Problem 24 - 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. \(a=4, b=8, c=5\)

Step-by-Step Solution

Answer

Use the Law of Cosines; angles are approximately 22°, 28°, and 130°.

Step 1: Identify the Appropriate Law

First, identify which law to use. Since we have all three sides given and no angles, we should use the Law of Cosines. The Law of Sines requires at least one complete ratio (an angle with its opposite side), which we don't have.

Step 2: Apply the Law of Cosines to Find One Angle

Use the Law of Cosines to find one angle, let's start with angle A. The Law of Cosines is given by: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Replace the values with \( a = 4 \), \( b = 8 \), and \( c = 5 \) in the formula to find \( C \):\[ 5^2 = 4^2 + 8^2 - 2 \cdot 4 \cdot 8 \cdot \cos(C) \]Calculate:\[ 25 = 16 + 64 - 64 \cdot \cos(C) \]Simplifying gives:\[ 25 = 80 - 64 \cdot \cos(C) \]\[ 64 \cdot \cos(C) = 55 \]\[ \cos(C) = \frac{55}{64} \]Now find \( C \) using the inverse cosine function:\[ C \approx \cos^{-1}\left(\frac{55}{64}\right) \approx 28^\circ \]

Step 3: Find the Second Angle Using the Law of Cosines Again

Next, use the Law of Cosines to find another angle, such as \( B \). The formula is:\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]Plug in the known values:\[ 4^2 = 8^2 + 5^2 - 2 \cdot 8 \cdot 5 \cdot \cos(A) \]Calculate:\[ 16 = 64 + 25 - 80 \cdot \cos(A) \]\[ 16 = 89 - 80 \cdot \cos(A) \]\[ 80 \cdot \cos(A) = 73 \]\[ \cos(A) = \frac{73}{80} \]Find \( A \):\[ A \approx \cos^{-1}\left(\frac{73}{80}\right) \approx 22^\circ \]

Step 4: Determine the Final Angle

Now use the fact that the sum of angles in a triangle is 180° to find angle \( B \).\[ A + B + C = 180^\circ \]Substitute \( A = 22^\circ \) and \( C = 28^\circ \):\[ 22 + B + 28 = 180 \]Solve for \( B \):\[ B = 180 - 22 - 28 \]\[ B = 130^\circ \]

Key Concepts

Law of Sines

The Law of Sines is a very useful tool when solving triangles, especially when an angle and its opposite side are known. It states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant across all three sides and angles. This means:

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

This law is particularly easy to apply when you have one angle and its opposite side, which was not the case in our exercise. When given all sides but no angles, as in this example, one typically needs to start with the Law of Cosines. However, once any angle is solved using the Law of Cosines, the Law of Sines can sometimes be used to find the remaining angles when the situation allows it. This becomes a handy tool in triangle solving because it provides an alternative means of getting other unknowns once you have some known values.

Angle Measurement

Understanding how to measure angles is crucial in trigonometry and solving triangles. In our exercise, we began with sides and no angles. An angle can be found once we know the cosine (or sine, tangent) of that angle.To measure an angle:

• Start by using a trigonometric function, like cosine, which relates angle to side lengths.
• For example, \( \cos(C) = \frac{b^2 + a^2 - c^2}{2ab} \) for angle C.
• Once you calculate the cosine value, use the inverse cosine function (\( \cos^{-1} \)) to find the angle.

When rounding, it's common to round angle measurements to the nearest degree. This is because angles in degrees are easier to visualize and communicate. Precision in angle measurement plays a significant role in accurately solving triangle problems and ensuring correct final results.

Triangle Solving

Solving triangles involves finding unknown side lengths and angles based on given information. In our example, the process showcased this foundational trigonometry problem-solving skill. Steps to solve a triangle:

• Identify Given Information: Determine what sides and angles are known and what needs solving. This helps to decide which trigonometric law or method to apply, like choosing between the Law of Sines or Law of Cosines.
• Apply Trigonometric Laws: Use the Law of Cosines when all sides are known, like in our example. Once you solve one angle, alternative laws or angle sum properties help find the others.
• Sum of Angles: Remember, the sum of angles in any triangle is always 180°. If you find two angles, the third is simply \( 180° - (A + B) \).

These steps ensure a comprehensive approach to fully solve the given triangle, achieving a complete understanding of its dimensions and angles.