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
Vinil7 [7]
3 years ago
10

sin(θ) = − 9/41 where 3π/2 < θ < 2π how to find sin(2θ) sin(θ/2) cos(2θ) cos(θ/2) tan(2θ) tan(θ/2)

Mathematics
1 answer:
jenyasd209 [6]3 years ago
5 0
We know the angle <span>3π/2 < θ < 2π  that simply means that </span>θ is the IV quadrant, where the sine is negative, and the cosine is positive,

now, we know the sine is -9/41.... which is the negative?  well, the hypotenuse, the denominator, is never negative because is just a radius unit, so the negative has to be the numerator of 9.

\bf sin(\theta )=\cfrac{\stackrel{opposite}{-9}}{\stackrel{hypotenuse}{41}}\impliedby \textit{now let's find the \underline{adjacent side}}&#10;\\\\\\&#10;\textit{using the pythagorean theorem}\\\\&#10;c^2=a^2+b^2\implies \pm\sqrt{c^2-b^2}=a\qquad &#10;\begin{cases}&#10;c=hypotenuse\\&#10;a=adjacent\\&#10;b=opposite\\&#10;\end{cases}&#10;\\\\\\&#10;\pm\sqrt{41^2-(-9)^2}=a\implies \pm\sqrt{1600}=a&#10;\\\\\\&#10;\pm 40=a\implies \stackrel{IV~quadrant}{+40=a}

now, we know what the sine and cosine are for θ, let's use that then,

\bf sin(2\theta)=2sin(\theta)cos(\theta)\\\\\\&#10;sin(2\theta)=2\left( \frac{-9}{41} \right)\left( \frac{40}{41} \right)\implies sin(2\theta)=-\cfrac{720}{1681}\\\\&#10;-------------------------------\\\\&#10;cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1+cos({{ \theta}})}{2}}&#10;\\\\\\&#10;cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1+\frac{40}{41}}{2}}\implies cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{\quad \frac{81}{41}\quad }{2}}&#10;\\\\\\&#10;

\bf cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{81}{82}}\implies cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \cfrac{\sqrt{81}}{\sqrt{82}}\implies cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \cfrac{9}{\sqrt{82}}&#10;\\\\\\&#10;\textit{and rationalizing the denominator we'd get}\qquad cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \cfrac{9\sqrt{82}}{82}

\bf -------------------------------\\\\&#10;tan\left(\cfrac{{{ \theta}}}{2}\right)=&#10;\begin{cases}&#10;\pm \sqrt{\cfrac{1-cos({{ \theta}})}{1+cos({{ \theta}})}}&#10;\\ \quad \\&#10;\cfrac{sin({{ \theta}})}{1+cos({{ \theta}})}&#10;\\ \quad \\&#10;\boxed{\cfrac{1-cos({{ \theta}})}{sin({{ \theta}})}}&#10;\end{cases}

\bf tan\left(\cfrac{{{ \theta}}}{2}\right)=\cfrac{1-\frac{40}{41}}{\frac{-9}{41}}\implies tan\left(\cfrac{{{ \theta}}}{2}\right)=\cfrac{\frac{1}{41}}{\frac{-9}{41}}\implies tan\left(\cfrac{{{ \theta}}}{2}\right)=\cfrac{1}{\underline{41}}\cdot \cfrac{\underline{41}}{-9}&#10;\\\\\\&#10;tan\left(\cfrac{{{ \theta}}}{2}\right)=-\cfrac{1}{9}
You might be interested in
Please help. Solve the triangle. Round ans to the nearest tenth.
White raven [17]

9514 1404 393

Answer:

  • C = 21°
  • a = 13.3
  • c = 5.4

Step-by-step explanation:

The third angle can be found from the sum of angles in a triangle.

  A + B + C = 180°

  C = 180° -62° -97°

  C = 21°

__

The remaining side lengths can be found using the Law of Sines.

  a/sin(A) = b/sin(B)

  a = sin(62°)(15/sin(97°)) ≈ 13.34

Similarly, ...

  c/sin(C) = b/sin(B)

  c = sin(21°)(15/sin(97°)) ≈ 5.42

The remaining side lengths are approximately ...

  a ≈ 13.3

  c ≈ 5.4

4 0
3 years ago
How do u Solve 2x2 + 2x2 + 2 - 2x2 =
Crank
<span>2x2 + 2x2 + 2 - 2x2 =  4 + 4 +2 -4
                             = 4+2   
                             = 6</span>
7 0
3 years ago
Read 2 more answers
30r + –20 = 760 r = _____
andrey2020 [161]

Answer:

r = <u>26</u>

Step-by-step explanation:

30r+(-20)=760\\30r-20=760\\30r=780\\r=26

6 0
3 years ago
Read 2 more answers
9. What date will it be? (One is done for you)
Licemer1 [7]

Answer:

A. 5 days after 15 June - June 20

B. 10 days before Independence day - 5 August

C. 7 days before 8 January - 1 January

D. 3 days before Republic day - 23 January

E. 14 days after 16 July - 30 july

F. 20 days after 1 August - 21 August

3 0
3 years ago
What is - 6<br><br> 5p - p when you simplify it
Darina [25.2K]
5p-p=4p. The question doesn't make sense because ther is no sign after the minus 6. If it is an X it would be -24p. If it is an add it would be -6+4p. If it was a - it would be -6-4p.
6 0
3 years ago
Other questions:
  • PLEASE ANSWER ASAP!!!! MUST BE CORRECT FOR POINTS GIVEN!!!!! THE QUESTION IS BELOW IN THE SCREENSHOT!!!!! VERY IMPORTANT!!!!!
    13·1 answer
  • Write the value of the underline digit in 0.496
    8·1 answer
  • citrus weekly wages are $20 less than twice what Mary earns in one week in citra makes $120 a week write and solve an equation t
    6·1 answer
  • Explain why the equation 2L + 2W = 4(L - W + 7) is true for the rectangle
    15·2 answers
  • The table shows the dimensions of 4 different rectangles. Each rectangle has an area of 36 square centimeters.
    9·2 answers
  • If four points are coplanar, then they lie on the same plane. True/False
    15·1 answer
  • Select three ratios that are equivalent to 8 : 5
    6·2 answers
  • Solve for the missing variable
    5·1 answer
  • 1.
    13·1 answer
  • Help please I will give brainiest​
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!