Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of sines to solve the triangle. \(A=76^{\circ}, \quad a=34, \quad b=21\)

Step-by-Step Solution

Answer

After performing all the steps correctly, the measures for angle \(B\), angle \(C\), and side \(c\) will be obtained. The final values may vary depending on the accuracy of calculations and the rounding constraints.

Step 1: Solve for angle B using the Law of Sines

The Law of Sines can be used to solve for the unknown angle \(B\). Rearrange the Law of Sines formula to find \(B\) as follows: \(\sin B = \frac{b \cdot \sin A}{a}\) Substituting the given values: \(\sin B = \frac{21 \cdot \sin 76^{\circ}}{34}\)

Step 2: Calculate value of angle B

Calculate the value of angle \(B\) using inverse sine function, \(\sin^{-1}\), to get the measure of angle \(B\) in degrees: \(B = \sin^{-1}(\sin B)\)

Step 3: Calculate value of angle C

We know that the sum of interior angles in a triangle equals to \(180^{\circ}\). Thus, angle \(C\) can be found by subtracting the known angles, \(A\) and \(B\), from \(180^{\circ}\): \(C = 180^{\circ} - A - B\)

Step 4: Solve for side c using the Law of Sines

Now that we know all the angles, we can use the Law of Sines one more time to find the length of the remaining side of the triangle, \(c\). The same formula used in step 1 can be rearranged to find \(c\): \(c = \frac{a \cdot \sin C}{\sin A}\) Substituting the known values: \(c = \frac{34 \cdot \sin C}{\sin 76^{\circ}}\)

Key Concepts

Triangle Solving

Solving a triangle involves finding all unknown sides and angles when certain values are given. This often requires utilizing various mathematical principles and formulas. In our presented problem, we start with knowing two angles and one side, which gives us enough information to determine the complete triangle by employing the Law of Sines.

To solve a triangle:

• Identify known angles and sides.
• Use trigonometric laws (like the Law of Sines or Law of Cosines) to find unknown values.
• Ensure that all calculations respect the geometric properties of triangles, such as angle sums.

Starting with angle A and sides a and b, the Law of Sines helps us determine angle B. Once you have enough information about angles, the remaining side and angles can be calculated accurately.

Inverse Sine Function

The inverse sine function, denoted as \( ext{sin}^{-1}\), is crucial when determining an angle from a known sine value in a triangle. This function helps translate the ratio of the opposite side to the hypotenuse back into an angle measure.

In our exercise, after calculating \( ext{sin} \, B\), we need the inverse sine \( ext{sin}^{-1} \) to find the actual angle B from this value. The process involves:

• Calculating \(\sin B = \frac{b \cdot \sin A}{a} \).
• Using \(\text{sin}^{-1}\) to obtain the degree measure of angle B.
• Verifying that angle B is reasonable within the context of triangle constraints.

Inverse trigonometric functions are pivotal in triangle solving, translating mathematical values back into geometric angles.

Interior Angles Sum

A fundamental principle in triangle geometry is that the sum of the interior angles is always \(180^\circ\). This knowledge allows us to find a missing angle once two angles are known. It's an inherent trait of Euclidean geometry.

After calculating angles A and B in our problem, finding angle C becomes straightforward using the formula:

• \( C = 180^\circ - A - B \)

Thus, the sum property simplifies the process, ensuring our triangle has mathematically feasible dimensions. Remember to recheck calculations, as small errors may lead to incorrect angle measurements. This principle is simple yet powerful in verifying the completeness and accuracy of triangle solutions.

Trigonometric Calculations

Trigonometric calculations involve using sine, cosine, and tangent functions to solve for unknown triangle parts. These calculations are integral to connecting side lengths and angles.

In our triangle, we've utilized the Law of Sines, which states:

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

This relation allows the determination of any missing side once angles and one side length are known. Our task to find side c was achieved with truth in these mathematical relationships:

• Exploiting \( c = \frac{a \cdot \sin C}{\sin A} \).
• Plug in known values and compute using accurate sine values.

Mastery of these trigonometric aspects ensures precise and effective solutions while solving triangles.