# Further Maths - Trigonometry Values

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/trigonometry-values/ · Updated: 2020-11-09 · Tags: further-maths, trigonometry, school, a-level

### 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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