Unit Circle Definition Formula Table and Solved Examples
Let us know about the circle, before going through what is unit circle? or try to define a unit circle. A circle is the locus of all points which lie on a perimeter of equal distance from a given point. The point from which all the points on a circle are equidistant will be called the centre of the circle, and the distance from that centre to all the points on the circle will be called the radius of the circle. A diagram is shown below.
[Image will be uploaded soon]
The circle above has shown its centre at point C and a radius of length r. By definition, all radii of a circle are equal, since all the points on a circle are the same distance from the centre, and all the radii of a circle have one endpoint on the centre of the circle and one at the perimeter.
Chord and Its Properties
A chord of a circle is a straight line segment whose starting and end points both lie on the circle. A secant line is the infinite line extension of a chord and a chord is a line segment that attaches any two points on the curve, for example, an ellipse. The word chord is from the Latin ‘chorda’ and it implies bowstring.
[Image will be uploaded soon]
Chord Has Some Following Important Properties
Chords will be equidistant to the centre only if their lengths are equal.
Equal chords subtend equal angles to the centre of the circle.
A chord that passes through the centre of a circle will be called the diameter of the circle and it will be the longest chord.
Perpendicular drawn from the centre to the chord will divide the chord into two equal parts and vice versa.
What is Unit Circle?
Define Unit Circle - The unit circle is a circle of unit radius, that is a radius of 1. The unit circle in the trigonometry is the circle of radius 1 centred at the origin (0, 0) in the cartesian coordinate system of the euclidean plane.
[Image will be uploaded soon]
Consider if (x, y) is a point on the unit circles circumference, then |x| and |y| are the length of the legs of a right triangle, whose hypotenuse length is equal to 1. Thus, according to the Pythagorean theorem, a and b satisfies the equation
y2 + x2 = 1
Trigonometric Functions on the Unit Circle
The trigonometric functions like cos and sine of an angle are defined on the unit circle as given below.
If (x, y) is a point on the unit circle and a ray from the origin (0, 0) touches the point (x, y), then it makes an angle θ from the positive x-axis. In this clockwise turning is positive, then Cos θ = x and Sin θ = y.
x2 + y2 = 1 is the equation, which gives the relation
Cos2 θ + Sin2 θ = 1
The unit circle also shows that sine and cosine are periodic functions, with the identities
Cos θ = Cos(2π k + θ)
Sin θ = Sin(2π k + θ)
For any integer k.
[Image will be uploaded soon]
Diagram of the unit circle showing coordinate points.
Uses of Unit Circle
While working on the right triangles sine, cosine and other trigonometric functions are valid for angle measure more than zero or less than π/2. These functions produce meaningful values, when defined on the unit circles, for any real-valued angle measures. For example even those, greater than 2π. All the trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, including archaic functions like versine and exsecant, can be defined geometrically in terms of a unit circle.
The unit circle is very useful, to calculate the values of any trigonometric functions, other than those labelled already, without using the calculator. Only by using the angle sum and difference formula, it can be calculated.
Facts About Circle
A circle has the shortest perimeter compared to all the shapes of the same area.
The shape of the circle has led to one of the very important invention wheels, which enabled our modern automotive system.
The shape of the circle looks attractive to humans, so it can be seen in many architectural structures.
FAQs on Unit Circle in Trigonometry Complete Guide
1. What is the unit circle in trigonometry?
The unit circle is a circle with radius 1 centered at the origin (0, 0) on the coordinate plane. In trigonometry, it is used to define the trigonometric functions for all real numbers.
- Equation of the unit circle: x² + y² = 1
- Each point on the circle corresponds to an angle θ measured from the positive x-axis.
- The coordinates of a point are (cos θ, sin θ).
2. What is the equation of the unit circle?
The equation of the unit circle is x² + y² = 1. This comes from the standard circle equation (x − h)² + (y − k)² = r².
- Center: (0, 0), so h = 0 and k = 0
- Radius: r = 1
- Therefore: x² + y² = 1² = 1
3. How do you find sine and cosine using the unit circle?
On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate of the point corresponding to angle θ.
- Locate angle θ from the positive x-axis.
- Find the point where the terminal side intersects the circle.
- Coordinates are (cos θ, sin θ).
4. Why is the unit circle important in trigonometry?
The unit circle is important because it defines all trigonometric functions for any real angle θ. Unlike right triangle trigonometry, it works for angles greater than 90° and negative angles.
- Defines sine and cosine as coordinates.
- Extends trig functions beyond acute angles.
- Supports understanding of radians and periodic functions.
5. What are the key unit circle values to memorize?
The key unit circle values are the sine and cosine of special angles such as 0°, 30°, 45°, 60°, and 90° (and their radian forms).
- 0° (0): (1, 0)
- 30° (π/6): (√3/2, 1/2)
- 45° (π/4): (√2/2, √2/2)
- 60° (π/3): (1/2, √3/2)
- 90° (π/2): (0, 1)
6. How do you convert degrees to radians on the unit circle?
To convert degrees to radians, multiply by π/180. This works because 180° equals π radians.
- Formula: radians = degrees × (π/180)
- Example: 60° × (π/180) = π/3
7. How do you find tangent using the unit circle?
On the unit circle, tan θ = sin θ / cos θ. Since sin θ is the y-coordinate and cos θ is the x-coordinate, tangent equals y/x.
- Formula: tan θ = y/x
- Example: At 45° (π/4), sin = √2/2 and cos = √2/2
- So tan(π/4) = 1
8. What is the difference between the unit circle and a regular circle?
The difference is that the unit circle has radius 1 and center (0, 0), while a regular circle can have any radius and center.
- Unit circle equation: x² + y² = 1
- General circle equation: (x − h)² + (y − k)² = r²
- Used specifically for defining trig functions.
9. How do you find the coordinates of a point on the unit circle?
The coordinates of a point on the unit circle at angle θ are (cos θ, sin θ).
- Step 1: Measure angle θ from the positive x-axis.
- Step 2: Determine cos θ and sin θ.
- Step 3: Write coordinates as (cos θ, sin θ).
10. What are the quadrants of the unit circle and the signs of trig functions?
The unit circle is divided into four quadrants, and the signs of sine, cosine, and tangent depend on the quadrant.
- Quadrant I: sin, cos, tan are positive.
- Quadrant II: sin positive, cos and tan negative.
- Quadrant III: tan positive, sin and cos negative.
- Quadrant IV: cos positive, sin and tan negative.
