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Find the coordinates x(cos), y(sin), or tan values of an angle on the unit circle.

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This unit circle calculator helps to calculate trigonometric values (sine, cosine, and tangent) for any angle on the unit circle. Simply enter the angle in degrees, radians, or pie (π) radians and the calculator will determine:

- The coordinates (sine and cosine) of the point on the unit circle corresponding to the given angle
- Tangent values

**“A unit circle is a circle on a cartesian plane having a ****radius**** equal to 1, centered at the origin (0,0)”**

By marking the endpoint (terminal point) of a rotation from the positive x-axis (1, 0) on the unit circle, we can calculate the values of the cosine, sine, and tangent for a specific angle θ. The unit circle is very helpful when you have to work with core trigonometry functions and need to find the angle measurements.

While the equation of the unit circle can be derived using the basic circle equation and the Pythagorean theorem, Let's see how: The equation of a circle, having center (x1, y1) and radius r can be written as:

**\(\ (x-\ x_{1})^{2}\ \ +\ \ (y-\ y_{1})^{2}=\ r^{2}\)**

Where,

- (x, y) are the coordinates of the unit circle point

According to the Pythagorean theorem, the radius of the unit circle is always 1 so the equation will be transformed as:

\(\ (x-\ 0)^{2}\ \ +\ \ (y-\ 0)^{2}=\ r^{2}\) \(\ (x-\ 0)^{2}\ \ +\ \ (y-\ 0)^{2}=\ 1\) \(\ x^{2}\ \ +\ \ y^{2}=\ 1\)

Therefore,

\(\cos^{2}\theta\ \ +\ \sin^{2}\theta =\ 1\)

In trigonometry, the unit circle is used to visualize and derive trigonometric functions and their relationships. Let's see how the unit circle helps us derive the trigonometric ratios!

Sine and cosine relationship with unit circle:

- Sine is the y-coordinate
- Cosine is the x-coordinate

**Explanation:**

- Let's take a look at the point A in the above image
- Xx and Yy are the coordinates of the point P. As it’s a unit circle the radius is equal to 1
- After projecting the radius on the x and y coordinates, you will get the right triangle
- The horizontal side(adjacent to the unit circle angle) has a length equal to the x-coordinate
- The vertical side of this triangle (opposite to the angle) has a length equal to the y coordinate
- The hypotenuse (the diagonal side that creates the right angle) is the radius, which remains equal to 1

**\(\ sin\ \alpha =\frac{ Opposite }{ Hypotenuse }=\frac{y}{1}=\ y\)**

**\(\ cos\ \alpha =\frac{ Adjacent }{ Hypotenuse } =\frac{x}{1}=\ x\)**

As we have discussed above the equation of the unit circle that comes from the Pythagorean theorem is as follows:

**\(\ x^{2}\ \ +\ \ y^{2}=\ 1\)** **\(\ cos^{2}\theta\ \ +\ \ sin^{2}\theta=\ 1\)**

According to the definition of tangent, it's the ratio of opposite sides and adjacent sides to an angle in the right triangle.

**\(\tan\alpha =\frac{opposite}{adjacent}\)**

**\(\tan\alpha =\frac{y}{x}\)**

The tangent of an angle (α) is calculated using the sine (y) and cosine (x) values from the unit circle or a right triangle using the following formula:

**\(\tan\alpha =\frac{\ sine\ (y)}{cose\ (x)}\)**

**\(\tan\alpha =\frac{\ sine\ (y)}{cose\ (x)}=\frac{y}{x}\) **(when x = 0, the tangent is undefined)

- Determine the given point on the unit circle. The coordinates of the unit circle are \(\ (sin\theta,\ cos\theta)\), where \(\theta\) is the angle between the positive x-axis and the line that connects the point with the origin
- Find the \(\ cos\theta\) as it’s the x-coordinate
- Find the \(\ sin\theta\) as it’s the y-coordinate
- Get the value of tangent by dividing the \(\ cos\theta\) by \(\ sin\theta\), \(\ tan\theta=\frac{\ cos\ \theta}{sin\ \theta}\)

The unit circle coordinates calculator eliminates the need for stepwise trig function (coordinates) calculation from an angle.

