Problem 20 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$$

Step-by-Step Solution

Answer

The triangle angles A, B, C and sides a, b, c are calculated using the rule of sines and the sum of triangle angles knowing that it totals 180 degrees. The values are then approximated to two decimal places.

Step 1:Convert the given angle to decimal form

Firstly, we're given the angle B in degrees and minutes. We need to convert this to a decimal. The conversion of minutes to degrees is done by dividing the number of minutes by 60. Hence, the angle B becomes; \(B = 2 + 45/60 = 2.75^\circ\).

Step 2: Apply the Law of Sines to find angle C

We apply the Law of Sines, in the form sine(A)/a = sine(B)/b = sine(C)/c, using angle B and side b to find angle C. Rearranging and plugging in the values we have; \(\sin C = \sin B × (c / b) = \sin 2.75^\circ × (5.8 / 6.2)\) Solve for C by taking the inverse sine on both sides, we get: \(C = \sin^{-1} (\sin 2.75^\circ × (5.8 / 6.2))\)

Step 3: Compute for angle A

The sum of all interior angles in a triangle is 180 degrees. Thus we can find angle A as; \(A = 180^\circ - B - C\)

Step 4: Calculate Side a Using the Law of Sines

We can compute for the side a using the rule of sines again, but this time round using angle A and side b or c with their corresponding angles. Hence, a is given by; \(a = b × (\sin A) / (\sin B) = 6.2 × \sin A / \sin 2.75^\circ\) where A is the angle that we calculated in step 3.

Key Concepts

Solving Triangles

Solving a triangle involves finding all the missing sides and angles of the triangle given some initial information. In the Law of Sines, which applies to any triangle (not just right-angled ones), we use the proportionality between the sides and the sines of the opposite angles. This powerful mathematical tool allows us to calculate unknown sides when we know an angle and its opposite side, and just as importantly, find unknown angles when we know the length of their opposite side.

To apply the law effectively, we need to identify known values accurately and establish the correct ratio, as seen with the given angle B of 2 degrees and 45 minutes, and opposite side length b.

Angle Conversion

Angles can be measured in degrees, minutes, and seconds, but for solving triangles using the Law of Sines, we need to convert these into decimal degrees. One degree has 60 minutes and one minute has 60 seconds. The conversion from minutes to degrees is straightforward — divide the number of minutes by 60. A similar process applies for seconds, dividing by 3600 (60 x 60), though seconds are not present in the given problem. An angle of 2 degrees and 45 minutes converts to 2.75 degrees, a more manageable figure for calculations.

Why Convert to Decimal?

Using decimal degrees simplifies the computations involved in trigonometric functions, allowing for easier entry into calculators and reducing the chance of errors.

Inverse Sine

Inverse sine, or sin^-1, is a function that does the opposite of sine, used to find an angle when the sine of the angle is known. It's particularly useful in scenarios where we're solving for an angle of a triangle and we already have the ratio of the opposite side to the hypotenuse. In this case, we used the Law of Sines to find the value of sin C, and then applied inverse sine to find the measure of angle C itself, given the proportion from angle B and its opposite side b. Knowing how to apply inverse sine correctly is crucial in triangulation problems.

Triangle Interior Angles

The sum of interior angles in any triangle is always 180 degrees. This fundamental truth is indispensable for solving triangles. Once one or two angles are known, we can deduct them from 180 degrees to determine the unknown angle(s). In our problem, after finding angle C using the Law of Sines and inverse sine, we subtracted the measures of angles B and C from 180 degrees to get angle A. This relationship between interior angles forms the basis for many of the steps in solving triangle problems and is a key check to ensure that calculated angles are plausible.