// Numbas version: exam_results_page_options {"name": "Trigonometry", "metadata": {"description": "

11 questions that test trig equations including use of identities and those needing stretches and translations. Also some questions related to trig graph shapes and applications.

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Solving tanx in given interval. With random variation and worked solutions.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

Simpler Trig Questions

", "advice": "

WORKED SOLUTIONS

\\begin{align}
tan(x)&=\\var{t} \\\\\\\\
x_1&=tan^{-1}\\var{t}=\\var{t_1_3sf}^c \\\\\\\\
x_2&=\\var{t_1_5sf}+\\pi=\\var{t_2_3sf}^c
\\end{align}

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"can_override": false}, "first_soln_5": {"name": "first_soln_5", "group": "Part 1 sin", "definition": "sigformat(first_soln,5)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "part c tan", "definition": "random(-3 .. 3#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "t_1": {"name": "t_1", "group": "part c tan", "definition": "arctan({t})", "description": "", "templateType": "anything", "can_override": false}, "t_2": {"name": "t_2", "group": "part c tan", "definition": "{t_1}+pi", "description": "", "templateType": "anything", "can_override": false}, "t_1_5sf": {"name": "t_1_5sf", "group": "part c tan", "definition": "sigformat({t_1,5)", "description": "", "templateType": "anything", "can_override": false}, "t_1_3sf": {"name": "t_1_3sf", "group": "part c tan", "definition": "sigformat({t_1},3)", "description": "", "templateType": "anything", "can_override": false}, "t_2_3sf": {"name": "t_2_3sf", "group": 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Solve $tan(x)=\\var{t}$ between $-\\frac{\\pi}{2}\\leq x \\leq \\frac{3\\pi}{2}$

Write your answers below in ascending order and to 3 significant figures.

$x=$[[0]]$^c$ or [[1]]$^c$

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Solving sinx in given interval. With random variation and worked solutions.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

Simpler Trig Questions

", "advice": "

WORKED SOLUTIONS

\\begin{align}
\\var{coeffsin}sin{x} &=\\var{RHS} \\\\\\\\
sin{x} &=\\frac{\\var{RHS}}{\\var{coeffsin}} \\\\\\\\
x &=\\arcsin{\\frac{\\var{RHS}}{\\var{coeffsin}}} \\\\\\\\
x_1 &=\\var{first_soln_3}^o \\text { to 3 sig fig}\\\\\\\\
x_2 &=180-\\var{first_soln_5}=\\var{second_soln_3}^o\\text { to 3 sig fig}
\\end{align}

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Solve $\\var{coeffsin}sin(x)=\\var{RHS}$ between $0 \\leq x<360^o$

Write your answers here with the smallest first and giving your answer to 3 significant figures.

$x=$[[0]]$^o$ or [[1]]$^o$

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Solving cosx in given interval. With random variation and worked solutions.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

Simpler Trig Questions

", "advice": "

WORKED SOLUTIONS

\\begin{align}
\\simplify{{a}cos(x)+{b}}&=\\var{c} \\\\\\\\
cos{x}&=\\frac{\\var{c}-\\var{b}}{\\var{a}} \\\\\\\\
x &=cos^{-1}\\frac{\\simplify{{c}-{b}}}{\\var{a}} \\\\\\\\
x_1 &=\\var{x_1_3sf}^c \\text { to 3 sig fig} \\\\\\\\
x_2 &=2\\pi-\\var{x_1_6sf}=\\var{x_2_3sf}^c\\text { to 3 sig fig} \\\\\\\\
x_3&=\\var{x_1_6sf}+2\\pi=\\var{x_3_3sf}^c\\text { to 3 sig fig}
\\end{align}

Solve $\\simplify{{a}cos(x)+{b}}=\\var{c}$ between $0<x<3 \\pi$

Write your answers below in ascending order and to 3 significant figures.

$x=$ [[0]]$^c$ or [[1]]$^c$ or [[2]]$^c$

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Solve trig equation in a given internval with stretch and worked solutions

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

Trig Equation with a stretch

", "advice": "

\\begin{align}
cos(2x) & =\\var{a} \\\\\\\\
2x & = cos^{-1}\\var{a}=\\var{x1x2_5} \\text{ or } 2\\pi-\\var{x1x2_5} \\\\\\\\
\\text{(To find a second angle you could also use $-\\var{x1x2_5}$ and then add $2\\pi$)} \\\\\\\\
2x & = \\var{x1x2_5} \\text{ or } \\var{x2x2_5} \\\\\\\\

\\text{You can now add $2\\pi$ to both of these angles to find extra angles} \\\\\\\\

