How Does the Unit Circle Simplify Trigonometry?
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.
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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.
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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.
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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.
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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 Explained: Formula, Properties & Uses
1. What defines a unit circle in trigonometry, and what is its standard equation?
A unit circle is a circle with a radius of exactly 1 unit, centred at the origin (0, 0) of a Cartesian coordinate system. Its simplicity is fundamental for understanding trigonometric functions. The standard equation of a unit circle is derived from the Pythagorean theorem, which is x² + y² = 1. Any point (x, y) that lies on the circumference of the unit circle satisfies this equation.
2. How are the coordinates of a point on the unit circle related to sine and cosine functions?
For any point P(x, y) on the unit circle that corresponds to an angle θ (measured counter-clockwise from the positive x-axis), the coordinates are directly defined by the cosine and sine of that angle. Specifically, the x-coordinate is equal to the cosine of the angle (x = cos θ), and the y-coordinate is equal to the sine of the angle (y = sin θ). This relationship is the foundation for defining trigonometric functions for all real numbers.
3. How is the unit circle divided into four quadrants, and what do they represent?
The unit circle is divided into four equal sections called quadrants by the x-axis and y-axis. These quadrants are numbered counter-clockwise from the top right:
- Quadrant I: 0° to 90° (0 to π/2 radians), where both x and y coordinates are positive.
- Quadrant II: 90° to 180° (π/2 to π radians), where x is negative and y is positive.
- Quadrant III: 180° to 270° (π to 3π/2 radians), where both x and y are negative.
- Quadrant IV: 270° to 360° (3π/2 to 2π radians), where x is positive and y is negative.
These quadrants are crucial for determining the signs of trigonometric functions for any given angle.
4. Why is the radius of the unit circle specifically chosen to be 1?
The radius is chosen as 1 for simplification and convenience. When the radius (hypotenuse of the reference triangle) is 1, the definitions of sine and cosine become much simpler. In a right-angled triangle, sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse. On a unit circle, the hypotenuse is always 1, so the definitions reduce to sin θ = opposite (the y-coordinate) and cos θ = adjacent (the x-coordinate). This elegant simplification makes it much easier to derive trigonometric identities, like the fundamental identity cos²θ + sin²θ = 1.
5. How can we determine the value of the tangent function (tan θ) using the unit circle?
The tangent of an angle θ is defined as the ratio of the sine to the cosine: tan θ = sin θ / cos θ. Since on a unit circle sin θ = y and cos θ = x, the tangent can be calculated as the ratio of the y-coordinate to the x-coordinate (tan θ = y/x). This also means that the tangent is undefined when the x-coordinate is 0 (at 90° and 270°), as this would involve division by zero.
6. What is the importance of using radians instead of degrees when working with the unit circle?
While both degrees and radians measure angles, radians are the more natural unit for mathematics, especially in calculus and higher-level topics. On a unit circle (where r=1), the arc length subtended by a central angle is equal to the angle's measure in radians (Arc Length = r * θ = 1 * θ = θ). This direct relationship between arc length and angle simplifies many important formulas in calculus, such as the derivatives of trigonometric functions.
7. How do the signs of sin, cos, and tan change across the different quadrants of the unit circle?
The signs of the trigonometric functions are determined by the signs of the x (cos θ) and y (sin θ) coordinates in each quadrant. A common mnemonic to remember this is 'All Students Take Calculus' (ASTC), starting from Quadrant I:
- Quadrant I (All): All three functions (sin, cos, tan) are positive.
- Quadrant II (Students/Sine): Only sin is positive, while cos and tan are negative.
- Quadrant III (Take/Tangent): Only tan is positive, while sin and cos are negative.
- Quadrant IV (Calculus/Cosine): Only cos is positive, while sin and tan are negative.
8. Beyond finding basic trigonometric values, what is another key application of the unit circle?
A key application of the unit circle is in defining and visualising periodic functions. As you travel around the circle, the values of sine and cosine repeat every 360° (or 2π radians). This cyclical nature is fundamental to modelling real-world phenomena that are periodic, such as sound waves, light waves, alternating current (AC) circuits, and simple harmonic motion in physics. The unit circle provides a visual and mathematical foundation for understanding these repeating patterns.
