Sine, Cosine, Tangent Calculator Guide

The trigonometric functions sine, cosine, and tangent are fundamental in mathematics, particularly in triangles and wave functions. This guide will help you understand these functions and how to calculate them.

Understanding Sine, Cosine, and Tangent

In a right triangle, the sine, cosine, and tangent of an angle are defined as ratios of the sides:

• Sine (sin): Opposite / Hypotenuse
• Cosine (cos): Adjacent / Hypotenuse
• Tangent (tan): Opposite / Adjacent

Formulas

For a right triangle with an angle θ:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = sin(θ) / cos(θ) = Opposite / Adjacent

Examples

For a right triangle with an angle of 30°, Hypotenuse = 2, Opposite = 1, and Adjacent = √3:

• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3

Values Table

Angle (Degrees)	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	0.5	√3/2 ≈ 0.8660	0.5773
45°	√2/2 ≈ 0.7071	√2/2 ≈ 0.7071	1
60°	√3/2 ≈ 0.8660	0.5	√3 ≈ 1.7320
90°	1	0	Undefined (because cos(90°) is 0)

FAQs

What is the main difference between sine, cosine, and tangent?

Sine, cosine, and tangent are trigonometric functions that relate the angles of a right triangle to the lengths of its sides. Specifically, sine represents the ratio of the opposite side to the hypotenuse, cosine represents the ratio of the adjacent side to the hypotenuse, and tangent represents the ratio of the opposite side to the adjacent side.

How are sin, cos, and tan used in real life?

These trigonometric functions have various applications in real life, including in fields like engineering, physics, architecture, and navigation. For instance, engineers might use them to calculate forces in bridges or buildings, while pilots and sailors use them for navigation.

Why is the tangent of 90° undefined?

The tangent function is the ratio of sine to cosine. At 90°, the cosine value becomes zero, making the division by zero undefined. This is why the tangent of 90° is considered undefined.

How do I remember the values of sine, cosine, and tangent for common angles?

Many students use mnemonic devices to remember these values. A popular mnemonic for this is "SOHCAHTOA," which stands for "Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent."

Can I use these trigonometric functions for angles greater than 90°?

Yes, sine, cosine, and tangent functions can be defined for any angle, not just those between 0° and 90°. Their values are determined using the unit circle and can be found using a scientific calculator or trigonometry tables.

What is the significance of the sine, cosine, and tangent?

These trigonometric functions are foundational for studying triangles, analyzing wave functions, and modeling periodic phenomena in various scientific fields.

Are these functions limited to right triangles?

While the initial definitions are based on right triangles, sine, cosine, and tangent can be extended to any angle using the unit circle approach.

How can I remember the functions' ratios?

A popular mnemonic is "SOHCAHTOA", where SOH stands for "Sine = Opposite over Hypotenuse", CAH stands for "Cosine = Adjacent over Hypotenuse", and TOA stands for "Tangent = Opposite over Adjacent".