Solving trigonometric equations using identities

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Simon Thomas 6 years, 2 months ago

Created this as a copy of trig identity solving equation.

Name	Status	Author	Last Modified
Trigonometric Equations 1 - Simple (Degrees)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 4 - Intermediate (Radians)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 6 - Complex (Radians)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 3 - Intermediate (Degrees)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 5 - Complex (Degrees)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 2 - Simple (Radians)	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 7 - Using Identities	Ready to use	Katie Lester	01/06/2016 09:47
Trigonometric Equations 2 - Simple (Radians multiple angles0)	Ready to use	steve kilgallon	23/01/2022 10:45
FY001 EXAM 1 Trigonometric Equations 2 - Simple (Radians)	draft	steve kilgallon	30/08/2016 15:13
steve's copy of FY001 EXAM 1 Trigonometric Equations 2 - Simple (Radians)	draft	steve kilgallon	30/08/2016 15:21
Ida Friestad's copy of Trigonometric Equations 7 - Using Identities	draft	Ida Friestad Pedersen	08/11/2016 14:22
Solving trigonometric equations using identities	draft	Simon Thomas	14/02/2019 10:46
Simon's copy of Solving trigonometric equations using identities	draft	Simon Thomas	13/05/2019 13:42
Exam05 - Ttrigonometric Equations	draft	Michael Proudman	07/12/2021 09:54
nuExam05 - Ttrigonometric Equations	draft	Michael Proudman	07/12/2021 09:54
Trigonometric Equations 1 - Simple (Degrees)	draft	Xiaodan Leng	11/07/2019 06:30
Trigonometric Equations 2 - Simple (Radians)	draft	Xiaodan Leng	11/07/2019 06:31
Trigonometric Equations 3 - Intermediate (Degrees)	draft	Xiaodan Leng	11/07/2019 06:31
Trigonometric Equations 4 - Intermediate (Radians)	draft	Xiaodan Leng	11/07/2019 06:32
Trigonometric Equations 5 - Complex (Degrees)	draft	Xiaodan Leng	11/07/2019 06:32
Trigonometric Equations 6 - Complex (Radians)	draft	Xiaodan Leng	11/07/2019 06:33
Trigonometric Equations 7 - Using Identities	draft	Xiaodan Leng	11/07/2019 06:33
T2	Ready to use	Shaheen Charlwood	16/05/2022 14:22
Trigonometric Equations - Simple (Radians)	Ready to use	Shaheen Charlwood	27/10/2022 18:52
Trigonometric Equations - Core (Degrees)	Ready to use	Shaheen Charlwood	27/10/2022 18:52
Trigonometric Equations - Complex (Radians)	Ready to use	Shaheen Charlwood	07/09/2023 15:03

There are 119 other versions that do you not have access to.

Question statement

Solve the following equations in radians, for $\theta$ in the range $0\leq \theta\leq2\pi$ .

The square root of $a$ is written as sqrt( $a$ ). $\theta$ is written as theta.

You may want to use the following trigonometric identities.

$\tan(x)=\frac{\sin(x)}{\cos(x)}$

$\sin^2(x)+\cos^2(x)=1$

$\cos(2x)=\cos^2(x)-\sin^2(x)=2\cos^2(x)-1=1-2\sin^2(x)$

$\sin(2x)=2\sin(x)\cos(x)$

For angles, insert values in increasing order. DO NOT write your answer in an exact form with $\pi$ but rather in decimal form to up 3 decimal places.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
s11	number	1.2309594173
s12	number	4.3725520709

			Name	Type	Generated Value

b	integer	4
d	integer	1
c	integer	2
positivex	number	0.5930703308
negativex	number	-0.8430703308
theta1	number	0.9359294557
theta2	number	2.5737632807
theta3	number	5.3472558515
theta4	number	3.7094220265

			Name	Type	Generated Value

x	integer	5
tanx	integer	-1
Rangenegative	integer	-90
Rangepositive	integer	90
Answerforx	number	-45
Multipliedrangenegative	integer	-450
Multipliedrangepositive	integer	450
Answer	number	-9

			Name	Type	Generated Value

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random(2..9)

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Generated value: `integer`

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$\sin\theta-\sqrt{\var{a}}\cos\theta=0$

$\tan\theta=$ Gap 0.

$\theta=$ Gap 1. or Gap 2.

Advice

a) Rearrange the equation to give $\sin\theta=\sqrt{\var{a}}\cos\theta$

Divide both sides by $\cos\theta$ to give $\tan\theta=\sqrt{\var{a}}$

Take $\tan^{-1}\sqrt{\var{a}}$ to give $\theta=\var{precround(s11,3)}$ . The second solution in the range $0\leq \theta\leq2\pi$ is given by $\theta=\var{precround(s11,3)}+\pi = \var{precround(s12,3)}$

b) Rearrange the equation to give $\var{b}\cos^2\theta+\var{d}\cos\theta-\var{c}=0$

This is a quadratic in $cos\theta$ which we can solve using the quadratic formula to give $cos\theta =\frac{-\var{d}\pm\sqrt{\var{d}^2+4\times{\var{b}\times\var{c}}}}{2\times\var{b}} = \var{precround(positivex,3)}$ or $\var{precround(negativex,3)}$

Take $\cos^{-1}\var{precround(positivex,3)}=\var{precround(theta1,3)}$ and $\cos^{-1}\var{precround(negativex,3)}=\var{precround(theta2,3)}$ .

Now recall that the periodicity of cos means that if $\theta$ is a solution then so is $\ 2\pi-\theta$ . Hence we obtain 4 solutions in the range $0\leq \theta\leq2\pi:$

$\theta = \var{precround(theta1,3)}$ or $2\pi-\var{precround(theta1,3)}=\var{precround(theta3,3)}$ or $\var{precround(theta2,3)}$ or $2\pi-\var{precround(theta2,3)}=\var{precround(theta4,3)}$

c) Use the identity $\cos^2\theta=1-\sin^2\theta$ and substituting this in place of $\cos^2\theta$ gives $1-\sin^2\theta-\var{a}\sin^2\theta=1$

Hence $\var{a+1}\sin^2\theta=0$

So $\sin^2\theta=0$ and in turn $\sin\theta=0$ , which has 3 solutions $\theta = 0, \pi, 2\pi$ in the range $0\leq \theta\leq2\pi$

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This question is used in the following exam:

Using trig identities by Simon Thomas in Trigonometry.

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Solving trigonometric equations using identities

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Simon Thomas 6 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Solving trigonometric equations using identities

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`