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 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:

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

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.

Frequently Asked Questions

What is sine?

Sine of an angle in a right triangle is the ratio of the side OPPOSITE the angle to the hypotenuse. sin(theta) = opposite/hypotenuse.

What is cosine?

Cosine of an angle in a right triangle is the ratio of the side ADJACENT to the angle to the hypotenuse. cos(theta) = adjacent/hypotenuse.

What is tangent?

Tangent of an angle in a right triangle is the ratio of the side OPPOSITE the angle to the side ADJACENT. tan(theta) = opposite/adjacent.

What is SOH-CAH-TOA?

A mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Remembering SOH-CAH-TOA gives all three trig functions at once.

Which side is the hypotenuse?

The longest side of a right triangle, always opposite the 90-degree angle. The other two sides are legs.

Which side is opposite vs adjacent?

Opposite the angle: the leg ACROSS from it (not touching the angle). Adjacent: the leg NEXT to the angle (touching it, but not the hypotenuse).

Walk through computing sin(30 degrees) in a 30-60-90 triangle.

The 30-60-90 triangle has side ratios 1 : sqrt(3) : 2. The side opposite the 30-degree angle is 1 and the hypotenuse is 2. So sin(30) = 1/2.

What is tan(45 degrees)?

In a 45-45-90 triangle, the two legs are equal, so opposite/adjacent = 1. So tan(45°) = 1.

Can sin be greater than 1?

No. Since sine = opposite/hypotenuse, and the hypotenuse is the longest side, the ratio is always between 0 and 1 (inclusive of endpoints at 0 and 90 degrees).

Where are sine, cosine, and tangent used?

Surveying (heights and distances), navigation, physics (resolving forces into components), engineering, computer graphics, music synthesis, and many other applications.

Related Lessons You May Like

• Significance of the unit circle
• How to use right-triangle trigonometry
• Ambiguous case (SSA) in trigonometry
• How to find similar figures
• How to use the Pythagorean Theorem

For a workbook on trigonometry, Trigonometry for Beginners covers the unit circle, identities, and applications. Pre-Calculus for Beginners is the natural next step.