Calculator with Sin Cos Tan

Calculate the sine, cosine, or tangent of an angle.

Understanding Trigonometric Functions (Sine, Cosine, Tangent)

Trigonometry is a fundamental branch of mathematics that studies the relationships between the angles and sides of triangles. At its core are the three primary trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions are defined for any angle and are essential in fields ranging from physics and engineering to computer graphics and navigation.

The Unit Circle and Definitions

The most intuitive way to understand sine, cosine, and tangent is through the unit circle. A unit circle is a circle with a radius of 1 centered at the origin (0,0) of a Cartesian coordinate system. For any angle $\theta$ measured counterclockwise from the positive x-axis, the terminal side of the angle intersects the unit circle at a point (x, y).

• Sine (sin θ): Defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle. So, sin θ = y.
• Cosine (cos θ): Defined as the x-coordinate of the point where the terminal side of the angle intersects the unit circle. So, cos θ = x.
• Tangent (tan θ): Defined as the ratio of the sine to the cosine, or the slope of the line segment forming the angle. So, tan θ = sin θ / cos θ = y / x (where cos θ ≠ 0).

Right-Angled Triangles

For acute angles within a right-angled triangle, these functions can be defined using the sides of the triangle relative to the angle:

• Hypotenuse (H): The side opposite the right angle (always the longest side).
• Opposite (O): The side opposite the angle in question.
• Adjacent (A): The side next to the angle in question (not the hypotenuse).

The mnemonic SOH CAH TOA is often used:

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

Units: Degrees vs. Radians

Trigonometric functions can accept angles measured in either degrees or radians.

• Degrees (°): A full circle is 360°. A right angle is 90°.
• Radians (rad): A full circle is 2π radians. A right angle is π/2 radians. Radians are often preferred in higher mathematics and physics because they simplify many calculus formulas. The relationship is: 180° = π radians.

Use Cases

Trigonometric functions are ubiquitous:

• Physics: Analyzing wave motion (sound, light, water), projectile motion, oscillations, and alternating currents.
• Engineering: Designing structures, analyzing forces, signal processing, and control systems.
• Navigation: Calculating distances and bearings using celestial or terrestrial observations.
• Computer Graphics: Rotating objects, creating animations, and rendering 3D scenes.
• Surveying: Determining heights and distances of inaccessible points.

Example Calculation

Let's calculate the sine of 30 degrees.

Using the calculator:

• Angle Value: 30
• Unit: Degrees
• Function: Sine (sin)

Result: 0.5 (or 1/2, as expected from the 30-60-90 triangle).

Now, let's calculate the cosine of π/4 radians.

• Angle Value: 3.14159 / 4 (approximately 0.7854)
• Unit: Radians
• Function: Cosine (cos)

Result: Approximately 0.7071 (which is √2 / 2).

This calculator provides a quick way to find these fundamental trigonometric values.