A Trick to Memorizing Trig Special Angle Values Table

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Memorizing trigonometric special angle values is crucial for calculus, as evaluations often involve these angles. A suggested method is to create a table with angles in both degrees and radians, filling in sine, cosine, and tangent values systematically. The sine function starts at zero and follows a pattern based on the square roots of integers divided by four. Understanding the unit circle and basic triangle geometry aids in deriving these values. This approach simplifies learning and retention of essential trigonometric functions.
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In calculus classes when you are asked to evaluate a trig function at a specific angle, it’s 99.9% of the time at one of the so-called special angles we use in our chart. Since you are likely to have learned degrees first I’ll include degree angles in the first chart, but after that, it’s going to be radian only.
Begin by setting up the table on scratch paper as follows:
$$\begin{array}{ l| |c|c|c|c|c } \theta & 0 = 0º & \tfrac{\pi}{6} = 30º & \tfrac{\pi}{4}=45º & \tfrac{\pi}{3}=60º & \tfrac{\pi}{2}=90º \\ \hline\hline \sin\theta &   &   &  &   &    \\ \hline \cos\theta &   &    &    &    &    \\ \hline \tan\theta &    &    &    &   &    \\ \hline \end{array} $$
Then remember ##\sin\theta## starts at zero, fill in the pattern
$$\begin{array}{ l| |c|c|c|c|c } \theta & 0 & \tfrac{\pi}{6} & \tfrac{\pi}{4} & \tfrac{\pi}{3} & \tfrac{\pi}{2} \\ \hline\hline\sin\theta & \tfrac{\sqrt{0}}{2} & \tfrac{\sqrt{1}}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{3}}{2} & \tfrac{\sqrt{4}}{2} \\...

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Likes yucheng, sysprog, Wrichik Basu and 4 others
I think I'll make this compulsory reading for my Maths students!
 
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Likes symbolipoint, benorin and Greg Bernhardt
In our junior classes, we learned it in a similar way:
##\theta~\rightarrow##​
0° = 0​
30° = ##\dfrac{\pi}{6}##​
45° = ##\dfrac{\pi}{4}##​
60° = ##\dfrac{\pi}{3}##​
90° = ##\dfrac{\pi}{2}##​
##\sin \theta##​
##\sqrt{\dfrac{0}{4}}##​
##\sqrt{\dfrac{1}{4}}##​
##\sqrt{\dfrac{2}{4}}##​
##\sqrt{\dfrac{3}{4}}##​
##\sqrt{\dfrac{4}{4}}##​
##\cos \theta##​
##\sqrt{\dfrac{4}{4}}##​
##\sqrt{\dfrac{3}{4}}##​
##\sqrt{\dfrac{2}{4}}##​
##\sqrt{\dfrac{1}{4}}##​
##\sqrt{\dfrac{0}{4}}##​
##\tan \theta##​
##\sqrt{\dfrac{0}{4 - 0}}##​
##\sqrt{\dfrac{1}{4 - 1}}##​
##\sqrt{\dfrac{2}{4 - 2}}##​
##\sqrt{\dfrac{3}{4 - 3}}##​
##\sqrt{\dfrac{4}{4 - 4}}##​
 
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"Trick"? The very basics of triangle Geometry and the Pythagorean Theorem, and The UNIT CIRCLE.

Easily enough done, drawing a Unit Circle and judging Sines and Cosines, and whichever other of the functions to derive what you need. Degree measures 30, 45, 60, 0, and 90, and 180 are the easy ones and are commonly used "Reference" angles.
 
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