Find The Values Of The Trigonometric Functions Of θ Calculator

Find sin(θ), cos(θ), tan(θ) & more

What are the Trigonometric Functions of θ?

The trigonometric functions of an angle θ are mathematical functions that relate the angles of a right triangle to the ratios of its side lengths. They can also be defined using a point (x, y) on the terminal side of an angle θ in standard position (vertex at the origin, initial side on the positive x-axis) and the distance 'r' from the origin to that point. This calculator helps you find the values of the trigonometric functions of θ using the coordinates (x, y).

The six fundamental trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Knowing the x and y coordinates of a point on the terminal side of θ allows us to calculate these values, along with the angle θ itself. This find the values of the trigonometric functions of θ calculator is useful for students, engineers, and anyone working with angles and their relationships in geometry and other sciences.

Common misconceptions include thinking these functions only apply to angles between 0 and 90 degrees. However, by using the x, y, and r definitions, they apply to angles of any measure, including negative angles and angles greater than 360 degrees. The find the values of the trigonometric functions of θ calculator handles these general cases.

Trigonometric Functions of θ Formula and Mathematical Explanation

Given a point (x, y) on the terminal side of an angle θ in standard position, and the distance r = √(x² + y²) from the origin (0, 0) to the point (x, y) (where r > 0), the six trigonometric functions are defined as:

• Sine (sin θ): y / r
• Cosine (cos θ): x / r
• Tangent (tan θ): y / x (undefined if x = 0)
• Cosecant (csc θ): r / y (undefined if y = 0)
• Secant (sec θ): r / x (undefined if x = 0)
• Cotangent (cot θ): x / y (undefined if y = 0)

The distance r is always positive. The angle θ can be found using the arctangent function, specifically atan2(y, x), which gives the angle in radians, and can then be converted to degrees.

Variables Table

Variable	Meaning	Unit	Typical Range
x	The x-coordinate of a point on the terminal side of θ	–	Any real number
y	The y-coordinate of a point on the terminal side of θ	–	Any real number
r	The distance from the origin to (x, y), r = √(x² + y²)	–	r > 0 (unless x=0 and y=0)
θ	The angle in standard position	Degrees or Radians	Any real number
sin θ, cos θ	Sine and Cosine values	–	-1 to 1
tan θ, cot θ	Tangent and Cotangent values	–	Any real number (with exceptions)
csc θ, sec θ	Cosecant and Secant values	–	|value| ≥ 1 (with exceptions)

Table of variables used in calculating trigonometric functions.

Practical Examples (Real-World Use Cases)

Example 1: Point in the First Quadrant

Suppose a point on the terminal side of θ is (3, 4).

Inputs: x = 3, y = 4

Calculations:

• r = √(3² + 4²) = √(9 + 16) = √25 = 5
• sin θ = 4 / 5 = 0.8
• cos θ = 3 / 5 = 0.6
• tan θ = 4 / 3 ≈ 1.333
• csc θ = 5 / 4 = 1.25
• sec θ = 5 / 3 ≈ 1.667
• cot θ = 3 / 4 = 0.75
• θ ≈ 53.13 degrees

The find the values of the trigonometric functions of θ calculator would show these results.

Example 2: Point in the Second Quadrant

Suppose a point on the terminal side of θ is (-1, √3).

Inputs: x = -1, y = √3 ≈ 1.732

Calculations:

• r = √((-1)² + (√3)²) = √(1 + 3) = √4 = 2
• sin θ = √3 / 2 ≈ 0.866
• cos θ = -1 / 2 = -0.5
• tan θ = √3 / -1 = -√3 ≈ -1.732
• csc θ = 2 / √3 ≈ 1.155
• sec θ = 2 / -1 = -2
• cot θ = -1 / √3 ≈ -0.577
• θ = 120 degrees

Our find the values of the trigonometric functions of θ calculator accurately computes these values for any quadrant.

