The Relationship between Sin2B and SinB in Triangle ABC: A Mathematical Analysis

In the context of triangle ABC, where the sides are in the ratio AB:BC:AC 11:16:11, we delve into the complex relationship between Sin2B and SinB. This analysis involves the use of trigonometric rules and the cosine rule to derive these relationships accurately.

Introduction

The concept of trigonometry is fundamental in solving complex problems related to triangles. The problem at hand involves the usage of the cosine rule to find the measure of angle B in triangle ABC, and subsequently, determine the relationship between Sin2B and SinB. This article breaks down the steps used to obtain the solution and provides a deeper understanding of the underlying mathematical principles.

Given Information and Cosine Rule Application

Given the sides of the triangle ABC are in the ratio AB:BC:AC 11:16:11, we can denote the sides as follows:

AB 11x BC 16x AC 11x

Here, x is a common multiplier. Using the cosine rule, we can find the cosine of angle B:

cosB (AB2 AC2 - BC2) / (2 * AB * AC)

Calculating the Cosine of Angle B

Substituting the given values:

cosB (11x)2 (11x)2 - (16x)2 / (2 * 11x * 11x)

Simplifying the expression:

cosB (121x2 121x2 - 256x2) / (2 * 121x2) (242x2 - 256x2) / 242x2 -256 / 242 -1.0582

However, we notice an error in the above calculation as the cosine rule yields a value within the range -1 to 1. Let's correct the calculations:

-256/-352 cosB

cosB 0.727272

Finding the Measure of Angle B

Using the inverse cosine function:

B arccos(0.727272)

Calculating the value of B:

B 43.34°

Calculating SinB

Using the identity sin2B cos2B 1:

sinB √(1 - cos2B) √(1 - 0.7272722)

Computing the value:

sinB √(1 - 0.529008) √0.470992 ≈ 0.68633

Calculating Sin2B

Using the double angle formula for sine, sin2B 2sinBcosB:

sin2B 2 * 0.68633 * 0.727272 ≈ 0.9983

Determining the Ratio Sin2B : SinB

To determine the ratio, we divide sin2B by sinB:

sin2B / sinB 0.9983 / 0.68633 ≈ 1.45

Hence, the ratio Sin2B : SinB is approximately 1.45 : 1.

By following this detailed breakdown, we are able to establish the relationship between Sin2B and SinB in the given triangle ABC, thus adding to our knowledge in applied trigonometry.

Conclusion

This analysis highlights the importance of accurate application of trigonometric identities and the cosine rule when dealing with complex geometrical problems. Understanding and mastering these concepts can significantly aid in solving a wide range of mathematical and real-world applications involving triangles and trigonometric functions.