Problem 13 - Algebra 2

Question

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=20, c=24, B=47^{\circ}\)

Step-by-Step Solution

Answer

Use Law of Cosines first, followed by Law of Sines: \(A \approx 35^\circ\), \(C \approx 98^\circ\), \(b \approx 25.6\).

Step 1: Analyzing Given Information

The given triangle has two sides and one angle: \(a = 20\), \(c = 24\), and \(B = 47^\circ\). To determine which law to use first, notice that the angle given is not included between the two known sides, which suggests beginning with the Law of Sines.

Step 2: Using the Law of Sines

Apply the Law of Sines to find angle \(A\): \[\frac{\sin(A)}{a} = \frac{\sin(B)}{b}\]Currently, \(b\) is unknown, so we instead use: \[\frac{\sin(A)}{20} = \frac{\sin(47^\circ)}{b}\]We will first calculate \(b\) using the Law of Cosines.

Step 3: Applying the Law of Cosines

Use the Law of Cosines to find the side \(b\): \[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \]\[ b^2 = 20^2 + 24^2 - 2(20)(24) \cdot \cos(47^\circ) \]Calculate the values:\[ b^2 = 400 + 576 - 480 \cdot \cos(47^\circ) \]\[ b^2 \approx 976 - 320.87 \]\[ b^2 \approx 655.13 \]\[ b \approx \sqrt{655.13} \approx 25.6 \]

Step 4: Solving for Angle A

Now that we know \(b = 25.6\), use Law of Sines:\[ \frac{\sin(A)}{20} = \frac{\sin(47^\circ)}{25.6} \]\[ \sin(A) = 20 \cdot \frac{\sin(47^\circ)}{25.6} \]\[ \sin(A) \approx 20 \cdot 0.73135 / 25.6 \]\[ \sin(A) \approx 0.5717 \]\[ A \approx \arcsin(0.5717) \approx 35^\circ \]

Step 5: Solving for Angle C

With angles \(A\) and \(B\) known, use the angle sum property of triangles:\[ C = 180^\circ - A - B \]\[ C = 180^\circ - 35^\circ - 47^\circ \]\[ C = 98^\circ \]

Key Concepts

Law of Sines

The Law of Sines is an important principle that helps us find unknown sides or angles in a triangle. It states that the ratios of each side of a triangle to the sine of its opposite angle are equal. This can be written as:

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

This law is particularly helpful in cases where we know two angles and one side, or when we know two sides and a non-enclosed angle.
If we go back to our triangle, we initially considered the Law of Sines to find angle A.
However, because we did not know the length of side b, we postponed its use.
Keep in mind that one important application of this law is solving what is often called the "AAS" (angle-angle-side) or "ASA" (angle-side-angle) configurations of triangles.

Law of Cosines

The Law of Cosines extends the Pythagorean theorem to non-right triangles, making it useful in solving for unknown parts in any triangle. For a triangle with sides a, b, and c and opposite angles A, B, and C respectively, the law is expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)
• Similarly, you can write the equation for the other sides: \( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \)
• \( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \)

This law is essential when you have two sides and the included angle, commonly known as the "SAS" (side-angle-side) situation.
In our example, after analyzing, we used the Law of Cosines to solve for side b since we have two sides and an angle not enclosed.
This approach led us to find side b and then allowed us to return to our Law of Sines strategy.

Angle sum property of triangles

The angle sum property is a basic yet crucial rule in trigonometry, stating the sum of the angles in any triangle is always 180 degrees.
This is often the final step in solving triangles, particularly when two angles are already known and it's necessary to find the third.

• Formula: \( A + B + C = 180^\circ \)

In our scenario, once angles A (35 degrees) and B (47 degrees) were determined, calculating angle C became straightforward using this property.
It's a simple subtraction from 180 degrees, avoiding the complexity of further trigonometric calculations.
Remember, this property is universal for all triangles and acts as a check to ensure calculations are on the right track.