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
Assoli18 [71]
3 years ago
14

If cos() = − 2 3 and is in Quadrant III, find tan() cot() + csc(). Incorrect: Your answer is incorrect.

Mathematics
1 answer:
nydimaria [60]3 years ago
6 0

Answer:

\tan(\theta) \cdot \cot(\theta) + \csc(\theta) = \frac{5 - 3\sqrt 5}{5}

Step-by-step explanation:

Given

\cos(\theta) = -\frac{2}{3}

\theta \to Quadrant III

Required

Determine \tan(\theta) \cdot \cot(\theta) + \csc(\theta)

We have:

\cos(\theta) = -\frac{2}{3}

We know that:

\sin^2(\theta) + \cos^2(\theta) = 1

This gives:

\sin^2(\theta) + (-\frac{2}{3})^2 = 1

\sin^2(\theta) + (\frac{4}{9}) = 1

Collect like terms

\sin^2(\theta)  = 1 - \frac{4}{9}

Take LCM and solve

\sin^2(\theta)  = \frac{9 -4}{9}

\sin^2(\theta)  = \frac{5}{9}

Take the square roots of both sides

\sin(\theta)  = \±\frac{\sqrt 5}{3}

Sin is negative in quadrant III. So:

\sin(\theta)  = -\frac{\sqrt 5}{3}

Calculate \csc(\theta)

\csc(\theta) = \frac{1}{\sin(\theta)}

We have: \sin(\theta)  = -\frac{\sqrt 5}{3}

So:

\csc(\theta) = \frac{1}{-\frac{\sqrt 5}{3}}

\csc(\theta) = \frac{-3}{\sqrt 5}

Rationalize

\csc(\theta) = \frac{-3}{\sqrt 5}*\frac{\sqrt 5}{\sqrt 5}

\csc(\theta) = \frac{-3\sqrt 5}{5}

So, we have:

\tan(\theta) \cdot \cot(\theta) + \csc(\theta)

\tan(\theta) \cdot \cot(\theta) + \csc(\theta) = \tan(\theta) \cdot \frac{1}{\tan(\theta)} + \csc(\theta)

\tan(\theta) \cdot \cot(\theta) + \csc(\theta) = 1 + \csc(\theta)

Substitute: \csc(\theta) = \frac{-3\sqrt 5}{5}

\tan(\theta) \cdot \cot(\theta) + \csc(\theta) = 1 -\frac{3\sqrt 5}{5}

Take LCM

\tan(\theta) \cdot \cot(\theta) + \csc(\theta) = \frac{5 - 3\sqrt 5}{5}

You might be interested in
Dylan hikes 90 miles in 3 days. At that rate, how many days does it take him to hike a total of 240 miles?
zysi [14]

Answer:

8 days

Step-by-step explanation:

90/3 = 30 mi/day

240/30 = 8 days

8 0
3 years ago
Read 2 more answers
Please help!!!!!!!!!
zloy xaker [14]
Line point L would be the midpoint of question 3.
6 0
3 years ago
Read 2 more answers
I need number 2 I need help fast
Semmy [17]

Answer:

B. 70

Step-by-step explanation:

<h3><u>look at the smaller triangle</u></h3>

its dimensions are :

45 , 35 , 24

<h3><u>look at the bigger triangle ( this includes the dimensions of the smaller one)</u></h3>

its dimensions are :

45 + 45 ,  x , 24 + 24

= 90 , x  , 48

from the dimensions given we can notice that :

<em>bigger triangle dimensions are  = 2 * smaller triangle dimensions</em>

<em></em>

therefore to find the value of x :

x = 35 * 2

<h3><u>x = 70</u></h3>

<u />

<h3><u>hence the answer is B ) 70 </u></h3>
4 0
3 years ago
Look at the image. (calculus)
mash [69]
<h3>Answer: Choice H)  2</h3>

=============================================

Explanation:

Recall that the pythagorean trig identity is \sin^2 x + \cos^2x = 1

If we were to isolate sine, then,

\sin^2 x + \cos^2x = 1\\\\\sin^2 x = 1-\cos^2x\\\\\sin x = \sqrt{1-\cos^2x}\\\\

We don't have to worry about the plus minus because sine is positive when 0 < x < pi/2.

Through similar calculations, \cos x = \sqrt{1-\sin^2x}\\\\

Cosine is also positive in this quadrant.

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

So,

\frac{\sqrt{1-\cos^2x}}{\sin x}+\frac{\sqrt{1-\sin^2x}}{\cos x}\\\\\frac{\sin x}{\sin x}+\frac{\cos x}{\cos x}\\\\1+1\\\\2

Therefore,

\frac{\sqrt{1-\cos^2x}}{\sin x}+\frac{\sqrt{1-\sin^2x}}{\cos x}=2

is an identity as long as 0 < x < pi/2

5 0
2 years ago
Read 2 more answers
Which equation represents a line that has a slope of 3/4 and passes through the point (2,1)?
ValentinkaMS [17]
The general equation of a line is:
\frac{y- y_{1}}{x- x_{1}} =m
When m is the gradient and x1 and y1 are the coordinates of the points on the line
Therefore:
\frac{y-1}{x-2}=3/4
4y-4=3x-6
4y=3x-2
5 0
3 years ago
Other questions:
  • What is the answer, I’m having a hard time understanding it.:(
    15·1 answer
  • What are expressions
    8·2 answers
  • Which of the following steps were applied to ∆ABC obtain ∆A'B'C'?
    8·1 answer
  • if it is cloudy on a given day it is 40% likely to rain. in general the chance of clouds on any given day is 50%. The probabilit
    12·1 answer
  • What is 20/3 equal to?
    14·1 answer
  • Cherie is jogging around a circular track. She
    12·1 answer
  • -2 1/3-(-5) can someone help me
    8·1 answer
  • Which expressions are equivalent to -7+3(-4e-3)
    5·1 answer
  • Order these list of decimals from smallest to largest -8.91 4.56 -2.8 0.62
    15·1 answer
  • Written as a rate, 100 miles in 5 hours would be ____ miles per hour.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!