Proving the Trigonometric Identity: sin(180°-x) sin x

Investigating the trigonometric identity sin(180°-x) sin x involves a deep dive into the properties of the sine function and the unit circle. This article will explore different methods to prove this identity, providing a comprehensive understanding of its validity for all values of x.

Proof Using the Unit Circle

Gleaning insights from the unit circle, we can prove that sin(180°-x) sin x. Here is how:

Understanding Angles in the Unit Circle:

The angle x is measured from the positive x-axis counterclockwise. The angle 180° - x is measured from the positive x-axis in the clockwise direction.

Coordinates on the Unit Circle:

The coordinates of a point on the unit circle corresponding to angle x are (cos x, sin x). The coordinates for 180° - x can be derived as:

The angle 180° corresponds to the point (-1, 0).

The angle 180° - x is in the second quadrant where the x-coordinate is negative and the y-coordinate is positive. Hence, the coordinates for 180° - x are (-cos x, sin x).

Finding sin(180° - x):

The sine function gives the y-coordinate of the point on the unit circle. Therefore, sin(180° - x) corresponds to the y-coordinate of the point at angle 180° - x.

Hence, sin(180° - x) sin x.

Using the Angle Sum Formula

An alternate method to prove the identity involves the sine subtraction formula:

sin(180° - x) sin(180°)cos(x) - cos(180°)sin(x)

0 ? cos(x) - (-1) ? sin(x)

sin(x)

Conclusion

In conclusion, we have demonstrated the validity of the identity sin(180° - x) sin x. This identity holds true for all values of x.

Further Exploration

To dive deeper into this topic, consider familiarizing yourself with the properties of the sine, cosine, and unit circle. You can explore more complex trigonometric identities and their proofs.

References

[1] Sine Function and Unit Circle Properties, Wikipedia

[2] Trigonometric Identities, Wikipedia

[3] Angle Sum and Difference Identities, Math Is Fun