Proving sin 90° 1 Using Various Mathematical Approaches

Trigonometry is a fundamental branch of mathematics dealing with the relationships between the sides and angles of triangles. One of the most important functions in trigonometry is the sine function, which can be defined in several ways. In this article, we will explore how to prove that sin 90° 1 using the unit circle, right triangle, and trigonometric identities. Understanding these methods can provide a deeper insight into the nature of trigonometric functions.

The Unit Circle Definition

The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. It is a powerful tool in trigonometry, as it allows us to define the sine and cosine functions for any angle. For an angle (theta), the corresponding point on the unit circle is ((x, y)).

According to the unit circle definition, the sine of an angle (theta) is the y-coordinate of this point. Specifically, for (theta 90°) or (frac{pi}{2}) radians, the point on the unit circle is ((0, 1)). Therefore, we can conclude that:

[sin 90° y 1]

Right Triangle Definition

The right triangle definition of sine relates the sine of an angle to the ratios of the sides of a right triangle. In a right triangle, the sine of an angle (x) is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

When the angle (x) approaches (90°), the opposite side of the triangle becomes the hypotenuse itself. This is because as (x) gets closer to (90°), the adjacent side (the base of the triangle) approaches zero, and the opposite side becomes equal to the hypotenuse. Thus, we have:

[sin 90° frac{text{opposite}}{text{hypotenuse}} frac{h}{h} 1]

Here, (h) represents the length of the hypotenuse.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domain. One useful identity is the co-function identity, which states:

[sin(90° - theta) cos(theta)]

Using this identity, we can set (theta 0°) to get:

[sin(90° - 0°) cos(0°)]

Since (sin(90°)) is the left-hand side of the identity and (cos(0°)) is the right-hand side, we have:

[sin 90° cos 0° 1]

Conclusion

All these methods confirm that sin 90° 1. Depending on your preference for understanding or the context of your work, you can choose the approach that suits you best. Understanding these different proofs can enrich your knowledge of trigonometry and its applications.

Additional Proofs and Insights

The question of why sin 90° 1 can be explored further through the perspective of a grade 8 student. A simpler way to express this is:

Consider a right triangle with one angle approaching (90°). In such a triangle, the opposite side becomes the hypotenuse, making the sine of the angle equal to 1.

Another interesting proof can be derived using the identity:

sin 90° 2 sin 45° cos 45°

Since:

sin 45° cos 45° (frac{1}{sqrt{2}})

We can calculate:

[sin 90° 2 times frac{1}{sqrt{2}} times frac{1}{sqrt{2}} 2 times frac{1}{2} 1]

This showcases not only the elegance of the identity but also the interconnectedness of trigonometric functions.

Keywords

trigonometry, sin 90°, right triangle