How the Unit Circle Unlocks Trigonometry Concepts
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.
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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.
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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.
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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.
FAQs on Explore the Interactive Unit Circle
1. What is an interactive unit circle and what is its fundamental equation?
A unit circle is a circle with its center at the origin (0,0) of the Cartesian plane and a radius of exactly 1 unit. It is a fundamental tool in trigonometry because it helps visualise the relationship between angles and their trigonometric function values. The equation of the unit circle is x² + y² = 1, which is derived from the standard equation of a circle where the centre (h,k) is (0,0) and radius 'r' is 1.
2. How does the unit circle define the sine and cosine of an angle?
For any point (x, y) on the circumference of the unit circle, an angle θ is formed by the radius to that point and the positive x-axis. The trigonometric functions are defined by the coordinates of this point:
The cosine of the angle, cos(θ), is equal to the value of the x-coordinate.
The sine of the angle, sin(θ), is equal to the value of the y-coordinate.
This direct relationship (cos θ = x, sin θ = y) applies to all angles, including those beyond 360°.
3. What are the coordinates for key angles like 0°, 90°, 180°, and 270° on the unit circle?
The coordinates at these four quadrantal angles are crucial reference points:
0° (or 0 radians): The point is at (1, 0). Therefore, cos(0°) = 1 and sin(0°) = 0.
90° (or π/2 radians): The point is at (0, 1). Therefore, cos(90°) = 0 and sin(90°) = 1.
180° (or π radians): The point is at (-1, 0). Therefore, cos(180°) = -1 and sin(180°) = 0.
270° (or 3π/2 radians): The point is at (0, -1). Therefore, cos(270°) = 0 and sin(270°) = -1.
4. Why is the radius of the unit circle specifically 1? What is the importance of this?
The radius is set to 1 to create the simplest possible relationship between an angle and its trigonometric values. In any right-angled triangle, sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse. By making the hypotenuse (the radius) equal to 1, the definitions are simplified to:
sin(θ) = opposite/1 = opposite side (the y-coordinate)
cos(θ) = adjacent/1 = adjacent side (the x-coordinate)
This removes the need for division, making the (x, y) coordinates on the circle directly represent the (cos θ, sin θ) values.
5. How can you determine the signs (positive or negative) of trigonometric functions in different quadrants using the unit circle?
The signs of sin(θ), cos(θ), and tan(θ) depend on the signs of the x and y coordinates in each quadrant. A helpful mnemonic is "All Students Take Calculus" (ASTC Rule), starting from Quadrant I:
Quadrant I (0° to 90°): All functions are positive (since x and y are both positive).
Quadrant II (90° to 180°): Sine (and its reciprocal, csc) is positive (since y is positive, x is negative).
Quadrant III (180° to 270°): Tangent (and its reciprocal, cot) is positive (since x and y are both negative, so y/x is positive).
Quadrant IV (270° to 360°): Cosine (and its reciprocal, sec) is positive (since x is positive, y is negative).
6. How does the Pythagorean theorem apply to the unit circle to give the main trigonometric identity?
For any point (x,y) on the unit circle, a right-angled triangle can be formed with the origin. The horizontal side has length 'x', the vertical side has length 'y', and the hypotenuse is the radius, which is 1. Applying the Pythagorean theorem (a² + b² = c²) gives us x² + y² = 1². Since we know that on the unit circle x = cos(θ) and y = sin(θ), substituting these values gives the fundamental Pythagorean identity: cos²(θ) + sin²(θ) = 1.
7. Beyond sine and cosine, how can the unit circle be used to find tan(θ), sec(θ), and csc(θ)?
Once you have the coordinates (x, y) for an angle θ on the unit circle, where x = cos(θ) and y = sin(θ), you can easily find the other trigonometric functions using their definitions:
Tangent (tan θ) = sin(θ) / cos(θ) = y / x
Secant (sec θ) = 1 / cos(θ) = 1 / x
Cosecant (csc θ) = 1 / sin(θ) = 1 / y
It's important to note that these functions will be undefined whenever their denominator is zero.
