Peter's copy of Exact values for sin, cos, tan (0 to 330, degrees)

Error

There was an error loading the page.

Contributors

Peter Johnston

History

○

Peter Johnston 8 years ago

Created this as a copy of Exact values for sin, cos, tan (0 to 330, degrees).

Name	Status	Author	Last Modified
Exact values for sin, cos, tan (acute, degrees)	Ready to use	Ben Brawn	13/07/2017 14:57
Exact values for csc, sec, cot (acute, radians)	Ready to use	Ben Brawn	13/07/2017 14:56
Exact values for csc, sec, cot (acute, degrees)	Ready to use	Ben Brawn	16/02/2023 01:42
Exact values for sin, cos, tan (acute, radians)	Ready to use	Ben Brawn	13/07/2017 14:56
Exact values for sin, cos, tan (0 to 330, degrees)	Ready to use	Ben Brawn	16/02/2023 01:43
Exact values for sin, cos, tan (-1080 to 1080, degrees)	Ready to use	Ben Brawn	16/02/2023 01:47
Exact values for sin, cos, tan (0 to 330, degrees)	draft	Marlon Arcila	31/08/2016 17:41
Exact values for csc, sec, cot (acute, radians)	draft	Marlon Arcila	19/09/2016 16:49
Exact values for sin, cos, tan (acute, radians)	draft	Marlon Arcila	19/10/2016 13:18
Peter's copy of Exact values for sin, cos, tan (0 to 330, degrees)	draft	Peter Johnston	30/03/2017 09:13
Exact values for sin, cos, tan (0 to 2pi, radians)	Ready to use	Ben Brawn	16/02/2023 02:49
Exact values for sin, cos, tan (-6pi to 6pi, radians)	Ready to use	Ben Brawn	17/10/2023 01:13
Terry's copy of Exact values for sin, cos, tan (-1080 to 1080, degrees)	draft	Terry Young	03/04/2019 17:51
Maria's copy of Exact values for sin, cos, tan (-6pi to 6pi, radians)	draft	Maria Aneiros	25/05/2019 08:00
Maria's copy of Exact values for sin, cos, tan (0 to 2pi, radians)	draft	Maria Aneiros	25/05/2019 08:01
Maria's copy of Exact values for sin, cos, tan (acute, degrees)	draft	Maria Aneiros	29/05/2019 07:52
Exact values for sin, cos, tan (-1080 to 1080, degrees)	draft	Xiaodan Leng	11/07/2019 01:05
Exact values for sin, cos, tan (acute, degrees)	draft	Xiaodan Leng	11/07/2019 02:08
Exact values for csc, sec, cot (acute, degrees)	draft	Xiaodan Leng	11/07/2019 02:08
Exact values for csc, sec, cot (acute, radians)	draft	Xiaodan Leng	11/07/2019 02:09
Exact values for sin, cos, tan (acute, radians)	draft	Xiaodan Leng	11/07/2019 02:09
Exact values for sin, cos, tan (0 to 330, degrees)	draft	Xiaodan Leng	11/07/2019 02:10
Exact values for sin, cos, tan (0 to 2pi, radians)	draft	Xiaodan Leng	11/07/2019 02:24
Exact values for sin, cos, tan (-6pi to 6pi, radians)	draft	Xiaodan Leng	11/07/2019 02:25
Leonardo's copy of Exact values for sin, cos, tan (0 to 330, degrees)	draft	Leonardo Juliano	28/08/2019 18:24

There are 55 other versions that do you not have access to.

Question statement

Often we prefer to work with exact values rather than approximations from a calculator. In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example to input the exact value of $\sin(60^\circ)$ , which is $\dfrac{\sqrt{3}}{2}$ , you would input sqrt(3)/2

Give any introductory information the student needs.

			Name	Type	Generated Value

theta

integer

180

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(0,90,120,135,150,180,210,225,240,270,300,315,330)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

180

→ Used by:

"Unnamed part" - prompt
"Unnamed part" → "Gap 0." - Correct answer
"Unnamed part" → "Gap 1." - Correct answer
"Unnamed part" → "Gap 2." - Custom matrix expression
"Unnamed part" → "Gap 3." - Correct answer

Parts

Gap-fill

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

The exact value of $\sin(\var{theta}^\circ)$ is Gap 0..

The exact value of $\cos(\var{theta}^\circ)$ is Gap 1..

Is $\tan(\var{theta}^\circ)$ defined? Gap 2.

The exact value of $\tan(\var{theta}^\circ)$ is Gap 3..

Advice

By drawing the following triangles we can determine the exact values of $\sin$ , $\cos$ and $\tan$ (and their reciprocals $\csc$ , $\sec$ , $\cot$ ) for the angles $30^\circ$ , $45^\circ$ and $60^\circ$ .

Alternatively, one can memorise the following table:

	$30^\circ$	$45^\circ$	$60^\circ$

$\sin$	$\dfrac{1}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{\sqrt{3}}{2}$

$\cos$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{2}$

$\tan$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$

That combined with the unit circle definitions:

$\cos\theta$ is the $x$ coordinate of the point on the unit circle $\theta$ degrees anti-clockwise from the positive $x$ -axis
$\sin\theta$ is the $y$ coordinate of the point on the unit circle $\theta$ degrees anti-clockwise from the positive $x$ -axis
$\tan\theta$ is the slope of the line through the origin and the point on the unit circle $\theta$ degrees anti-clockwise from the positive $x$ -axis, or equivalently, $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

and some understanding of congruent triangles:

allows us to work out $\sin$ , $\cos$ and $\tan$ for certain angles regardless of what quadrant the point is in. Because whatever angle we are asked about, we can always use the triangle in the first quadrant to determine the side lengths and then consider the signs of the coordinates separately.

For example, to determine $\sin(210^\circ)$ , $\cos(210^\circ)$ and $\tan(210^\circ)$ we first draw the following:

From this diagram, we can see that $\cos(210^\circ)=-\cos(30^\circ)$ , and $\sin(210^\circ)=-\sin(30^\circ)$ since the triangles are congruent and we are in the 3rd quadrant where both the $x$ and $y$ values (and hence the $\cos$ and $\sin$ values) are negative.

But given we know these exact values, we can conclude

$\cos(210^\circ)=-\cos(30^\circ)=-\dfrac{\sqrt{3}}{2},$

$\sin(210^\circ)=-\sin(30^\circ)=-\dfrac{1}{2},$ and finally

$\tan(210^\circ)=\dfrac{\sin(210^\circ)}{\cos(210^\circ)}=\dfrac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\dfrac{1}{\sqrt{3}}.$

Give a worked solution to the whole question.

URL

Text alternative
URL

Text alternative

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Numbas Error: Part a: Error in part Part a custom script Part a: Part is not defined

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is not used in any exams.

Loading...

Error

Peter's copy of Exact values for sin, cos, tan (0 to 330, degrees)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Peter Johnston 8 years ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Peter's copy of Exact values for sin, cos, tan (0 to 330, degrees)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Peter Johnston 8 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`