# Maths - Multiple Trig Solutions

> Source: https://ollybritton.com/notes/a-level/maths/topics/multiple-trigonometry-solutions/ · Updated: 2024-08-25 · Tags: maths

See also:
- [Further Maths - Trigonometry Values](https://ollybritton.com/notes/a-level/further-maths/topics/trigonometry-values/)

##### What's the easiest way to answer a question about mutliple trigonometry solutions??
Draw an accurate graph.

##### After how many degrees does $\sin(\theta)$ repeat??
$$
360^{\circ}
$$

##### After how many degrees does $\cos(\theta)$ repeat??
$$
360^{\circ}
$$

##### After how many degrees does $\tan(\theta)$ repeat??
$$
180^{\circ}
$$

##### When drawing a $\tan$ graph, what should you sketch first??
The asymptotes.

##### Where are the asymptotes on a $\tan$ graph, in degrees??
Start at $90^\circ$ and then repeat every $180^{\circ}$.

##### What is the value of $\sin$ at the origin??
$$
0
$$

##### What is the value of $\cos$ at the origin??
$$
1
$$

##### What is the value of $\tan$ at the origin??
$$
0
$$

##### How could you write $\tan\theta$ in terms of $\sin$ and $\cos$??
$$
\frac{\sin\theta}{\cos\theta}
$$

##### If $\cos 90^{\circ}$ is zero, why is $\tan 90^{\circ}$ undefined??
Because $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and you can't divide by $0$.
##### ![PHOTO TRI SHALLOW ADJ](paste-69411fd27eb2f60c337d618283edf0ec87329bf5.jpg) This diagram shows a triangle where $\theta$ is approaching $0$. What is the value of $\cos\theta$ approaching??
$$
1
$$

##### ![PHOTO TRI SHALLOW OPP](paste-27e33dd0da0b0a20e76da97fc707a7f6563d3d4c.jpg) This diagram shows a triangle where $\theta$ is approaching $0$. What is the value of $\sin\theta$ approaching??
$$
0
$$

##### ![PHOTO TRI HYP 1](paste-ff7092cfb55feafbeea4141fc40282ff6b070b41.jpg) What proportion of the hypotenuse is the opposite side??
$$
\sin\theta
$$

##### If your calculator gives you $x$ as a solution to $\sin^{-1}$, what are the three other solutions??
$$
180 - x
$$

$$
-180 - x
$$

$$
-360 + x
$$

##### If your calculator gives you $x$ as a solution to $\cos^{-1}$, what are the three other solutions??
$$
360 - x
$$

$$
-x
$$

$$
-(360 - x)
$$

##### $\cos^{-1}$ has solutions $x$ and $360 - x$. Because $\cos\theta = \cos-\theta$, what are the other two solutions??
$$
-x
$$

$$
-(360 - x)
$$

##### If your calculator gives you $x$ as a solution to $\tan^{-1}$, what are the other two solutions??
$$
x + 180
$$

$$
x - 180
$$

##### How could you find one solution of $\sin(2\theta) = 0.1$??
$$
\frac{\sin^{-1}(0.1)}{2}
$$

##### How could you find one solution of $\sin(\theta + 30^{\circ}) = 0.1$??
$$
\sin^{-1}(0.1) - 30^{\circ}
$$

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