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2 years ago

Which trigonometric functions are negative in the fourth (IV) quadrant?

Mathematics

1 answer:

anzhelika [568]2 years ago

7 0

<h3>Answers are: sine, tangent, cosecant, cotangent</h3>

Explanation:

On the unit circle we have some point (x,y) such that x = cos(theta) and y = sin(theta). The sine corresponds to the y coordinate of the point on the circle. Quadrant IV is below the x axis which explains why sine is negative here, since y < 0 here.

Since sine is negative, so is cosecant as this is the reciprocal of sine

csc = 1/sin

In quadrant IV, cosine is positive as x > 0 here. So the ratio tan = sin/cos is going to be negative. We have a negative over a positive when we divide.

Because tangent is negative, so is cotangent.

The only positive functions in Q4 are cosine and secant, which is because sec = 1/cos.

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Part 4: Identify the vertex, focus and directrix of each. Then sketch the graph.

Nezavi [6.7K]

Answer: 1) Vertex: (6, -2) Focus: (6, -7/4) Directrix: y = -9/4

2) Vertex: (-2, -1) Focus: (-7/4, -1) Directrix: x = -9/4

<u>Step-by-step explanation:</u>

Rewrite the equation in vertex format y = a(x - h)² + k or x = a(y - k)² + h by completing the square. Divide the b-value by 2 and square it - add that value to both sides of the equation.

(h, k) is the vertex
p is the distance from the vertex to the focus
-p is the distance from the vertex to the directrix

$\bullet\quad a=\dfrac{1}{4p}$

1) y = x² - 12x + 34

$y-34=x^2-12x\\\\y-34+\bigg(\dfrac{-12}{2}\bigg)^2=x^2-12x+\bigg(\dfrac{-12}{2}\bigg)^2\\\\\\y-34+36=(x-6)^2\\\\y+2=(x-6)^2\\\\y=(x-6)^2-2\qquad \rightarrow \qquad a=1\quad (h,k)=(6,-2)\\$

$a=\dfrac{1}{4p}\quad \rightarrow \quad 1=\dfrac{1}{4p}\quad \rightarrow \quad p=\dfrac{1}{4}\\\\\\\text{Focus = Vertex + p}\\.\qquad \quad =\dfrac{-8}{4}+\dfrac{1}{4}\\\\.\qquad \quad =-\dfrac{7}{4}\quad \rightarrow \quad\text{Focus}=\bigg(6,-\dfrac{7}{4}\bigg)\\\\\\\text{Directrix: y= Vertex - p}\\.\qquad \qquad y=\dfrac{-8}{4}-\dfrac{1}{4}\\\\.\qquad \qquad y=-\dfrac{9}{4}$

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2) x = y² + 2y - 1

$x+1=y^2+2y\\\\x+1+\bigg(\dfrac{2}{2}\bigg)^2=y^2+2y+\bigg(\dfrac{2}{2}\bigg)^2\\\\\\x+1+1=(y+1)^2\\\\x+2=(y+1)^2\\\\x=(y+1)^2-2\qquad \rightarrow \qquad a=1\quad (h,k)=(-2,-1)\\$

$a=\dfrac{1}{4p}\quad \rightarrow \quad 1=\dfrac{1}{4p}\quad \rightarrow \quad p=\dfrac{1}{4}\\\\\\\text{Focus = Vertex + p}\\.\qquad \quad =\dfrac{-8}{4}+\dfrac{1}{4}\\\\.\qquad \quad =-\dfrac{7}{4}\quad \rightarrow \quad\text{Focus}=\bigg(-\dfrac{7}{4},-1\bigg)\\\\\\\text{Directrix: x= Vertex - p}\\.\qquad \qquad y=\dfrac{-8}{4}-\dfrac{1}{4}\\\\.\qquad \qquad x=-\dfrac{9}{4}$

4 0

2 years ago

7. A cylindrical tank can hold 2279.64 cubic feet of water. The radius of the tank is 11 feet. What is the height of the tank?

aliina [53]

Answer:

r = 8.12

Step-by-step explanation:

The volume for a cylinder is V=πr^2h therefore,

v = 2279.64

h = 11ft

r = ?

2279.64 = πr^2 * 11

$\frac{2279.64}{11}= {\pi r^2} \\$

207.24 = πr^2

$\frac{207.24}{\pi} =r^2\\\\66 = r^2\\\\\sqrt{66} = r\\\\\\8.12 = r$

3 0

1 year ago

Write the ordered pair for the point S.

beks73 [17]

Given:

The location of point S on a coordinate plane.

To find:

The ordered pair for the point S.

Solution:

A point is defined as (x,y), where, |x| is the distance between the point and y-axis, and |y| be the distance between the point and x-axis. Signs of coordinates depend on the quadrant.

From the given graph it is clear that,

Distance between S and y-axis = 3.5

Distance between S and x-axis = 5.5

Point S lies in 3rd quadrant, it means x- and y-coordinates are negative.

Therefore, the ordered pair of point S is (-3.5,-5.5).

8 0

2 years ago

The management of a large airline wants to estimate the average time after takeoff taken before the crew begins serving snacks a

Dahasolnce [82]

Answer:

The answer is "Option a, Option b, and Option d".

Step-by-step explanation:

In the given question it is used to stratifying the sampling if the population of this scenario it flights takes off when it is divided via some strata.

In option a, In this case, it stratified the sampling, as the population of planes taking off has been divided into the days of the week.
In option b, It also used as the case of stratified sampling.
In options c, it is systematic sampling, that's why it is wrong.
In option d, It is an example of stratifying the sampling.

3 0

2 years ago

Evaluate the expression<br>7 + 10 /5

Anestetic [448]

Answer:

Step-by-step explanation:

Use PEMDAS (Parenthesis, exponent, multiplication, devision, addition, subtraction)

So....

7 + 2 .... Simplified ( 10/5 =2)

9 ..... Added

Hope this helps!

4 0

3 years ago

Read 2 more answers