Unit Circle
- Introduction
- Definition of Unit Circle
- Equation of a Unit Circle
- How to Find Trigonometric Ratios Using Unit Circle
- Unit Circle Values in Radians
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
By definition, a unit circle is a circle with a unit radius. A closed geometric form with no sides or angles is called a circle. All of the characteristics of a circle are present in a unit circle, and its equation is also derived from that of a circle. Additionally, the standard angle values of all trigonometric ratios can be obtained using a unit circle.
In this lesson, we will study the unit circle equation and discover how to use the trigonometric ratios of `cosθ` and `sinθ` to represent each point on the unit circle's circumference.
Definition of Unit Circle
A unit circle is described as a circle with a radius of one unit, centered at the point `(0, 0)` on the coordinate plane, which is commonly used to indicate the origin of a cartesian coordinate system.
The unit circle is a common tool in trigonometry used to define trigonometric functions including cosine, sine, tangent, secant, cotangent, and cotangent. The ratios of the different side lengths of right triangles encircling the unit circle constitute the definition of these functions.
In complex analysis and physics, the unit circle is also important because it gives complex numbers and their connections to trigonometric functions as a visual representation.
Equation of a Unit Circle
The equation of a unit circle in cartesian coordinates is:
\( x^2 + y^2 = 1 \)
Here, \( x \) and \( y \) represent the coordinates of any point \( Q \) on the circle. Since the unit circle is centered at the origin `(0, 0)`, any point \( Q \) on the unit circle will satisfy this equation.
This equation can also be expressed in polar coordinates as:
\( r = 1 \)
Where \( r \) represents the distance from the origin to the point \( Q\) on the unit circle, which is always `1` unit.
How to Find Trigonometric Ratios Using Unit Circle
To find trigonometric ratios (sine, cosine, tangent, etc.) using the unit circle, we can follow these steps:
`1`. Draw the Unit Circle: Begin by drawing a coordinate plane with the unit circle centered at the origin `(0, 0)`. The unit circle is a circle with a radius of `1` unit.
`2`. Identify the Angle: Determine the angle \( \theta \) for which you want to find the trigonometric ratios. This angle is measured counterclockwise from the positive `x`-axis to the terminal side of the angle.
`3`. Locate the Point on the Circle: Find the point where the terminal side of the angle intersects the unit circle. The coordinates of this point are \( (x, y) \), where \( x \) is the cosine of the angle and \( y \) is the sine of the angle.
`4`. Calculate Ratios: Once you have the coordinates of the point, you can calculate the trigonometric ratios using these values:
Sine (sin): \( \sin(\theta) = y \)
Cosine (cos): \( \cos(\theta) = x \)
Tangent (tan): \( \tan(\theta) = \frac{y}{x} \)
Cosecant (csc): \( \csc(\theta) = \frac{1}{y} \)
Secant (sec): \( \sec(\theta) = \frac{1}{x} \)
Cotangent (cot): \( \cot(\theta) = \frac{x}{y} \)
`5`. Use Reference Angles: If the angle \( \theta \) is greater than `90` degrees or less than `0` degrees, you may need to use reference angles to find the trigonometric ratios. Reference angles are angles formed by the terminal side of \( \theta \) and the `x`-axis or `y`-axis, and they can help you determine the sign of the trigonometric ratios.
`6`. Account for Quadrants : Depending on the quadrant in which the angle \( \theta \) lies, the signs of the trigonometric ratios may change. Remember that in the first quadrant, all trigonometric ratios are positive; in the second quadrant, only sine is positive; in the third quadrant, only tangent is positive; and in the fourth quadrant, only cosine is positive.
Unit Circle Values in Radians
When we use the unit circle to calculate trigonometric values in radians, we measure angles not in degrees but in radians. A unit of angular measurement determined by the radius of a circle is called a radian. The angle subtended at a circle's center by an arc whose length is equal to the circle's radius is known as a radian.
In the context of trigonometry, the unit circle is a circle with a radius of `1` unit, centered at the origin of a cartesian coordinate system. It plays a crucial role in defining trigonometric functions and their values in radians.
`1`. Radians vs. Degrees: In radians, a complete revolution around the unit circle corresponds to an angle of \(2\pi\) radians. This is equivalent to \(360^\circ\) in degrees.
`2`. Angles and Points on the Unit Circle: Each angle measured in radians corresponds to a unique point on the unit circle. For example, an angle of `pi/6` radians corresponds to the point `(\cos(\pi/6), \sin(\pi/6)) = (\sqrt{3}/2, 1/2)` on the unit circle.
`3`. Common Radian Values: Some common radian values and their corresponding points on the unit circle include:
- \(0\) radians: Corresponds to the point \((1, 0)\) on the unit circle.
- `pi/6` radians (\(30^\circ\)): Corresponds to the point `(\sqrt{3}/2, 1/2)`.
- `pi/4` radians (\(45^\circ\)): Corresponds to the point `(\sqrt{2}/2, \sqrt{2}/2)`.
- `pi/3` radians (\(60^\circ\)): Corresponds to the point `(1/2, \sqrt{3}/2)`.
- `pi/2` radians (\(90^\circ\)): Corresponds to the point \((0, 1)\).
- \(\pi\) radians (\(180^\circ\)): Corresponds to the point \((-1, 0)\).
