November 15, 2024

Understanding \(\cos \dfrac{\pi}{3}=\cos 60^\circ\)

Introduction to Radian and Degree Conversion Trigonometry often uses radians as a way to measure angles, although degrees are also common in many applications. Knowing how to convert between these units is essential. The main relationship between radians and degrees is: \[\begin{align*} \pi \text{ radians} = 180^\circ \end{align*}\] If we divide both sides by 3…

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Cosine of $\pi$

Cosine of $\pi$ is $\boxed{-1}$. 1 Unit Circle There are multiple ways to find $\cos \pi$. One option is to use the unit circle: Notice that $\cos \pi = \cos 180^\circ$. Recall that $180^\circ$ is a straight angle, so the terminal side in standard position would intersect the unit circle on the $x$-axis at $(-1,0)$.…

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Understanding the Unit Circle

Unit Circle The unit circle is defined by $x^2 + y^2 = 1$, which is a circle with radius $1$, centered at the origin $(0, 0)$. An example of the unit circle is below: Standard Position of an Angle An angle is in standard position if its initial side is on the positive $x$-axis, its…

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Implicit Differentiation Guide

What is Implicit Differentiation? In many calculus problems, you’ll see equations that don’t exactly represent functions. Normally, we’re used to seeing equations like $$y = 2x + 1$$ where $y$ is isolated on one side. However, what if \(x\) and \(y\) are mixed together, like $$x^2 + y^2 = 1 { or } 3x^2y -4x\cos…

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Blank Unit Circle

The unit circle is defined as $x^2 + y^2 = 1$, which is a circle with radius $1$, centered at the origin $(0, 0)$. It is used in trigonometry to simplify finding values of trig functions. An empty unit circle is below with markings to be filled in: Blank_Unit_Circle.PDF Practice Problems Find the radian value…

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