2x & = \\var{x1x2_5} \\text{ or } \\var{x2x2_5} \\text{ or } (\\var{x1x2_5}+2\\pi) \\text{ or } (\\var{x2x2_5}+2\\pi) \\\\\\\\
2x & = \\var{x1x2_5} \\text{ or } \\var{x2x2_5} \\text{ or } \\var{x3x2_5} \\text{ or } \\var{x4x2_5} \\\\\\\\
x &= \\var{x1_3}^c \\text{ or }\\var{x2_3}^c \\text{ or }\\var{x3_3}^c \\text{ or }\\var{x4_3}^c \\text{ to 3 sig fig}
\\end{align}

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Solve $cos(2x)=\\var{a}$ between $0 \\leq x \\leq 2\\pi$

Enter your answers below giving the angles in ascending order and to 3 significant figures.

$x=$ [[3]]$^c$ or [[0]]$^c$ or [[1]]$^c$ or [[2]]$^c$

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solve trig equation involving a translation in given internval. with worked solutions

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

Trig equation with a translation

", "advice": "

\\begin{align}

sin \\left(x-\\frac{\\pi}{2} \\right) & =\\var{a} \\\\\\\\
\\left(x-\\frac{\\pi}{2} \\right) & = sin^{-1}(\\var{a})=\\var{x1n} \\\\\\\\
\\text{for $sinx$ you can find a second solution by going $\\pi-x_1$} \\\\\\\\
\\left(x-\\frac{\\pi}{2} \\right) & = \\pi-\\var{x1n}=\\var{x2n} \\\\\\\\

\\end{align}

As the range of solution goes down to $-2\\pi$ you can find two more angles by taking away $2\\pi$ from the first two angles.

\\begin{align}

x_3 &=\\var{x1n}-2\\pi=\\var{x3n} \\\\\\\\
x_4 &=\\var{x2n}-2\\pi=\\var{x4n}

\\end{align}

In ascending order your three solutions so far are

\\begin{align}
\\left(x-\\frac{\\pi}{2} \\right) & =\\var{x3n} \\text{ or } \\var{x4n} \\text{ or } \\var{x1n} \\text{ or } \\var{x2n}
\\end{align}

To find $x$ all you need to do now is add $\\frac{\\pi}{2}$

\\begin{align}
x=\\var{x3} \\text{ or } \\var{x4} \\text{ or } \\var{x1} \\text{ or } \\var{x2}
\\end{align}

To 3 significant figures this gives

\\begin{align}
x=\\var{x3_3}^c \\text{ or } \\var{x4_3}^c \\text{ or } \\var{x1_3}^c \\text{ or } \\var{x2_3}^c
\\end{align}

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Solve $sin \\left(x-\\frac{\\pi}{2} \\right)=\\var{a}$ between $-2\\pi<x<2\\pi$

Give your answers in radians in ascending order and to 3 significant figures.

$x_1$=[[2]]

$x_2$=[[3]]

$x_3$=[[0]]

$x_4$=[[1]]

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solve trig equation involving sin2x=2sinxcosx in a given interval

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "

First replace $sin2x$ with $2sinxcosx$ and rearrange

\\begin{align}

\\var{a}sin2x & =\\simplify{0+{b} cosx} \\\\\\\\
\\var{a}\\times 2sinxcosx & =\\simplify{0+{b} cosx} \\\\\\\\
\\simplify{2{a}}sinxcosx \\simplify{0-{b}cosx} & =0 \\\\\\\\
cosx(\\simplify{2{a}}sinx-\\var{b}) &=0
\\end{align}

This can now be solved to give

\\[ cosx=0 \\text{ or } sinx=\\frac{\\var{b}}{\\simplify{2{a}}} \\]

Taking $cosx=0$ and remembering that for $cosx$ a second angle can be found by talking the first away from $2 \\pi$

\\begin{align}

cosx=0 \\implies x=cos^{-1}(0)=\\frac{\\pi}{2} \\text{ or } \\frac{3 \\pi}{2}

\\end{align}

Next taking $cosx=\\frac{\\var{b}}{\\simplify{2{a}}}$ and remembering that for $sinx$ a second angle can be found by talking the first away from $ \\pi$

\\begin{align}

sinx=\\frac{\\var{b}}{\\simplify{2{a}}} \\implies x=sin^{-1}\\left(\\frac{\\var{b}}{\\simplify{2{a}}}\\right)=\\var{x1} \\text{ or } \\var{x3}

\\end{align}

To 3 significant figures and in radians the answers are therefore

$\\var{x1_3} \\text{ and } \\var{x2_3} \\text{ and } \\var{x3_3} \\text{ and } \\var{x4_3}$

Solve the equation $\\var{a}sin2x=\\simplify{0+{b} cosx}$ in the range $0\\leq x \\leq 2\\pi$

Give your answers below in ascending order and to 3 significant figures.