How to Use This Trigonometric Functions of θ Calculator

1. Enter Coordinates: Input the x-coordinate and y-coordinate of a point that lies on the terminal side of the angle θ.
2. Calculate: The calculator will automatically update the results as you type, or you can click the "Calculate Functions" button.
3. View Results: The calculator displays:
   • The primary trigonometric functions: sin(θ), cos(θ), and tan(θ).
   • Intermediate values: r (distance), csc(θ), sec(θ), cot(θ).
   • The angle θ in both degrees and radians.
   • A visual representation on a coordinate plane.
4. Interpret: The signs of sin(θ), cos(θ), and tan(θ) depend on the quadrant in which the terminal side of θ lies. The visual chart helps identify the quadrant.
5. Reset/Copy: Use the "Reset" button to clear inputs to default values and "Copy Results" to copy the calculated values.

This find the values of the trigonometric functions of θ calculator is designed to be intuitive and provide comprehensive results.

Key Factors That Affect Trigonometric Function Values

• The x-coordinate: Its sign and magnitude influence cos(θ), tan(θ), sec(θ), and cot(θ), as well as r.
• The y-coordinate: Its sign and magnitude influence sin(θ), tan(θ), csc(θ), and cot(θ), as well as r.
• The Quadrant: The signs of x and y determine the quadrant of θ, which in turn dictates the signs of the trigonometric functions (ASTC rule: All, Sine, Tangent, Cosine positive in quadrants I, II, III, IV respectively).
• The Value of r: As r = √(x² + y²), it depends on both x and y and is always positive (for points other than the origin). It normalizes x and y in sin and cos.
• Angle Unit: While the trig function values are the same, the angle θ itself can be expressed in degrees or radians. Our find the values of the trigonometric functions of θ calculator shows both.
• Zero Values for x or y: If x=0, tan(θ) and sec(θ) are undefined (angle is 90° or 270° or coterminal). If y=0, cot(θ) and csc(θ) are undefined (angle is 0° or 180° or coterminal).

Frequently Asked Questions (FAQ)

1. What if r = 0?

The distance r can only be 0 if both x=0 and y=0 (the origin). In this case, the angle θ is undefined, and the trigonometric functions are not defined based on r, as division by zero would occur. Our find the values of the trigonometric functions of θ calculator assumes r > 0.

2. What happens if x=0 or y=0?

If x=0 (point on the y-axis), tan(θ) and sec(θ) are undefined. If y=0 (point on the x-axis), cot(θ) and csc(θ) are undefined. The find the values of the trigonometric functions of θ calculator will indicate "Undefined" for these cases.

3. What are radians and degrees?

Radians and degrees are two different units for measuring angles. 360 degrees = 2π radians. The calculator provides the angle θ in both units.

4. What is the unit circle?

The unit circle is a circle with a radius of 1 centered at the origin. If the point (x, y) is on the unit circle, then r=1, and sin(θ) = y, cos(θ) = x. It's a useful tool for understanding trigonometric functions for all angles.

5. How are trigonometric functions periodic?

The values of trigonometric functions repeat every 360 degrees (or 2π radians) because adding 360° to an angle results in the same terminal side. For example, sin(θ) = sin(θ + 360°).

6. Can I enter the angle directly instead of x and y?

This specific find the values of the trigonometric functions of θ calculator uses x and y coordinates. If you have the angle, you can find a point by choosing r=1, then x=cos(θ) and y=sin(θ) and use those values here, or use a calculator that takes the angle directly.

7. Why use x, y, and r instead of opposite, adjacent, and hypotenuse?

The x, y, r definitions extend the trigonometric functions to angles beyond 0° to 90° (0 to π/2 radians), which is the limit when using only right triangles with positive side lengths (opposite, adjacent, hypotenuse). This find the values of the trigonometric functions of θ calculator uses the more general definition.

8. How do I know which quadrant my angle is in based on x and y?

Quadrant I: x > 0, y > 0; Quadrant II: x < 0, y > 0; Quadrant III: x < 0, y < 0; Quadrant IV: x > 0, y < 0. The chart in our find the values of the trigonometric functions of θ calculator visualizes this.