Maths - Multiple Trig Solutions

Pearson Edexcel Mathematics 2022

Created: November 25, 2020 | Updated: August 25, 2024 | Read markdown | About these notes | View in context | Study these flashcards |

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Approach to multiple solutions

What’s the easiest way to answer a question about multiple trigonometry solutions?

multiple-solutions-draw-graph

Draw an accurate graph.

Periods of the trig functions

After how many degrees does $\sin(\theta)$ repeat?

sin-period-degrees
\[360^{\circ}\]

After how many degrees does $\cos(\theta)$ repeat?

cos-period-degrees
\[360^{\circ}\]

After how many degrees does $\tan(\theta)$ repeat?

tan-period-degrees
\[180^{\circ}\]

Sketching the tan graph

When drawing a $\tan$ graph, what should you sketch first?

tan-graph-sketch-asymptotes-first

The asymptotes.

Where are the asymptotes on a $\tan$ graph, in degrees?

tan-graph-asymptote-positions

Start at $90^\circ$ and then repeat every $180^{\circ}$.

Values at the origin

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

sin-at-origin
\[0\]

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

cos-at-origin
\[1\]

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

tan-at-origin
\[0\]

Relating tan to sin and cos

How could you write $\tan\theta$ in terms of $\sin$ and $\cos$?

tan-in-terms-of-sin-cos
\[\frac{\sin\theta}{\cos\theta}\]

If $\cos 90^{\circ}$ is zero, why is $\tan 90^{\circ}$ undefined?

tan-90-undefined-reason

Because $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and you can’t divide by $0$.

Small-angle intuition from triangles

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

cos-approaching-zero-angle
\[1\]

This diagram shows a triangle where $\theta$ is approaching $0$. What is the value of $\sin\theta$ approaching?

sin-approaching-zero-angle
\[0\]

What proportion of the hypotenuse is the opposite side?

opposite-over-hypotenuse-is-sin
\[\sin\theta\]

Finding all solutions from one

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

other-solutions-from-arcsin
\[180 - x\] \[-180 - x\] \[-360 + x\]

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

other-solutions-from-arccos
\[360 - x\] \[-x\] \[-(360 - x)\]

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

cos-even-other-two-solutions
\[-x\] \[-(360 - x)\]

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

other-solutions-from-arctan
\[x + 180\] \[x - 180\]

Solutions for transformed arguments

How could you find one solution of $\sin(2\theta) = 0.1$?

solution-sin-double-angle
\[\frac{\sin^{-1}(0.1)}{2}\]

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

solution-sin-shifted-argument
\[\sin^{-1}(0.1) - 30^{\circ}\]


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