$x_1=$[[0]]

$x_2=$[[1]]

$x_3=$[[2]]

$x_4=$[[3]]

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The first setp is to replace $sin2x$ with $2sinxcosx$

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solve trig equation that requires use of s^s+c^2=1. with worked solutions.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "

\\begin{align}

\\simplify{{a}sin^2(x)+({a}{c}-{b})cos(x)+({b}{c}-{a})}&=0 \\\\\\\\

\\text{first replace $sin^2(x) \\text{ with } 1-cos^2(x)$} \\\\\\\\

\\simplify{{a}(1-cos^2 ( x) )+({a}{c}-{b})cos(x)+({b}{c}-{a})}&=0 \\\\\\\\

\\text{then move everything to the other side and simplify} \\\\\\\\

0&=\\simplify{{a}cos^2(x)+({b}-{a}{c})cos(x)+(-{b}{c})} \\\\\\\\

\\text{now you can factorise} \\\\\\\\

0&=(\\simplify{{a}cos(x)+{b}})(cos(x)-\\var{c}) \\\\\\\\

cos(x)&=-\\frac{\\var{b}}{\\var{a}} \\text{ or } \\var{c} \\\\\\\\

\\text{there are no solutions to $cos(x)=\\var{c}$ so this leaves only}\\\\\\\\

x_1 &=cos^{-1}-\\frac{\\var{b}}{\\var{a}}= \\var{x1_3} \\\\\\\\

x_2 &=2\\pi-\\var{x1_3}=\\var{x2_3}

\\end{align}

Solve

$\\simplify{{a}sin^2(x)+({a}{c}-{b})cos(x)+({b}{c}-{a})}=0$ betwwen $0\\leq x\\leq 360^o$

Give your answers below in ascending order and to 3 significant figures

[[0]]$^o$ or [[1]]$^o$

trig equation that uses cos2x=c^2-s^s

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "

Solve $\\simplify{2{a}{d}+2{b}{c}}sinx=\\simplify{{a}{c}cos2x-{a}{c}-2{b}{d}}$

The first step is to replace $cos2x$ with $1-2sin^2x$ as below

\\[ \\simplify{2{a}{d}+2{b}{c}}sinx=\\simplify{{a}{c}(1-2sin^2x)-{a}{c}-2{b}{d}} \\]

If you expand the brackets and rearrange you will get the following.

\\[ \\simplify{ {a}{c}sin^2x +{a}{d}sinx+{b}{c}sinx+{b}{d}} \\]

which factorises to

\\[ (\\simplify{{a}sinx+{b}})(\\var{c}sinx+\\var{d})=0 \\]

You can now take each bracket separately

\\[ sinx=\\frac{\\var{absb}}{\\var{a}} \\text{ or } sinx=\\frac{-\\var{d}}{\\var{c}} \\]

Taking each separately

\\[ x=sin^{-1}\\frac{\\var{absb}}{\\var{a}} = \\var{x1} \\text{ or } \\pi-\\var{x1} \\]

\\[ x=sin^{-1}\\frac{-\\var{d}}{\\var{c}} = \\var{x3} \\text{ or } \\pi-\\var{x3} \\]

To 3 significant figures and in ascending order this gives

\\[ x=\\var{x1_3} \\text{ or }\\var{x2_3} \\text{ or }\\var{x3_3} \\text{ or }\\var{x4_3} \\]

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Solve $\\simplify{2{a}{d}+2{b}{c}}sinx=\\simplify{{a}{c}cos2x-{a}{c}-2{b}{d}}$ between $0<x<2\\pi$

Give your answers to 3 significant figures and in ascending order.

[[0]]

[[1]]

[[2]]

[[3]]

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The first step is to replace $cos2x$ with $1-2sin^2x$

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given plot of a graph. find amplitude, period and phase shift

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What are the period, amplitude and phase shift (from $y=sinx^o$) of the graph shown below?

If needed you can zoom and pan the graph to find suitable points.

{geogebra_applet('https://www.geogebra.org/m/uetnepha',defs)}

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The amplitude is the vertical height of the graph from the central line to the highest point. In this case that equals $\\var{a}$.

