Deducing the Sine of 2A from the Cosine of A

In trigonometry, understanding the relationships between different trigonometric functions is crucial for solving complex equations. Let's explore how to determine the value of sin(2A) when cos(A) is known, specifically when cos(A) 1/2.

Understanding the Given Information

We start with the given information:

cos(A) 1/2

This information alone does not provide a unique angle for A, as the cosine function has the same value in the first and fourth quadrants. Therefore, the possible values for A are:

A 60° or π/3 radians A 300° or 5π/3 radians

Using the Double Angle Formula

The double angle formula for sine can be expressed as:

sin(2A) 2sin(A)cos(A)

To determine sin(2A), we need to first find the value of sin(A).

Case 1: A 60° or π/3 Radians

Here, sin(60°) √3/2.

Substituting into the double angle formula:

sin(2A) 2sin(A)cos(A) 2 × (√3/2) × (1/2) √3/2

Case 2: A 300° or 5π/3 Radians

Here, sin(300°) -√3/2.

Substituting into the double angle formula:

sin(2A) 2sin(A)cos(A) 2 × (-√3/2) × (1/2) -√3/2

Conclusion

Thus, the value of sin(2A) can be either √3/2 or -√3/2, depending on the angle A.

Finding the Sine using Pythagorean Identity

An alternative approach is to use the Pythagorean identity:

sin^2(A) cos^2(A) 1

Substituting the given cos(A) 1/2:

sin^2(A) (1/2)^2 1

sin^2(A) 1/4 1

sin^2(A) 3/4

sin(A) ±√(3/4) ±√3/2

Using the double angle formula again:

sin(2A) 2sin(A)cos(A) 2 × (±√3/2) × (1/2) ±√3/2

Summary of Key Points

cos(A) 1/2 indicates that A could be 60° or 300°. The sine of 2A will be negative in the case of A 300° and positive in the case of A 60°. Using the Pythagorean identity to find sin(A) before applying the double angle formula.

Additional Considerations

Understanding the quadrants of trigonometric functions can help in predicting the sign of sin(2A). For instance, if A is in the first quadrant, 2A will also be in the first or second quadrant, which is where the sine function is positive. Conversely, if A is in the fourth quadrant, 2A could be in the third or fourth quadrants, where the sine function is negative.