Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$A=42^{\circ}, \quad B=35^{\circ}, \quad c=1.2$$

Step-by-Step Solution

Answer

After calculations, the third angle C will be 103 degrees. The length of side a using law of sines will be approximately 0.83 units and the length of side b will be approximately 0.75 units.

Step 1: Determine the Missing Angle

Since we know two of the angles (42 degrees and 35 degrees), we can calculate the third angle. The sum of the angles in a triangle is always 180 degrees. So, to find angle C, the calculation is \(C = 180 - A - B = 180 - 42 - 35 = 103\) degrees.

Step 2: Apply the Law of Sines to Find Side a

Now we can use the Law of Sines to solve for the missing sides. We have side c and angle C, so we can start with those to solve for side a. The Law of Sines is as follows: \(\frac{a}{\sin{A}} = \frac{c}{\sin{C}}\). So to find a, the calculation is \(a = c * \frac{\sin{A}}{\sin{C}} = 1.2 * \frac{\sin{42}}{\sin{103}}\). Calculate this to find the length of side a.

Step 3: Apply the Law of Sines to Find Side b

Similarly, use the Law of Sines to find side b. To find b, the calculation is \(b = c * \frac{\sin{B}}{\sin{C}} = 1.2 * \frac{\sin{35}}{\sin{103}}\). Calculate this to find the length of side b.

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in trigonometry used to find unknown sides or angles in a triangle. It is particularly useful for solving oblique triangles, which are triangles that do not include a right angle. The law states:

• The ratio of a side of a triangle to the sine of its opposite angle is equal for all three sides and angles: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
• This means if you know one side and its opposite angle, along with another angle, you can find the missing side easily.
• In our exercise, we used this law to find the lengths of sides \(a\) and \(b\) after identifying all angles.

The Law of Sines is essential when working with triangles where a side-opposite angle pair is known, making it incredibly helpful in cases where the triangle is not right-angled.

Law of Cosines

The Law of Cosines is another valuable tool in trigonometry, especially when dealing with triangles that do not have a right angle. This law becomes handy when you have different combinations of known sides and angles:

• It relates the lengths of the sides of a triangle to the cosine of one of its angles: \( c^2 = a^2 + b^2 - 2ab \cdot \cos C \).
• This formula helps when you know two sides and the included angle, or when all three sides are known but not the angles.
• It's similar to the Pythagorean theorem but adjusted for non-right triangles.

Although the Law of Cosines was not used in our specific exercise, it is important for triangles where all three sides are known, or when you're looking to find an angle from known sides.

Triangle Angles

Understanding triangle angles is crucial for solving many trigonometry problems.

• The sum of the interior angles of any triangle will always equal 180 degrees. This is a fundamental principle used to find a missing angle once two angles are known.
• In the provided exercise, we used this rule to determine the measure of angle \(C\), which was initially unknown.
• This knowledge allows for subsequent use of either the Law of Sines or Law of Cosines effectively.

Knowing this principle helps in making sense of more complex problems, as it allows you to focus on finding specific measures using the right trigonometric laws.

Sine Rule

The Sine Rule is another term often used to refer to the Law of Sines.

• This rule is particularly useful in scenarios involving non-right triangles.
• With the Sine Rule, you can solve for missing sides or angles when specific criteria are met, such as knowing two angles and one side.
• Applying this rule correctly helps in calculating unknown triangle measures effectively.

In practice, like in our exercise, you always start by ensuring you have the correct data such as angle measures and one side length, making the application of the Sine Rule straightforward. It serves as a bridge to solving various components of triangle problems simply and efficiently.