The period is the time for one wave. This is the distance from a point on the curve to the equivalent next time it occurs along the graph. The easiest way to do this is to look at points along the axis. For this graph one point is at $(\\simplify{0-{c}},0)$. Moving to the right, the next equivalent point is at $(\\simplify{{period}-{c}},0)$. The $x$ distance between these two points and hence the period is $\\var{period}$

The phase shift is how far to the left or right the point $(0,0)$ from the graph $y=sinx$ has moved. In this case the point $(0,0)$ has moved $\\var{abs_c}$ units to the $\\var{L_R}$ so the phase shift is $\\simplify{0-{c}}$.

Give the exact value for all answers (you should get an integer)
Do not write units in the answer
For the phase shift you need to write for example $-20$ or $10$. Do not use the words left of right.

Amplitude [[0]]

period [[2]]

phase shift [[1]]

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given trig equation. find period, amplitude and phase shift

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Amplitude: The number $\\var{a}$ next to the $cos$ tells you the amplitude. This is because it is the amount the original $y=cosx$ graph has been stretched in the y direction. You do not include the $\\var{e}$ in the amplitude as this only translates the graph in the y direction. It doesn't change the height of the graph from its middle point to its heighest or lowest value.

Period: The period of the function is the time for one wave. For the graph $y=cosx$ the period is $2 \\pi$. In the function above it has been stretched by a factor of $\\frac{\\var{c}}{\\var{b}}$ in the $x$ direction. This means that the period for function in the question is $2\\pi \\cdot \\frac{\\var{c}}{\\var{b}}$ which equals $\\var{period3sf}$

Phase shift: The phase shift is how far to the left or right the graph has moved from $y=cosx$. As the $x$ values have now been replaced by the values from $\\simplify{x+{d}}$ the graph moves to the $\\var{L_R}$ by $\\var{absd}$ units and so the phase shift is $\\simplify{0-{d}}$.

amplitude

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vertical translation

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negative of the phase shift

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What are the amplitude, period and phase shift (from $y=cosx^c$) for the following equation

\\[ y=\\var{a}cos \\left (\\frac{\\var{b}}{\\var{c}} (\\simplify{x^c+{d}}) \\right)+\\var{e} \\]

amplitude [[0]]

period [[2]]$^c$ Give your answer to 3 significant figures

phase shift [[1]]$^c$ Give your answer to 1 decimal place

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given trig function applied to tides. students need to find max depth, time of low tide and times between which a boat of given depth can use the port.

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Part a

Low tide will be at the minimum value on the graph. This will be the central line $\\var{a}$ minus the amplitude of $\\var{b}$ which gives $\\simplify{{a}-{b}}$

Part b

The graph is shown below.

It can be found by applying the following transformations to $y=cosx$

stretch in the $y$ direction by scale factor of $\\var{b}$
translate up by $\\var{a}$
stretch scale factor of $\\frac{6}{\\pi}$ in the $x$ direction

{geogebra_applet('https://www.geogebra.org/m/fzfgusaj',defs)}

From the graph you can see that the first high tide after midnight is at $t=12$ which is 12pm.

You could also have solved the equation equal to the maximum value as shown below

\\[\\simplify{{a}+{b}}=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]

Part c

To do this you need to solve as follows

\\[\\var{c}=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]

Rearranging this gives

\\begin{align}
cos \\left( \\frac{\\pi t}{6} \\right) & =\\left( \\frac{\\var{c}-\\var{a}}{\\var{b}} \\right) \\\\\\\\
\\left( \\frac{\\pi t}{6}\\right) & = cos^{-1}\\left( \\frac{\\var{c}-\\var{a}}{\\var{b}} \\right)=\\var{pt_6} \\text{ or }2\\pi-\\var{pt_6} \\\\\\\\
t & = \\var{t1} \\text{ or } \\var{t2}
\\end{align}

For $\\var{t1}$, you can multiply the $\\simplify{{t1}-{t1_hour}}$ by $60$ to get $\\var{t1_min}$ minutes
For $\\var{t2}$, you can multiply the $\\simplify{{t2}-{t2_hour}}$ by $60$ to get $\\var{t2_min}$ minutes

So the boat cannot use the port between $\\var{t1_min} \\text{ minutes past } \\var{t1_hour}$ and $\\var{t2_min} \\text{ minutes past } \\var{t2_hour}$

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The depth of water ($x$ m) in a port is given by the equation below where $t$ represents the time in hours after midnight

\\[x=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]

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What is the depth of water at the low tide? (you do not need to include the units in your answer)

[[0]]

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What is the time of the first high tide after $t=0$?

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What are the first two times after midnight and between which a boat with a draught of $\\var{c}$m is not able to use the port?

Fill in the gaps below giving each number as an integer

The boat is not able to use the port between [[2]] minutes past [[0]] and [[3]] minutes past [[1]]

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Please check you have answered all parts.

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