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alexgriva [62]
3 years ago
15

What values for θ (0≤θ≤2π) θ ( 0 ≤ θ ≤ 2 π ) satisfy the equation: cos(θ) − tan(θ)cos(θ) = 0

Mathematics
1 answer:
timama [110]3 years ago
6 0

When solving this problem any trigonometric equation, putting things in terms of sines and cosines is a good first step - mainly since most identities are built off these two functions.


cos \theta - (\frac{sin\theta}{cos\theta} * cos\theta) = 0


The cosines after the minus sign divide out leaving us with cos  \theta - sin \theta = 0.


If we subtract the sine to the other side, we are then asking a question: when are the cosine and the sine of angle equal? One common occurence is the 45 - 45 - 90 triangle when the cosine and sine of each angle is the same. So at 45 degrees, or \frac{\pi}{4} radians that happens. Saying this only tells half the story.


We now need to find when the sine and cosine have the same sign (since two negatives would work). Let's work quadrant by quadrant.

Quadrant I - sine and cosine are positive.

Quadrant II - sine is positive, cosine is negative

Quadrant III - cosine is negative, sine is negative

Quadrant IV - cosine is positive, sine is negative.


The 45-45-90 triangle is the Quadrant I response of 45 degrees. The quadrant III angle would be at 180 degrees plus that or 225 degrees. Since the problem is in radians, it would be [tex] \frac{\pi}{4} [/tex] and \frac{\5pi}{4} since we add π radians to get around the unit circle.



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Suppose the mean income of firms in the industry for a year is 95 million dollars with a standard deviation of 5 million dollars
GuDViN [60]

Answer:

Probability that a randomly selected firm will earn less than 100 million dollars is 0.8413.

Step-by-step explanation:

We are given that the mean income of firms in the industry for a year is 95 million dollars with a standard deviation of 5 million dollars. Also, incomes for the industry are distributed normally.

<em>Let X = incomes for the industry</em>

So, X ~ N(\mu=95,\sigma^{2}=5^{2})

Now, the z score probability distribution is given by;

         Z = \frac{X-\mu}{\sigma} ~ N(0,1)

where, \mu = mean income of firms in the industry = 95 million dollars

            \sigma = standard deviation = 5 million dollars

So, probability that a randomly selected firm will earn less than 100 million dollars is given by = P(X < 100 million dollars)

    P(X < 100) = P( \frac{X-\mu}{\sigma} < \frac{100-95}{5} ) = P(Z < 1) = 0.8413   {using z table]

                                                     

Therefore, probability that a randomly selected firm will earn less than 100 million dollars is 0.8413.

5 0
3 years ago
3. Solve the system using elimination (not substitution or matrices). negative 2 x plus y minus 2 z equals negative 8A N D7 x pl
riadik2000 [5.3K]

Elimination Method

\begin{gathered} -2X+Y-2Z=-8 \\ 7X+Y+Z=-1 \\ 5X+2Y-Z=-9 \end{gathered}

If we multiply the equation 3 by (-1) we obtain this:

\begin{gathered} -2X+Y-2Z=-8 \\ 7X+Y+Z=-1 \\ -5X-2Y+Z=9 \end{gathered}

If we add them we obtain 0, therefore there are infinite solutions. So, let's write it in terms of Z

1. Using the 3rd equation we can obtain X(Y,Z)

\begin{gathered} 5X=-9-2Y+Z \\ X=\frac{-9-2Y+Z}{5} \\  \end{gathered}

2. We can replace this value of X in the 1st and 2nd equations

\begin{gathered} -2\cdot(\frac{-9-2Y+Z}{5})+Y-2Z=-8 \\ 7\cdot(\frac{-9-2Y+Z}{5})+Y+Z=-1 \end{gathered}

3. If we simplify:

\begin{gathered} \frac{-9Y+12Z-63}{5}=-1 \\ \frac{9Y-12Z+18}{5}=-8 \end{gathered}

4. We can obtain Y from this two equations:

\begin{gathered} Y=-\frac{-12Z+58}{9} \\  \end{gathered}

5. Now, we need to obtain X(Z). We can replace Y in X(Y,Z)

\begin{gathered} X=\frac{-9-2Y+Z}{5} \\ X=\frac{-9-2(-\frac{-12Z+58}{9})+Z}{5} \end{gathered}

6. If we simplify, we obtain:

X=\frac{-3Z+7}{9}

7. In conclusion, we obtain that

(X,Y,Z) =

(\frac{-3Z+7}{9},-\frac{-12Z+58}{9},Z)

8 0
1 year ago
Tanya drove 3 1/2 hours and used 1/3 tank of gas. For how long could Tanya drive using​ 1/2 ​tank of gas?
creativ13 [48]
<span>
3 1/2  = 7/2

7/2  hours and used 1/3 tank of gas
</span><span>? hours and used 1/2 tank of gas

1/2 * 7/2
------------
    1/3

= 7/4
-------
  1/3

= 7/4 * 3
= 21/4
= 5 1/4 

answer: 5 1/4 hours </span><span>Tanya could drive using​ 1/2 ​tank of gas</span><span>

</span>
4 0
3 years ago
 What is the mathematical term for "the same"?
melisa1 [442]

Answer:

Equivalent / Equal

Step-by-step explanation:

5 0
3 years ago
Solve the linear system of equations <br><br>3x+2y=7<br>-3x+4y=5<br><br>X= <br>Y=​
Anon25 [30]
Answer :
X= 1 , Y = 2


Step-by-step explanation :

5 0
3 years ago
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