of the side length to the sine of its opposite angle is constant, i.e., $$\frac{a}{\sin{A}} = \frac{b}{\sin{B}}$$ We can rewrite it as: $$\sin{A} = \frac{a\sin{B}}{b}$$ Given $$a=15, b=12, \text{ and } B=20^{\circ}$$, we can plug in these values to find the angle A: $$\sin{A} = \frac{15\sin{20^{\circ}}}{12}$$ $$\sin{A} \approx 0.3940$$ $$A \approx 23.20^{\circ}$$

Step 3: Find angle C.

Since the sum of the interior angles of a triangle equals 180 degrees, we can find the angle C by subtracting the angles A and B from 180 degrees: $$C = 180^{\circ} - A - B$$ $$C = 180^{\circ} - 23.20^{\circ} - 20^{\circ}$$ $$C \approx 136.80^{\circ}$$ The complete solution for the triangle is: $$a=15, b=12, c\approx 14.46, A\approx 23.20^{\circ}, B=20^{\circ}, C\approx 136.80^{\circ}$$.

Key Concepts

Law of Sines

The Law of Sines is an important tool when solving triangles, especially in situations where the Law of Cosines alone isn't sufficient. This law provides a relationship between the angles and sides of a triangle by stating:

• The ratio of the length of a side to the sine of its opposite angle is the same for all three sides of a triangle: \( \frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}} \)
• This can be useful in scenarios where you know two sides and an angle (SSA) or two angles and a side (AAS), among others.

To apply the Law of Sines, one rearranges the formula depending on the given information and solves for the unknowns. In the example problem, this law is used to find angle A, given sides a and b, and angle B. The formula was rearranged to solve for \( \sin{A} \) and the known values were substituted leading to the calculation of angle A.

triangle solving

Solving a triangle is the process of finding the missing lengths of its sides or its angles. In any triangle, this is often done using a combination of the Law of Sines, Law of Cosines, and basic angle sum properties.

• Using the Law of Cosines is useful when knowing two sides and the included angle (SAS), as it helps find the third side directly.
• The Law of Sines comes into play when there are two known angles and a non-included side (AAS) or when there are two sides and a non-included angle (SSA).
• The angle sum property states that the sum of all angles in a triangle equals 180°. This property is used to find the remaining angle when two are known.

With these tools, one can effectively "solve" a triangle to find every unknown side and angle. In the example provided, side c was solved using the Law of Cosines, followed by using the Law of Sines to solve for angle A, and concluding with the angle sum property to determine angle C.

angle calculation

Calculating angles is a crucial part of solving triangles, and understanding how angles relate to the sides of a triangle is fundamental.

• First, identifying the given angle and using trigonometric identities or laws can help solve for other angles or sides, as seen with angle B in the problem.
• Utilizing trigonometric functions, such as sine, allows for the computation of unknown angles by rearranging and solving equations derived from the Law of Sines or Cosines.
• After finding one of the unknown angles using these laws, the last angle can often be found using the angle sum property of triangles, which states the angles must total 180°.

In our specific example, after identifying angle B and calculating angle A through the Law of Sines, angle C was straightforwardly calculated by subtracting known angles from 180°. This ensures that all triangle properties are satisfied and each angle and side appropriately complements one another.