The unit circles are the circles of unit radius. A circle is a closed mathematical figure with no sides or points. As we know, the unit circle has the same properties as a circle. We can use the equation of a circle to find the equation of a unit circle. Equation of a circle in a cartesian coordinate system with the center as (p,q) and radius as r, can be written as (x – p)2 +(y – q)2 = r2. But for the unit circle, the center coordinates are (0,0) and the radius is 1. Hence the equation can be written as (x – 0)2 +(y – 0)2 = 12 x2 + y2 = 1. This is the required equation of the unit circle.
Use of Unit Circle in Trigonometric Functions
Let us take a unit circle. Draw a right angle triangle within the circle such that the hypotenuse is equal to 1, the length of the base is equal to x, and height is equal to y. The angle subtended at the center of the circle by the base and the hypotenuse is θ.
Now we know that according to the pythagoras theorem, the sum of the squares of the base and height is equal to the square of the hypotenuse in a right angled triangle.
∴ Base2 + height2 = hypotenuse2 ⇒ x2 + y 2 = 1.
In trigonometry in a right angle triangle,
- sinθ = Opposite / Hypotenuse = y/1
- cosθ = Adjacent / Hypotenuse = x/1
- tanθ = opposite/adjacent = y/x
- cotθ = 1/tanθ = 1/(yx) = x/y
- secθ = 1/ cosθ = 1/ x
- cosecθ = 1/sinθ = 1/y
Proof of Pythagorean Identities using Unit Circle
The Pythagoras theorem states that in a right-angled triangle the sum of the squares of the base and height is equal to the square of the hypotenuse. For a better understanding of the concept, I referred to Cuemath classes.
Pythagorean identities of trigonometry are as follows:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
- For the same right angle triangle by applying the pythagoras theorem we get x2 + y 2 = 1.
- But as we found the value of sinθ = y and cosθ = x. By applying these values in the equation we get, sin2θ + cos2θ = 1.
- For the second identity we know that tanθ = y/x and secθ = 1/x. By substituting these values we get
- 1 + tan2θ = sec2θ 1+(y/x)2 = (1/x)2
- (x2+y2)/x2 = 1/x2 x2 + y 2 = 1 Hence proved.
- Similarly, For third identity cotθ = x/y and cosecθ = 1/y. By substituting these values we get
- 1 + cot2θ = cosec2θ 1+(x/y)2 = (1/y)2
- (y2+x2)/y2 = 1/y2 x2 + y 2 = 1 Hence proved.
You can also use the unit circle to measure the angle in radians or degrees. For solved problems based on unit circles, you must log in to the Cuemath website. Here you get all varieties of problems along with a detailed explanation of the application of the unit circle to measure the angle in radians or degrees.
I ordered a Pizza for my lunch. It's been a long time since I had pizza. The moment the delivery boy gave my order, I just hopped on it. After I relished it, I started wondering what the radius of the regular pizza would be? Have you ever wondered the same thing? Never right. I usually just concentrate on eating it but I got this thought when my son asked me to explain the unit circle last time during his exams. It is very important to learn the trigonometric functions in detail. I guess now you can calculate the radius of the pizza after going through this article