`4`. Trigonometric Functions in Radians: Trigonometric functions such as sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)) are defined based on the coordinates of points on the unit circle corresponding to specific angles measured in radians.
Understanding the values of angles in radians on the unit circle is fundamental for solving trigonometric problems, particularly those involving calculus, physics, and engineering.
Solved Examples
Example `1`. Find the sine and cosine of \( \frac{\pi}{6} \) radians.
Solution: From the unit circle, we know that at \( \frac{\pi}{6} \) radians, the point is \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \).
Sine (\( \sin \)): \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)
Cosine (\( \cos \)): \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
Example `2`. Determine the tangent of \( \frac{5\pi}{4} \) radians
Solution: At \( \frac{5\pi}{4} \) radians, the point is \( \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \).
Tangent (\( \tan \)): \( \tan\left(\frac{5\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \)
Example `3`. Calculate the secant of \( \frac{7\pi}{3} \) radians.
Solution: At \( \frac{7\pi}{3} \) radians, the point is \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \).
Secant (\( \sec \)): \( \sec\left(\frac{7\pi}{3}\right) = \frac{1}{\cos\left(\frac{7\pi}{3}\right)} = \frac{1}{-\frac{1}{2}} = -2 \)
Example `4`. Find the cosecant of \( \frac{\pi}{4} \) radians.
Solution: At \( \frac{\pi}{4} \) radians, the point is \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).
Cosecant (\( \csc \)): \( \csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \)
Example `5`. Determine the cotangent of \( \frac{3\pi}{2} \) radians.
Solution: At \( \frac{3\pi}{2} \) radians, the point is \( (0, -1) \).
Cotangent (\( \cot \)): \( \cot\left(\frac{3\pi}{2}\right) = \frac{\cos\left(\frac{3\pi}{2}\right)}{\sin\left(\frac{3\pi}{2}\right)} = \frac{0}{-1} = 0 \)
Practice Problems
Q`1`. What is the cosine of the angle \( \frac{\pi}{3} \) radians (`60` degrees) on the unit circle?
- \( \frac{1}{2} \)
- \( \frac{\sqrt{3}}{2} \)
- \( \frac{\sqrt{2}}{2} \)
- \( \frac{1}{\sqrt{2}} \)
Solution: a
Q`2`. If a point on the unit circle has coordinates \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \), what is the angle (in radians) corresponding to this point?
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{6} \)
- \( \frac{\pi}{2} \)
Solution: b
Q`3`. Which of the following is the correct trigonometric ratio for the angle \( \frac{3\pi}{4} \) radians (`135` degrees) on the unit circle?
- \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
- \( \cos\left(\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
- \( \tan\left(\frac{3\pi}{4}\right) = -1 \)
- \( \cot\left(\frac{3\pi}{4}\right) = -\sqrt{2} \)
Solution: a
Q`4`. If \( \tan(\theta) = \sqrt{3} \) and \( \theta \) lies in the third quadrant, what is the value of \( \theta \) (in radians)?
- \( \frac{5\pi}{3} \)
- \( \frac{2\pi}{3} \)
- \( \frac{4\pi}{3} \)
- \( \frac{\pi}{3} \)
Solution: c
Q`5`. What is the secant of the angle \( \frac{7\pi}{6} \) radians (`210` degrees) on the unit circle?
- \( -\frac{2}{\sqrt{3}} \)
- \( -\frac{2\sqrt{3}}{3} \)
- \( -\frac{2}{3} \)
- \( -\frac{\sqrt{3}}{2} \)
Solution: a
Frequently Asked Questions
Q`1`. What is the unit circle, and why is it important in trigonometry?
Answer: The unit circle is a circle with a radius of `1` unit, centered at the origin of a cartesian coordinate system. It serves as a fundamental tool in trigonometry because it provides a geometric representation of the relationships between trigonometric ratios (sine, cosine, tangent, etc.) and angles measured in radians.
Q`2`. How are trigonometric ratios related to points on the unit circle?
Answer: Each angle measured in radians corresponds to a unique point on the unit circle. The trigonometric ratios (sine, cosine, tangent, etc.) of an angle are defined based on the coordinates of the point where the terminal side of the angle intersects the unit circle.
Q`3`. How do I find trigonometric ratios using the unit circle?
Answer: To find trigonometric ratios using the unit circle, first identify the angle of interest. Then, locate the corresponding point on the unit circle. The cosine of the angle is equal to the `x`-coordinate of the point, and the sine of the angle is equal to the `y`-coordinate. Tangent can be found by dividing the sine by the cosine. Additionally, remember to account for the signs of the ratios based on the quadrant in which the angle lies.
Q`4`. Why are the trigonometric ratios positive or negative in different quadrants?
Answer: The signs of the trigonometric ratios (sine, cosine, tangent, etc.) depend on the signs of the coordinates of points on the unit circle in each quadrant. In the first quadrant, all ratios are positive. In the second quadrant, only sine is positive. In the third quadrant, only tangent is positive. In the fourth quadrant, only cosine is positive.
Q`5`. How can the unit circle help me solve trigonometric equations and problems?
Answer: The unit circle provides a visual and geometric approach to solving trigonometric equations and problems. By understanding the relationships between angles, trigonometric ratios, and points on the unit circle, you can easily determine the values of trigonometric functions and apply them to various mathematical and real-world problems.