Further Maths - Trigonometry Values

Created: November 09, 2020

Degrees

What is $\sin 0^{\circ}$?
\[0\]

What is $\sin 30^{\circ}$?
\[\frac{1}{2}\]

What is $\sin 45^{\circ}$?
\[\frac{\sqrt{2}}{2}\]

What is $\sin 60^{\circ}$?
\[\frac{\sqrt{3}}{2}\]

What is $\sin 90^{\circ}$?
\[1\]

What is $\cos 0^{\circ}$?
\[1\]

What is $\cos 30^{\circ}$?
\[\frac{\sqrt{3}}{2}\]

What is $\cos 45^{\circ}$?
\[\frac{\sqrt{2}}{2}\]

What is $\cos 60^{\circ}$?
\[\frac{1}{2}\]

What is $\cos 90^{\circ}$?
\[0\]

What is $\tan 0^{\circ}$?
\[0\]

What is $\tan 30^{\circ}$?
\[\frac{\sqrt{3}}{3}\]

What is $\tan 45^{\circ}$?
\[1\]

What is $\tan 60^{\circ}$?
\[\sqrt{3}\]

What is $\tan 90^{\circ}$?
\[\text{undefined}\]

For what values are $\sin$ and $\cos$ the same?
\[45^{\circ}\]

Which $\sin$ and $\cos$ values swap over?
\[30^{\circ}, 60^{\circ}\]

#

What is special about the value under the square root for sine $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$?

It goes $1$, $2$, $3$.

#

What is special about the value under the square root for cosine $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$?

It goes $3$, $2$, $1$.

Radians

What is $\sin 0$?
\[0\]

What is $\sin \frac{\pi}{6}$?
\[\frac{1}{2}\]

What is $\sin \frac{\pi}{4}$?
\[\frac{\sqrt{2}}{2}\]

What is $\sin \frac{\pi}{3}$?
\[\frac{\sqrt{3}}{2}\]

What is $\sin \frac{\pi}{2}$?
\[1\]

What is $\cos 0$?
\[1\]

What is $\cos \frac{\pi}{6}$?
\[\frac{\sqrt{3}}{2}\]

What is $\cos \frac{\pi}{4}$?
\[\frac{\sqrt{2}}{2}\]

What is $\cos \frac{\pi}{3}$?
\[\frac{1}{2}\]

What is $\cos \frac{\pi}{2}$?
\[0\]

What is $\tan 0$?
\[0\]

What is $\tan \frac{\pi}{6}$?
\[\frac{\sqrt{3}}{3}\]

What is $\tan \frac{\pi}{4}$?
\[1\]

What is $\tan \frac{\pi}{3}$?
\[\sqrt{3}\]

What is $\tan \frac{\pi}{2}$?
\[\text{undefined}\]

What is special about the value under the square root for sine $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$?

It goes $1$, $2$, $3$.

What is special about the value under the square root for cosine $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$?

It goes $3$, $2$, $1$.

After how many radians does $\sin$ repeat?
\[2\pi\]

What’s another way of stating that $\sin$ repeats every $2\pi$ radians?
\[\sin(\theta) = \sin(\theta + 2\pi)\]

After how many radians does $\cos$ repeat?
\[2\pi\]

What’s another way of stating that $\cos$ repeats every $2\pi$ radians?
\[\cos(\theta) = \cos(\theta + 2\pi)\]

After how many radians does $\tan$ repeat?
\[\pi\]

What’s another way of stating that $\tan$ repeats every $\pi$ radians?
\[\tan(\theta) = \tan(\theta + \pi)\]

Because $\sin$ is the same going up as it comes down, what relation in radians can you write?
\[\sin(x) = \sin(\pi - x)\]

General Rules

What’s another way of writing $\sin(-\theta)$?
\[-\sin(\theta)\]

What’s another way of writing $-\sin(\theta)$?
\[\sin(-\theta)\]

What’s another way of writing $\cos(-\theta)$?
\[\cos(\theta)\]

2021-11-15

What’s

\[\sin\left(\frac{\pi}{12}\right)\]

?
\[\frac{\sqrt{6} - \sqrt{2}}{4}\]

What’s

\[\cos\left(\frac{\pi}{12}\right)\]

?
\[\frac{1 + \sqrt{3}}{2\sqrt{2}}\]

What’s

\[\tan\left(\frac{\pi}{12}\right)\]

?
\[\frac{\sqrt{3} - 1}{\sqrt{3} + 1}\]


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