Solving Trigonometric Equations: sinA-B 1/2 and cosAB 1/2

Trigonometric equations are a fascinating area in mathematics, particularly when they involve both sine and cosine functions. This article aims to explore the solution to the equations sinA - B 1/2 and cosAB 1/2. We will also cover the steps to find the values of A and B given the conditions 0A ≤ 90 degrees.

Understanding the Problem

We are given two equations:

sinA - B 1/2 cos(A B) 1/2

Our goal is to find the values of A and B.

Solving the Equations Step-by-Step

Step 1: Simplifying the Equations

We start by simplifying the equations. We know from the unit circle and trigonometric definitions that:

sinA - B sin 30° cos(A B) cos 60°

Therefore, we can write:

A - B 30° ………. 1 A B 60° ………. 2

Step 2: Adding and Subtracting the Equations

To find A and B, we can add equations 1 and 2:

2A 90°

Thus,

A 45° ………. 3

Substituting A 45° in equation 2:

45° B 60°

So,

B 15° ………. 4

Verification and Additional Insights

The values obtained A 45° and B 15° satisfy both the original equations:

sin45° - 15° sin 30° cos(45° 15°) cos 60°

Let’s verify these:

sin45° - 15° 0.5 sin30° cos60° 0.5

We see that our solutions are valid and correct.

General Solution

While the specific solution for 0A ≤ 90 is A 45° and B 15°, we can find the general solution using periodic properties of sine and cosine:

sin(A - B) 1/2

cos(A B) 1/2

From the periodicity of these functions, we have:

A - B 30° nπ - (1)^nπ/6, n ∈ Z

A B 2mπ ± π/3, m ∈ Z

Solving for A and B:

A (2mπ - 30° - nπ (1)^nπ/6) / 2

B (2mπ 30° - nπ - (1)^nπ/6) / 2

Where m, n ∈ Z, providing the general solution.

Conclusion

In conclusion, by solving the trigonometric equations sinA - B 1/2 and cosAB 1/2 using fundamental trigonometric identities and properties, we found that the values of A and B are 45° and 15° respectively, given the constraint 0A ≤ 90 degrees.

References

MathIsFun - Trigonometry