Maths - Multiple Trig Solutions

Created: November 25, 2020
Updated: August 25, 2024

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[[Further Maths - Trigonometry Values]]A

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 This diagram shows a triangle where $\theta$ is approaching $0$. What is the value of $\cos\theta$ approaching?
\[1\]

PHOTO TRI SHALLOW OPP This diagram shows a triangle where $\theta$ is approaching $0$. What is the value of $\sin\theta$ approaching?
\[0\]

PHOTO TRI HYP 1 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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