**#1: Find the trig functions if the angle of the unit circle is 60°.**

**Solution:**

Mark Point P on the unit circle where the angle is formed

Find coordinate on unit circle:

sin(60°) = y-coordinate of point P = \(\frac{\sqrt{3}}{2} = 0.86602…\) (positive because it is on the upper half of the circle)

cos(60°) = x-coordinate of point P = \(\frac{1}{2} =0.5\) (positive because it lies on the right half of the circle)

Calculate Tangent (tan(60°)):

\(\ tan(60°) =\frac{sin(60°)}{cos(60°)} = \frac{(\frac{\sqrt{3}}{2})}{(\frac{1}{2})} =\sqrt{3} = 1.7320…\)

**#2: ****Find trigonometric ratios**** for an angle of \(\frac{π}{3}\) radians on the unit circle.**

**Solution:**

Mark Point P on the unit circle as done in the previous example

Find the Coordinates: X-coordinate \(\cos(\frac{π}{3}) = \frac{1}{2}=\ 0.5\) (positive because it is present on the right half of the circle)

Y-coordinate (sin(π/3)) =\(\frac{\sqrt{3}}{2} =\ 0.86602 \) (positive because it is available on the upper half of the circle)

Calculate Tangent \(\ (tan(\frac{π}{3})\):

\(\ tan(\frac{π}{3}) =\frac{sin(\frac{π}{3})}{cos(\frac{π}{3})} = \frac{(\frac{\sqrt{3}}{2})}{(\frac{1}{2})} =\sqrt{3} = 1.7320…\)

**#3: ****Calculate the ****unit circle trig**** values for an angle of π radians.**

**Solution:**

π radians corresponds to 180 degrees, so for the unit circle radians the coordinates of point P will be:

X-coordinate (cos(π)) = P = -1 (negative because it is present on the left half of the circle)

Y-coordinate (sin(π)) = P = 0 (because it lies on the x-axis)

Tangent (tan(π)):

tan(π) = undefined

Also, you can use the unit circle trig calculator to find the exact values of sine, cosine, and tangent for any angle in radians, including π radians.

This chart shows a circle with angles marked in degrees or radians. With the help of this unit circle chart, you can easily find the sine (y-coordinate), cosine (x-coordinate), and tangent values on the circle's edge.

**Note:** A unit circle goes from 0 to 360 degrees (0 to 2 π radian). So whenever you get an angle bigger than 360 degrees, you should keep subtracting 360 until it reaches the normal value from 0 to 360 degrees.

The given table can be considered for calculating the coordinates (x, y) of the unit circle from the value of angle.

Angle (Degrees) | Angle (Radians) | Unit Circle Coordinates |
---|---|---|

30° | \(\frac{\pi}{6}\) | (\(\frac{\sqrt{3}}{2}\),\(\frac{1}{2}\)) |

45° | \(\frac{\pi}{4}\) | (\(\frac{\sqrt{2}}{2}\),\(\frac{\sqrt{2}}{2}\)) |

60° | \(\frac{\pi}{3}\) | (\(\frac{1}{2}\),\(\frac{\sqrt{3}}{2}\)) |

90° | \(\frac{\pi}{2}\) | (0, 1) |

120° | \(\frac{2\pi}{3}\) | (\(\frac{-1}{2}\),\(\frac{\sqrt{3}}{2}\)) |

135° | \(\frac{3\pi}{4}\) | (\(\frac{-{\sqrt{2}}}{2}\),\(\frac{\sqrt{2}}{2}\)) |

150° | \(\frac{5\pi}{6}\) | (\(\frac{-{\sqrt{3}}}{2}\),\(\frac{1}{2}\)) |

180° | π | (-1, 0) |

210° | \(\frac{7\pi}{6}\) | (\(\frac{-{\sqrt{3}}}{2}\),\(\frac{-1}{2}\)) |

225° | \(\frac{5\pi}{4}\) | (\(\frac{-{\sqrt{2}}}{2}\),\(\frac{-{\sqrt{2}}}{2}\)) |

270° | \(\frac{3\pi}{2}\) | (0, -1) |

300° | \(\frac{5\pi}{3}\) | (\(\frac{1}{2}\),\(\frac{-{\sqrt{3}}}{2}\)) |

315° | \(\frac{7\pi}{4}\) | (\(\frac{\sqrt{2}}{2}\),\(\frac{-{\sqrt{2}}}{2}\)) |

330° | \(\frac{11\pi}{6}\) | (\(\frac{\sqrt{3}}{2}\),\(\frac{-1}{2}\)) |

360° | 2π | (1, 0) |

Skip memorizing all the values! Find trigonometric ratios using the unit circle calculator.

The unit circle has a wide range of applications in real-world scenarios, including:

- Understanding Periodic Phenomena
- Solving Engineering Problems
- Computer Graphics and Animation
- Navigation and Positioning
- Signal Processing and Data Analysis

Positive angles on the unit circle are measured starting from the positive x-axis and rotating counterclockwise around the origin. They lie between the positive x-axis and the terminal side.

The special right triangles (30-60-90 and 45-45-90) serve as a roadmap for the unit circle. By scaling them to fit inside (hypotenuse = 1 unit), their side lengths directly give sine (y) and cosine (x) values for key angles (30°, 45°, 60°) on the circle. Using trigonometry, this foundation helps assign sine and cosine values to other angles.

**References:**

A source of Wikipedia: All you need to know about the unit circle

From the source of Khan Academy: Unit: Trigonometric functions, intro to radians & much more!

From the source of clarku: tangent to the circle

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