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

how to re arrange the trigonometric identity

Answer 1

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

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.!

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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

This method uses trig addition formulae.

#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^@#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

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

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.

#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:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7