Tan Sin Cos Calculator

Calculate trigonometric functions in degrees or radians instantly.

Select an angle and unit to see the results.

–Sine (sin)

–Cosine (cos)

–Tangent (tan)

What is a Tan Sin Cos Calculator?

A tan sin cos calculator is a tool designed to compute the three primary trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions are fundamental to trigonometry, a branch of mathematics that studies the relationships between the angles and side lengths of triangles. Specifically, for a right-angled triangle, these functions define the ratios of the lengths of its sides. [6] This calculator allows you to find these values for any given angle, specified in either degrees or radians. [1]

The Tan Sin Cos Formulas and Explanation

The core of trigonometry lies in the right-angled triangle. The ratios of the sides of a right-angled triangle are defined with respect to one of its acute angles (an angle less than 90°). The sides are named as follows:

• Opposite: The side across from the angle.
• Adjacent: The side next to the angle, which is not the hypotenuse.
• Hypotenuse: The longest side, opposite the right angle.

The formulas, often remembered by the mnemonic “SOHCAHTOA”, are: [2]

SOH: Sin(θ) = Opposite / Hypotenuse

CAH: Cos(θ) = Adjacent / Hypotenuse

TOA: Tan(θ) = Opposite / Adjacent

Variable Meanings

Variable	Meaning	Unit	Typical Range
θ (theta)	The angle of interest	Degrees or Radians	0-360° or 0-2π rad
Opposite	Length of the side opposite angle θ	Length (e.g., m, cm, in)	Positive number
Adjacent	Length of the side adjacent to angle θ	Length (e.g., m, cm, in)	Positive number
Hypotenuse	Length of the side opposite the right angle	Length (e.g., m, cm, in)	Positive number > Opposite & Adjacent

For more advanced topics, you might find a right triangle calculator useful.

Practical Examples

Example 1: Angle in Degrees

• Input Angle: 60°
• Unit: Degrees
• Results:
  • sin(60°) ≈ 0.866
  • cos(60°) = 0.5
  • tan(60°) ≈ 1.732

Example 2: Angle in Radians

• Input Angle: 0.7854 rad (approximately π/4)
• Unit: Radians
• Results:
  • sin(0.7854) ≈ 0.707
  • cos(0.7854) ≈ 0.707
  • tan(0.7854) ≈ 1.0

To understand the relationship between degrees and radians better, a radian to degree converter can be very helpful.

How to Use This Tan Sin Cos Calculator

1. Enter the Angle: Type the numerical value of the angle into the “Angle Value” field.
2. Select the Unit: Use the dropdown menu to choose whether your angle is in “Degrees” or “Radians”. The calculation depends heavily on this unit. [4]
3. View the Results: The calculator will automatically update and display the values for sine, cosine, and tangent for your specified angle.
4. Interpret the Chart: The bar chart provides a quick visual comparison of the magnitudes of sin, cos, and tan.
5. Copy or Reset: Use the “Copy Results” button to save the output or “Reset” to clear the fields.

Key Factors That Affect Tan, Sin, and Cos

• Angle’s Quadrant: The sign (positive or negative) of the sin, cos, and tan values depends on which quadrant of the unit circle the angle falls into. For example, cosine is positive in quadrants I and IV, while sine is positive in quadrants I and II. A unit circle calculator can illustrate this concept.
• Degrees vs. Radians: Using the wrong unit is a common mistake. 180 degrees is equal to π radians. Ensure your calculator is set to the correct mode. [10]
• Reference Angles: For angles greater than 90°, the trigonometric values are the same as for a corresponding acute angle (the reference angle), with the sign determined by the quadrant.
• Special Angles: Angles like 0°, 30°, 45°, 60°, and 90° have exact, well-known trigonometric values that are important to memorize.
• Undefined Values: The tangent function is undefined for angles where the cosine is zero, such as 90° and 270°. This is because tan(θ) = sin(θ)/cos(θ), which would involve division by zero. [4]
• Periodicity: Trigonometric functions are periodic. Sine and cosine repeat every 360° (2π radians), while tangent repeats every 180° (π radians). For example, sin(390°) is the same as sin(30°).

Frequently Asked Questions (FAQ)

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic device used to remember the definitions of sine, cosine, and tangent in a right-angled triangle: Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent. [6]

Why is tan(90°) undefined?

The tangent of an angle is its sine divided by its cosine (tan(θ) = sin(θ)/cos(θ)). At 90°, cos(90°) is 0. Since division by zero is undefined, tan(90°) is also undefined. [4]

How do I convert degrees to radians?

To convert degrees to radians, multiply the angle by π/180. For example, 90° * (π/180) = π/2 radians. [4]

What is a unit circle?

The unit circle is a circle with a radius of 1 centered at the origin of a graph. It’s a useful tool for visualizing the values of sine and cosine for any angle, where the x-coordinate of a point on the circle is the cosine of the angle and the y-coordinate is the sine. [2]

Can these values be negative?

Yes. The sign of sin, cos, and tan depends on the quadrant the angle is in. For example, in Quadrant II (91°-179°), sine is positive, but cosine and tangent are negative. You may want to study inverse trig functions for more details.

What’s the difference between sin and arcsin?

Sin (sine) takes an angle and gives you a ratio. Arcsin (inverse sine) takes a ratio and gives you an angle. For example, sin(30°) = 0.5, while arcsin(0.5) = 30°.

How are these functions used in the real world?

Trigonometry is used in many fields, including physics, engineering, architecture, video game design, and astronomy, often to solve for unknown distances or angles. For complex problems, engineers often consult a pythagorean theorem calculator.

Why does my calculator give a different answer?

Your calculator is likely in the wrong mode. Make sure it is set to “Degrees” or “Radians” to match the unit of your input angle. This is one of the most common errors when using a tan sin cos calculator.

Related Tools and Internal Resources

Explore these other calculators and guides to deepen your understanding of trigonometry and related mathematical concepts.