Solve sin(θ) tan(θ) − cos(θ) = 1 for 0 ≤ θ ≤ 360?

how to re arrange the trigonometric identity

3 Answers
Apr 13, 2018

:. theta=60^@, 180^@, 300^@

This method uses trig addition formulae.

Explanation:

sinthetatantheta-costheta=1

We know : tantheta= (sintheta)/(costheta)

So,

=>sintheta*(sintheta)/(costheta)-costheta=1

=>(sin^2theta)/(costheta)-costheta=1

=(sin^2theta- cos^2theta)/(costheta)=1

=>-(cos^2theta-sin^2theta)=costheta

Using addition formulae:

=>-cos2theta=costheta

=>cos2theta+costheta=0

=>2cos((3theta)/2)cos((theta)/2)=0

=>cos((3theta)/2)cos((theta)/2)=0

I'm kind of stuck in the last step. Please could anybody help? Thanks!

Edit:

:.cos(3/2theta)=0 or cos(1/2theta)=0

Start with cos(3/2theta)=0

3/2theta=90^@

theta=60^@, 300^@
The second result is from 360^@-theta_1, where #theta_1 is our first answer.

Take cos(1/2theta)=0

1/2theta=90^@

theta=180^@, 180^@ (this is a repeated root).

:. theta=60^@, 180^@, 300^@

Apr 13, 2018

theta=60^@, 180^@, 300^@

Explanation:

There are two main trig identities we will use:

tantheta=sintheta/costheta

sin^2theta+cos^2theta=1


Start:

sintheta tantheta - costheta=1

Using tantheta=sintheta/costheta

sintheta sintheta/costheta - costheta=1

sin^2theta/costheta-costheta=1

sin^2theta/costheta-cos^2theta/costheta=1

(sin^2theta-cos^2theta)/costheta=1

sin^2theta-cos^2theta=costheta

sin^2theta-costheta-cos^2theta=0


Notice how we nearly have a quadratic in costheta. All we need to do is to get rid of that sin^2theta then solve like a normal quadratic.

From sin^2theta+cos^2theta=1 =>sin^2theta=1-cos^2theta


sin^2theta-costheta-cos^2theta=0

Using sin^2theta=1-cos^2theta

(1-cos^2theta)-costheta-cos^2theta=0

1-costheta-2cos^2theta=0

2cos^2theta+costheta-1=0


This is a quadratic in costheta. From here, we can factorise directly, or make a substitution to make it easier. I will use a substitution to see the quadratic more easily#

Let u=costheta

2u^2+u-1=0

(2u-1)(u+1)=0

u=1/2 or u=-1

Remember u=costheta?

costheta=1/2 or costheta=-1

Take costheta=1/2

We get our first answer from doing cos^-1 on our calculator (or in this case, knowledge of special angles). For the second answer, we do 360^@-theta_1 where theta_1 is the first answer we got. The reason for this can be seen in symmetries in the cosine graph.

enter image source here

costheta=1/2

theta=60^@, 300^@

costheta=-1

theta=180^@, 180^@
This is a repeated root - look how 180 is on the line of symmetry.

:. theta=60^@, 180^@, 300^@

And for reference, the graph of y=sinx tanx -cosx-1 is:

enter image source here

Apr 13, 2018

theta in {60,180,300}.

Explanation:

Here is another way to crack the Problem :

sinthetatantheta-costheta=1.

:. sinthetatantheta=1+costheta=2cos^2(theta/2).

:. 2sin(theta/2)cos(theta/2)tantheta=2cos^2(theta/2)...(ast).

If, cos(theta/2)=0, then, since 0 le theta/2 le 180,

theta/2=90 rArr theta=180.

Now, if cos(theta/2)!=0, then dividing (ast) by 2cos^2(theta/2)!=0,

tan(theta/2)tantheta=1, or,

tan(theta/2)*(2tan(theta/2))/(1-tan^2(theta/2)}=1, i.e.,

2tan^2(theta/2)=1-tan^2(theta/2).

:. 3tan^2(theta/2)=1 rArr tan(theta/2)=+-1/sqrt3.

Selecting theta/2" from "[0,180], theta/2=30, 150.

rArr theta=60, 300.

Altogether, theta in {60,180,300}.

Enjoy Maths.!