How to Define Sine, Cosine, and Tangent

1. The Basics: Right Triangles and Trigonometric Functions

In a right triangle, you’ll find three angles and three sides. We typically label the angles as \(α\) (alpha), \(β\) (beta), and \(γ\) (gamma). The longest side, opposite the right angle (\(90\) degrees or \(\frac{π}{2}\) radians), is the hypotenuse. The remaining sides are the adjacent side (next to the angle of interest) and the opposite side (across from the angle of interest).

Sine, cosine, and tangent are trigonometric functions that establish relationships between these angles and sides.

2. Sine (sin)

The sine of an angle (\(sin\:α\), for example) in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically:

\(sin\:α =\frac{(Length\:of\:Opposite\:Side)}{(Length\:of\:Hypotenuse)}\)

3. Cosine (cos)

Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. In mathematical terms:

\(cos\:=\frac{(Length\:of\:Adjacent\:Side)}{(Length\:of\:Hypotenuse)}\)

4. Tangent (tan)

Tangent is the ratio of the sine of an angle to the cosine of that same angle. Essentially, it’s the ratio of the opposite side to the adjacent side. So, we write it as:

\(tan\:α =\frac{sin\:α}{cos\:α}=\frac{(Length\:of\:Opposite\:Side)}{(Length\:of\:Adjacent\:Side)}\)

Applying Sine, Cosine, and Tangent

To fully comprehend how these functions are applied, let’s use a practical example:

Imagine a right triangle with an angle \(α\) of \(30\) degrees (or \(\frac{π}{6}\) radians). The hypotenuse is \(10\) units long, and we want to find the lengths of the remaining sides.

1. Calculate the length of the opposite side using sine: \(sin\:30=\frac{(Length\:of\:Opposite\:Side)}{10}\)Solving for the opposite side gives us a length of \(5\) units.
2. Calculate the length of the adjacent side using cosine: \(cos\:30=\frac{(Length\:of\:Adjacent\:Side)}{10}\) Solving for the adjacent side gives us a length of approximately \(8.66\) units.
3. Verify using tangent: \(tan\:30\) should equal the length of the opposite side divided by the length of the adjacent side. Confirming this ensures that our calculations are accurate.

Beyond Right Triangles: Sine, Cosine, and Tangent in the Real World

While the immediate applications of sine, cosine, and tangent lie in geometry and trigonometry, these concepts extend far beyond. They play significant roles in physics, engineering, computer graphics, architecture, and more, underscoring the interconnectedness of mathematics and the real world.