# Interactive Unit Circle

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## Define Interactive Unit Circle

The interactive unit circle is what that joins the trigonometric functions - sine cosine, and tangent, and the unit circle. The unit circle is actually referred to as a circle of radius one suspended in a specific quadrant of the coordinate system. The radius of a unit circle can be taken at any point on the perimeter of the circle.

It forms a right-angled triangle. The angle between this interactive unit circle will be displayed by angle θ. In order to change a grade, you would simply need to click and drag the two control points.

### Functions of Interactive Unit Circle

This unit circle basically consists of 3 functions as follows:

• Sine

• Cosine

• Tangent

The interaction between this unit circle and its correlating functions is referred to as interactive unit circles.

### Sine, Cosine and Tangent

1. Sine

The second and another basic trigonometric function is sine represented by θ. In Mathematical terms, sine θ is computed by dividing the perpendicular of a right-angled triangle by its hypotenuse. Thus, we can compute the length of the sides or the angle of any structure with the help of the above relation. Hence, the formula to calculate Sineθ is as below;

Sine θ = Perpendicular/Hypotenuse

• Cosecant: With respect to cosine, the reciprocal of sineθ is referred to as cosecant θ. It is computed by reciprocating sine or just by dividing it with 1. Hence, Cosecant θ = 1/sin θ.

2. Cosine

In a right-angled triangle, the ratio between the base and hypotenuse of a triangle is referred to as cosineθ. It is actually one of the most crucial trigonometric functions of all. In Mathematical terms, cosine is obtained by dividing the base of a right-angled triangle with its hypotenuse. Hence, formula to calculate Cosineθ is as below;

Cosine = Base/Hyp

• Secant: The reciprocal of cosine which is known as secant θ is also used in some triangles. The secant θ is used in several numerical calculations and is calculated by reciprocating cosine θ. Thus, Secant = 1/cosine.

3. Tangent

Another and 3rd basic trigonometric function is referred to as tangent. As per sine θ and cosine θ, we can also calculate and get the answer for tangent in a right-angled triangle. In a right triangle, the perpendicular of a triangle is divided with its base, and we easily obtain the value of tangent θ.

The mathematical formula to calculate tangent θ is: Tang θ = Perp/Base.

• Cot: The reciprocal of Tangentθ is known as cot θ. The value of cot can be calculated by reciprocating the value of tangent. The mathematical form of this equation is as stated below: Cot θ = 1/Tang θ.

Thus, all the equations and the trigonometric functions can be understood by the interactive unit circle graph.

### Trigonometric Circle Interactive Simulation

Choose a Quadrant and drag the point in the simulation as shown in the figure to visualise the unit circle in all the four quadrants.

### Solved Examples

Example:

Why is cot 180° undefined?

Solution:

We know that

• cot θ = 1/tan θ

• tan θ = sin θ/cos θ

∴ cot θ = cos θ/sin θ

From the interactive unit circle graph chart, we are familiar with:

sin180° = 0

Since, division by 0 is ∞, cot 180° = ∞

Hence, cot 180° = ∞

Example:

Calculate the exact value of tan 210° using the interactive unit circle.

Solution:

We are familiar with:

tan210° = sin210° / cos210°

Making use of the unit circle chart:

• sin 210° =  -1/2

• cos 210° = -√3/2

Therefore,

tan 210°=sin 210°/cos 210°

=−1/2 / −√3/2

=1/√3

=√3/3

Therefore, tan210° = √3/3

### Key Facts

• The unit circle is referred to as a circle of radius 1 unit.

• The equation of a unit circle is x² + y² = 1.

• You can refer to the conversion table of angular measures to radian measures for finding important Sin Cos and Tan values of the 1st quadrant.

Q1. How do we Calculate Trigonometric Functions Using a Unit Circle?

Answer: We can calculate the important trigonometric functions like the sine, cosine and tangent using the unit circle. Let us use the Pythagoras theorem in an interactive unit circle for the purpose of understanding the trigonometric functions.

Thus, let’s take a right-angled triangle inscribed in a unit circle on the cartesian coordinate plane.

The trigonometric functions are shown below in the form of an interactive unit circle graph.

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

Q2. What is the Formula for Unit Circle?

Answer: Equation of a circle on the cartesian coordinate plane is given by the formula:

(x − h)² + (y − k)² = r² where

(h, k) = coordinates of the centre in the cartesian plane

r = the radius of the circle.

(x, y) are the points on the circumference of the circle, at a distance from the centre (h,k) and r (radius).

With reference to a unit circle, the centre lies at (0, 0) and the radius (r) is 1 unit.

Hence, the formula for the unit circle is: x² + y² = 1

Q3. What is the Interactive Unit Circle Graph of Trig Functions?

Answer: This interactivity takes an observation of the graphs of the trig functions such as sine, cosine, tangent, secant. That being said, first of all, take a look at the unit circle. Using the slider, You can also change the angle θ. This will move the point around the circle and get you the actual values.