The Unit Circle

A unit circle from the name itself defines a circle of unit radius. In the following guide, you will learn more about the unit circle.

The location of a point that is a distance of one unit from a fixed point is called a unit circle.

A step-by-step guide to the unit circle

A unit circle is a circle with a radius of $$1$$ unit. The unit circle is generally shown on the Cartesian coordinate plane.

The unit circle is represented algebraically using the second-degree equation with two variables $$x$$ and $$y$$.

The unit circle is used in trigonometry and is useful for finding the values of the trigonometric ratios of sine, cosine, and tangent.

Equation of a unit circle

The general equation of a circle is $$(x-a)^2+(y-b)^2=r^2$$, which represents a circle having the center $$(a, b)$$ and the radius $$r$$. This equation of a circle is simplified to show the equation of a circle.

A unit circle is formed with its center at the point$$(0, 0)$$, which is the origin of the coordinate axes and a radius of $$1$$ unit. So the equation of the unit circle is:

$$\color{blue}{x^2+y^2=1}$$

Note: the above equation satisfies all the points lying on the circle across the four quadrants.

Finding trigonometric functions using a unit circle

Consider a right triangle located in a unit circle on the Cartesian coordinate plane. The radius of the circle represents the hypotenuse of the right triangle.

The radius vector makes an angle $$θ$$ with the positive $$x$$-axis and the coordinates of the endpoint of the radius vector are $$(x, y)$$.

Here the values of $$x$$ and $$y$$ are the lengths of the base and the altitude of the right triangle. By applying this to trigonometry, we can find the values of the trigonometric ratio as follows:

$$\color{blue}{sin\:\theta=\frac{Altitude}{Hypoteuse}=\frac{y}{1}}$$

$$\color{blue}{cos\:\theta=\frac{Base}{Hypotenuse}=\frac{x}{1}}$$

Note: now we have $$sin\:\theta = y$$, $$cos\:\theta = x$$, and using this, we have $$tan\:\theta = \frac{y}{x}$$.

Unit circle and trigonometric identities

For a right triangle placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring $$1$$, $$x$$, $$y$$ units respectively, the unit circle identities can be given as:

• $$\color{blue}{sin\:\theta =\frac{y}{1}}$$
• $$\color{blue}{cos\:\theta =\frac{x}{1}}$$
• $$\color{blue}{tan\:\theta =\frac{sin\:\theta }{cos\:\theta }=\frac{y}{x}}$$
• $$\color{blue}{sec\:\left(\theta \right)=\frac{1}{x}}$$
• $$\color{blue}{csc\:\left(\theta \right)=\frac{1}{y}}$$
• $$\color{blue}{cot\:\left(\theta \right)=\frac{cos\:\theta }{sin\:\theta }=\frac{x}{y}}$$

Unit Circle – Example 1:

Does point $$A (\frac{1}{2}, \frac{1}{2})$$ lie on the unit circle?

Solution:

The equation of a unit circle is: $$x^2+y^2=1$$

Substituting $$x =\frac{1}{2}$$ and $$y = \frac{1}{2}$$, we get:

$$=\frac{1}{2}^2+\frac{1}{2}^2$$

$$=\frac{1}{4}+\frac{1}{4}$$

$$=\frac{2}{4}=\frac{1}{2}$$

$$≠ 1$$

Therefore $$(\frac{1}{2}, \frac{1}{2})$$ doesn’t lie on the unit circle.

Exercises for Unit Circle

Find the value of a function using a unit circle chart.

1. $$\color{blue}{sin\:900^{^{\circ }}}$$
2. $$\color{blue}{cos\:240^{^{\circ }}}$$
3. $$\color{blue}{tan\:225^{^{\circ }}}$$
1. $$\color{blue}{0}$$
2. $$\color{blue}{-\frac{1}{2}}$$
3. $$\color{blue}{1}$$

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