Pythagorean Identity: find sin given cos

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Ben Brawn 7 years, 8 months ago

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Created this as a copy of Pythagorean Identity: find cos given sin.

Name	Status	Author	Last Modified
Pythagoras' Theorem: find hypotenuse (using surds)	Ready to use	Ben Brawn	16/02/2023 00:36
Pythagoras' Theorem: find other side (using surds)	Ready to use	Ben Brawn	08/11/2021 00:51
Pythagorean Identity: find cos given sin	Ready to use	Ben Brawn	11/10/2023 23:07
Pythagorean Identity: find sin given cos	Ready to use	Ben Brawn	11/10/2023 23:06
Pythagoras' Theorem: find hypotenuse (using surds)	draft	Paul Howes	13/11/2016 23:46
Pythagoras' Theorem: find other side (using surds)	draft	Paul Howes	14/11/2016 17:48
Terry's copy of Pythagorean Identity: find sin given cos	draft	Terry Young	03/04/2019 17:51
Terry's copy of Pythagorean Identity: find cos given sin	draft	Terry Young	03/04/2019 17:52
Terry's copy of Pythagoras' Theorem: find hypotenuse (using surds)	draft	Terry Young	03/04/2019 17:52
Terry's copy of Pythagoras' Theorem: find other side (using surds)	draft	Terry Young	03/04/2019 17:53
Pythagorean Identity: find sin given cos	draft	Xiaodan Leng	11/07/2019 02:07
Pythagorean Identity: find cos given sin	draft	Xiaodan Leng	11/07/2019 02:07

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Question statement

You are told that $\cos\theta=\simplify{{a}/sqrt({cc})}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-3
c	string	$\simplify{sqrt({cc})\}$
aa	number	9
cc	integer	359
diff	number	0.9749303621

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Name

Data type

Value

xxxxxxxxxx
 
random(-11..11 except 0)

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Description

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

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Generated value: `integer`

-3

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

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

If $\theta$ is in the second quadrant, then the exact value of $\sin\theta$ is Unnamed gap.

Note: In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example, if your answer is $\frac{\sqrt{5}}{\sqrt{17}}$ , then enter sqrt(5)/sqrt(17)

Advice

The equation of the unit circle is

$x^2+y^2=1$

This equation is a consequence of Pythagoras' Theorem, $a^2+b^2=c^2$ , on a triangle in the unit circle with hypotenuse 1.

Given that $\cos\theta$ is the $x$ coordinate of a point on the unit circle, and that $\sin\theta$ is the $y$ coordinate of the same point, by substitution we have the following Pythagorean identity

$\cos^2\theta+\sin^2\theta=1$

where $\cos^2\theta$ and $\sin^2\theta$ is the notation used to represent $(\cos\theta)^2$ and $(\sin\theta)^2$ respectively.

Using the value of $\cos\theta$ given in the question we have that $\left(\simplify{{a}/sqrt({cc})}\right)^2+\sin^2\theta=1$ . We solve for $\sin\theta$ :

$\simplify{{aa}/{cc}}+\sin^2\theta$	$=$	$1$
$\sin^2\theta$	$=$	$1-\simplify{{aa}/{cc}}$
$\sin^2\theta$	$=$	$\simplify[fractionNumbers]{{diff}}$
$\sin\theta$	$=$	$\pm\simplify[fractionNumbers]{sqrt({diff})}$

Because we are told that $\cos\theta$ is positive, we know that $\theta$ is in the first or fourth quadrant (since $\cos\theta$ is the $x$ coordinate). If $\theta$ is in the first quadrant, then (since $\sin\theta$ is the $y$ coordinate) $\sin\theta$ is positive, that is, $\sin\theta=\simplify[fractionNumbers]{sqrt({diff})}$ . However, if $\theta$ is in the fourth quadrant, then (since $\sin\theta$ is the $y$ coordinate) $\sin\theta$ is negative, that is, $\sin\theta=-\simplify[fractionNumbers]{sqrt({diff})}$ . negative, we know that $\theta$ is in the second or third quadrant (since $\cos\theta$ is the $x$ coordinate). If $\theta$ is in the second quadrant, then (since $\sin\theta$ is the $y$ coordinate) $\sin\theta$ is positive, that is, $\sin\theta=\simplify[fractionNumbers]{sqrt({diff})}$ . However, if $\theta$ is in the third quadrant, then (since $\sin\theta$ is the $y$ coordinate) $\sin\theta$ is negative, that is, $\sin\theta=-\simplify[fractionNumbers]{sqrt({diff})}$ .

Give a worked solution to the whole question.

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This question is used in the following exams:

Test Exam by Erna Engelbrecht in Erna's workspace.
test 1 by Vijay Teeluck in Vijay's workspace.
Formative Test - Trigonometry by Andy Welch in Andy's workspace.
first exam by Sara Santos in Sara's workspace.
Trigonometry by Chris Templet in Chris's workspace.
Diagnostic Test by Matt Falk in Matt's workspace.
Trigonometry Identities: The Pythagorean Identity by Ben Brawn in Trigonometry.
EGD105_PST1 by Matt Falk in Matt's workspace.

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Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 8 years, 9 months ago

Ben Brawn 8 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Pythagorean Identity: find sin given cos

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 8 months ago

Ben Brawn 7 years, 8 months ago

Ben Brawn 8 years, 9 months ago

Ben Brawn 8 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`