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Last updated on October 7, 2025
In trigonometry, the unit circle is a fundamental concept that helps in understanding angles and their trigonometric functions. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. In this topic, we will learn the formulas related to the unit circle.
The unit circle is integral in trigonometry for defining sine, cosine, and tangent. Let’s learn the formulas associated with the unit circle.
The unit circle is a circle with a radius of 1 centered at the origin (0,0) of the coordinate plane. The equation of the unit circle is: \( x^2 + y^2 = 1 \) where \( x \) and \( y \) are the coordinates of any point on the circle.
Trigonometric functions can be derived from the unit circle: -
Sine function: \(( \sin(\theta) = y )
\)
Cosine function:\( ( \cos(\theta) = x )
\)
Tangent function: \(( \tan(\theta) = \frac{y}{x} ) (where ( x \neq 0 ))\)
Angles on the unit circle are typically measured in radians. Common angles and their corresponding coordinates are:
\(0 or ( 2\pi ): (1, 0) - ( \frac{\pi}{2} ): (0, 1) - ( \pi ): (-1, 0) - ( \frac{3\pi}{2} ): (0, -1)\)
The unit circle is crucial in trigonometry as it provides a geometric representation of trigonometric functions.
Students often find it challenging to remember unit circle values. Here are some tips:
Students often make errors when working with the unit circle. Here are some mistakes and how to avoid them.
Find the sine and cosine of \( \frac{\pi}{4} \).
The sine and cosine of \(( \frac{\pi}{4} ) \)are both \(( \frac{\sqrt{2}}{2} ).\)
At \( ( \frac{\pi}{4} ), \)the coordinates on the unit circle are \(( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) )\). Hence \(( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} ) \) and \( ( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} ).\)
Find the coordinates of the point on the unit circle at an angle of \( \frac{2\pi}{3} \).
The coordinates are \(( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) ).\)
At \(( \frac{2\pi}{3} ),\) the unit circle coordinates are \(( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) ),\) corresponding to cosine and sine respectively.
What is the tangent of \( \pi \)?
The tangent of\( ( \pi ) \)is 0.
At \(( \pi),\) the coordinates are (-1, 0). Therefore,\( ( \tan(\pi) = \frac{0}{-1} = 0 ).\)
Determine the cosine of \( \frac{3\pi}{2} \).
The cosine of \(( \frac{3\pi}{2} ) \)is 0.
At \(( \frac{3\pi}{2} ),\) the coordinates are (0, -1). Thus,\( ( \cos(\frac{3\pi}{2}) = 0 ).\)
Find the sine of \( \pi \).
The sine of \(( \pi ) \)is 0.
At \( \pi \), the coordinates are (-1, 0), so\( ( \sin(\pi) = 0 ).\)
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