Problem 20 - Precalculus : Building Concepts and Connections

Question

Solew the given triangles. The standard notation for labeling of triangles is used. Round answers to four decimal places. $$c=27, a=18, B=64^{\circ}$$

Step-by-Step Solution

Answer

Ensure to provide the triangle's calculated sides and angles, all rounded off to four decimal places.

Step 1: Application of Law of Cosines to find length of side b

Use the Law of Cosines, replacing a, c, and B with the given values and solve for b: \[ b = √(a² + c² - 2ac \cos{B})\]\[ b = √(18² + 27² - 2*18*27 \cos{64^{\circ}})\]

Step 2: Calculation of Side b

Calculate the value to get the length of side b. Make sure to round your answer to four decimal places.

Step 3: Application of Law of Sines to find angle A

Now, use the Law of Sines to solve for the measure of angle A, as follows: \[ A = \sin^{-1}{\frac{a*\sin{B}}{b}}\]\[ A = \sin^{-1}{\frac{18 * \sin{64^{\circ}}}{b}}\]

Step 4: Calculation of Angle A

Compute the value to get the measure of angle A, in degrees. Round your answer to four decimal places.

Step 5: Calculation of Angle C

Since the sum of the angles in a triangle always equals 180 degrees, we can find the measure of angle C as follows: \[ C = 180 - A - B\]\[ C = 180 - A - 64\]

Step 6: Calculation of Angle C

Calculate the measure of angle C, in degrees, and round your answer to four decimal places.

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental theorem used to solve triangles, especially when they are not right-angled. This is particularly useful when you know two sides and the included angle or when you want to find an unknown side.
For triangles labeled such that the sides opposite angles A, B, and C are a, b, and c respectively, the Law of Cosines states:

• \( c^2 = a^2 + b^2 - 2ab \cos{C} \)
• \( a^2 = b^2 + c^2 - 2bc \cos{A} \)
• \( b^2 = a^2 + c^2 - 2ac \cos{B} \)

In the given problem, you use this law to find side \( b \). With known values of \( a = 18 \), \( c = 27 \), and angle \( B = 64^{\circ} \), you apply:

• \( b = \sqrt{18^2 + 27^2 - 2 \cdot 18 \cdot 27 \cdot \cos{64^{\circ}}} \)

This helps in calculating the length of side \( b \), which is new information to solve the rest of the triangle. Round your final result to four decimal places for precision.

Law of Sines

After determining the side length using the Law of Cosines, the Law of Sines becomes handy to find missing angles in the triangle. The Law of Sines builds a relationship between the lengths of sides and the sines of their corresponding opposite angles.
It is given by:

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

Using the Law of Sines, you can find the measure of angle \( A \), once side \( b \) is known:

• \( \sin A = \frac{a \cdot \sin B}{b} \)
• \( A = \sin^{-1}\left( \frac{18 \cdot \sin{64^{\circ}}}{b} \right) \)

This gives the measure of angle \( A \), rounded to four decimal places. The Law of Sines is useful because it simplifies solving obtuse or non-right triangles when one side and its opposite angle are known.

Angle Calculation

Angles in a triangle always add up to 180 degrees, an essential concept to remember when solving for unknown angles.
Once the first two angles are known, calculating the third angle is straightforward using subtraction:

• \( C = 180 - A - B \)

In this triangle exercise:

• Given angle \( B = 64^{\circ} \)
• Calculated angle \( A \) from earlier

You simply compute:

• \( C = 180 - A - 64 \)

This approach ensures all angles are determined accurately and rounded to four decimal places. Angle calculations define completing the triangle and are a crucial skill for many geometry problems. Understanding how angles interrelate will make triangle solving much easier and more intuitive.