1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
zimovet [89]
3 years ago
7

Square root of 2tanxcosx-tanx=0

Mathematics
1 answer:
kobusy [5.1K]3 years ago
3 0
If you're using the app, try seeing this answer through your browser:  brainly.com/question/3242555

——————————

Solve the trigonometric equation:

\mathsf{\sqrt{2\,tan\,x\,cos\,x}-tan\,x=0}\\\\ \mathsf{\sqrt{2\cdot \dfrac{sin\,x}{cos\,x}\cdot cos\,x}-tan\,x=0}\\\\\\ \mathsf{\sqrt{2\cdot sin\,x}=tan\,x\qquad\quad(i)}


Restriction for the solution:

\left\{ \begin{array}{l} \mathsf{sin\,x\ge 0}\\\\ \mathsf{tan\,x\ge 0} \end{array} \right.


Square both sides of  (i):

\mathsf{(\sqrt{2\cdot sin\,x})^2=(tan\,x)^2}\\\\ \mathsf{2\cdot sin\,x=tan^2\,x}\\\\ \mathsf{2\cdot sin\,x-tan^2\,x=0}\\\\ \mathsf{\dfrac{2\cdot sin\,x\cdot cos^2\,x}{cos^2\,x}-\dfrac{sin^2\,x}{cos^2\,x}=0}\\\\\\ \mathsf{\dfrac{sin\,x}{cos^2\,x}\cdot \left(2\,cos^2\,x-sin\,x \right )=0\qquad\quad but~~cos^2 x=1-sin^2 x}

\mathsf{\dfrac{sin\,x}{cos^2\,x}\cdot \left[2\cdot (1-sin^2\,x)-sin\,x \right]=0}\\\\\\ \mathsf{\dfrac{sin\,x}{cos^2\,x}\cdot \left[2-2\,sin^2\,x-sin\,x \right]=0}\\\\\\ \mathsf{-\,\dfrac{sin\,x}{cos^2\,x}\cdot \left[2\,sin^2\,x+sin\,x-2 \right]=0}\\\\\\ \mathsf{sin\,x\cdot \left[2\,sin^2\,x+sin\,x-2 \right]=0}


Let

\mathsf{sin\,x=t\qquad (0\le t


So the equation becomes

\mathsf{t\cdot (2t^2+t-2)=0\qquad\quad (ii)}\\\\ \begin{array}{rcl} \mathsf{t=0}&\textsf{ or }&\mathsf{2t^2+t-2=0} \end{array}


Solving the quadratic equation:

\mathsf{2t^2+t-2=0}\quad\longrightarrow\quad\left\{ \begin{array}{l} \mathsf{a=2}\\ \mathsf{b=1}\\ \mathsf{c=-2} \end{array} \right.


\mathsf{\Delta=b^2-4ac}\\\\ \mathsf{\Delta=1^2-4\cdot 2\cdot (-2)}\\\\ \mathsf{\Delta=1+16}\\\\ \mathsf{\Delta=17}


\mathsf{t=\dfrac{-b\pm\sqrt{\Delta}}{2a}}\\\\\\ \mathsf{t=\dfrac{-1\pm\sqrt{17}}{2\cdot 2}}\\\\\\ \mathsf{t=\dfrac{-1\pm\sqrt{17}}{4}}\\\\\\ \begin{array}{rcl} \mathsf{t=\dfrac{-1+\sqrt{17}}{4}}&\textsf{ or }&\mathsf{t=\dfrac{-1-\sqrt{17}}{4}} \end{array}


You can discard the negative value for  t. So the solution for  (ii)  is

\begin{array}{rcl} \mathsf{t=0}&\textsf{ or }&\mathsf{t=\dfrac{\sqrt{17}-1}{4}} \end{array}


Substitute back for  t = sin x.  Remember the restriction for  x:

\begin{array}{rcl} \mathsf{sin\,x=0}&\textsf{ or }&\mathsf{sin\,x=\dfrac{\sqrt{17}-1}{4}}\\\\ \mathsf{x=0+k\cdot 180^\circ}&\textsf{ or }&\mathsf{x=arcsin\bigg(\dfrac{\sqrt{17}-1}{4}\bigg)+k\cdot 360^\circ}\\\\\\ \mathsf{x=k\cdot 180^\circ}&\textsf{ or }&\mathsf{x=51.33^\circ +k\cdot 360^\circ}\quad\longleftarrow\quad\textsf{solution.} \end{array}

where  k  is an integer.


I hope this helps. =)

You might be interested in
What is the slope below
raketka [301]

Answer:

-1

Step-by-step explanation:

We can find the slope by change in y of change in x

change in y

---------------------

change in x

The y changes  -3  ( goes down 3)

the x changes +3 ( to the right 3)

-3

----

3

This simplifies to -1

4 0
3 years ago
Read 2 more answers
Please help quickly, it’s more than one answer
fiasKO [112]

Answer:

It is B, D and F

Step-by-step explanation:

because red 2 is very specific and not putting it back in the deck makes it independent

6 0
3 years ago
Read 2 more answers
Simplify the expression.<br> I NEED HELP PLEASE!!!
tatiyna

Answer:

-27/64p^3  

Tell me if you want a step by step. :)

7 0
2 years ago
What is a real situation that would be represented by <br> a+3
spin [16.1K]
A is just a variable to cover up any number you want. <span>a can be anything you want, such as the money in your bank account.
a+3
a=200
200+3</span>
7 0
3 years ago
The number of people in thousands who own a cell phone is a function of the time in years starting in 1990 where p(t) = 20(1.075
Julli [10]
Try to look it up tbh
4 0
2 years ago
Other questions:
  • (2w^2 x^3 y)^2 (x^4 y^5)
    5·1 answer
  • Explain the different ways it it possible to add 2 rational numbers and get a negative number
    15·1 answer
  • What's the answer to question 10? I don't understand.
    10·1 answer
  • 2.
    13·1 answer
  • A health club charges an initial fee of $100 to join, plus a fee of $33 per month. Write an algebraic equation. How much will yo
    9·2 answers
  • what time is it when the hour hand and the minute hand are both pointing to the same number ? Aiden says it is 6:30 . Camila say
    5·1 answer
  • What numbers multiply to -24 but add up to -2
    14·1 answer
  • Convert 7200 seconds into weeks. Round your answer to the nearest hundredth.
    10·1 answer
  • Which lists all of the x-intercepts of the graphed function?
    14·1 answer
  • HELPPP MEEE PLZZZZZ ASAP
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!