The variable z is directly proportional to x. When x is 6, z has the value 60. What is the value of z when x = 11?

Answer:

20

4x will be 80

5x will be 100

Step-by-step explanation:

So those two how pink arrows means those opposite sides are parallel.

The side at the bottom is acting as a transversal through the parallel lines.

The angle that has measurement 5x and the one that has 4x are actually same-side interior angles; some people like to call it consecutive angles.

These angles add up to be 180 degrees when dealing with parallel lines.

So we have 5x+4x=180

which means 9x=180

Divide both sides by 9 giving us x=180/9

x=180/9=20.

The angle labeled 4x will then be 4(20) which is 80.

The angle labeled 5x will then by 5(20) which is 100.

To solve for x in vector problems, identify the axes, decompose each vector into its components using trigonometric functions, combine the components to find the resultant vector, and ensure that the solution is reasonable. Be sure to use radians for angles in calculations.

To find the value of x, we need to follow specific steps when dealing with vectors and their components. The given information suggests we have vectors A and B with specific magnitudes and angles relative to the x-axis. Here's how to proceed:

For example, given A = 53.0 m, θA = 20.0°, B = 34.0 m, and θB = 63.0°, we can find the x-components as Ax = A cos θA. It is important to use radians when calculations involve angles.

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Answer:

Answer choice C is the correct answer.

Step-by-step explanation:

Answer choice A, 4/9, is equal to roughly .44.

Answer choice B, 2/5, is equal to exactly .4.

Answer choice C, 3/6, is equal to exactly .5.

Answer choice D, 5/12, is equal to roughly .42.

A proportion implies that the fractions will equal each other in value. The fraction 7/14 is equal to 1/2 or .5. Answer choice C, when simplified, is equal to 1/2 or .5, making it the correct answer.

Answer:

  B.  $14,984

Step-by-step explanation:

The multiplier is ...

  (1 +r/n)^(nt) . . . . where r is the nominal annual rate, n is the number of times interest is compounded per year, and t is the number of years.

Here, that multiplier is ...

  (1 +.08/2)^(2·8) = 1.04^16 ≈ 1.87298

Then Lana will be paying Tina ...

  $8000×1.87298 ≈ $14984

at the end of 8 years.

Final answer:

In propositional logic, the statement translates to ‘m → (e ∨ p)’, which denotes that seeing the movie 'm' can happen if the person is over 18 'e' or has parental permission 'p'.

Explanation:

The student is asking to translate a given statement into propositional logic. The statement in question is "You can see the movie only if you are over 18 years old or you have the permission of a parent." Using the propositions provided, where m represents "You can see the movie," e represents "You are over 18 years old," and p represents "You have the permission of a parent," we can express the logical relation in the following way:

In terms of logic, the statement can be written as a conditional: if you can see the movie (m), then you are over 18 years old (e) or you have the permission of a parent (p). This can be expressed as:

m → (e ∨ p)

This states that seeing the movie is the consequence of the condition of being over 18 years old or having parental permission. The conditional statement here establishes a necessary and sufficient relationship, while the use of or indicates that at least one of the conditions (being over 18 or having permission) must be met for the consequence (seeing the movie) to be true.

Answer:

  x = 5

Step-by-step explanation:

Ordinarily, one would not need the distributive property to solve this equation. It is quickly and easily solved by making use of the multiplication property of equality: multiply both sides of the equation by 1/3.

  3x(1/3) = 15(1/3)

  x = 5

___

To use the distributive property, we need the sum of two terms that have a common factor. We can get that form by subtracting 15 from both sides of the equation (subtraction property of equality):

  3x - 15 = 15 - 15 . . . . subtract 15

  3x -15 = 0 . . . . . . . . .simplify

Now, we can apply the distributive property to remove a factor of 3:

  3(x -5) = 0

And we can use the multiplication property of equality to multiply by 1/3:

  3(1/3)(x -5) = 0(1/3)

  x -5 = 0 . . . . . . . . . . simplify

Finally, we can add 5 to both sides of the equation (addition property of equality):

  x -5 +5 = 0 +5

  x + 0 = 5 . . . . . . simplify

  x = 5 . . . . . . . . . .simplify more

Answer:

X=5

Step-by-step explanation:

i believe this is the correct answer!

Answer:

  a) J, K, W

  b) any of X, Y, or P

  c) 1 line

Step-by-step explanation:

a) The figure shows points J, K and W on line m.

b) Any points in the diagram other than J, K, and W are not on line m. Those include points X, Y, and P. Your answer will be one or more of these.

c) Two points define 1 line. There is only one line through any given pair of points. It can be named many ways, but it is still the same (one) line.

Answer:

 1) J, K, W

 2) any of X, Y, or P

 3) Only one line  (pyx)

Step-by-step explanation:

1) The figure shows points J, K and W on line m.

2) All points in the diagram except for j, w, and k are not on line m. These include points x, y, and p. Your answer should be one or more of these.

3) Three points define 1 line. There is only one line through any given group of points. It can be named many ways, but it will remain (one) line. The line would run through the points p, y, and x.

Hope this helps

- Que

Answer:

y = 5x - 11

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

here slope m = 5, hence

y = 5x + c ← is the partial equation

To find c substitute (1, - 6) into the partial equation

- 6 = 5 + c ⇒ c = - 6 - 5 = - 11

y = 5x - 11 ← equation of line

the first one

the degree of the polynomial in the numerator is 2.

the degree of the polynomial in the denominator is 2.

when the top and bottom have the same degree, like in this case, the horizontal asymptotes that that can afford us is simply the value of their coefficients.

[tex]\bf \cfrac{x^2-16}{x^2+2x+1}\implies \cfrac{1x^2-16}{1x^2+2x+1}\implies \stackrel{\textit{horizontal asymptote}}{y=\cfrac{1}{1}\implies y=1}[/tex]

for the second one

well, the degree of the numerator is 3.

the degree of the denominator is 2.

when the numerator has a higher degree than the denominator, there are no horizontal asymptotes, however, when the degree of the numerator is exactly 1 degree higher than that of the denominator, the rational has an oblique or slant asymptote, and its equation comes from the quotient of the whole expression, check the picture below, the top part.

for the third one

this one is about the same as the one before it, the numerator has exactly one degree higher than the denominator, so we're looking at an oblique asymptote, check the picture below, the bottom part.

Answer:

It’s A, D, and E

The magnetic field lines flow from Earth’s geographic South Pole to Earth’s

geographic North Pole.

The magnetic field is generated in Earth’s core.

The magnetic field is similar to the magnetic field of a bar magnet.

thank youhave a good day

Answer:

It’s A, D, and E.

Step-by-step explanation: Try and find out :)

Jk Jk it's right.

Answer:

3

Step-by-step explanation:

I don't see the graph.

So I'm going to do algebra and should be x-coordinate of the intersection you see in front of you.

If y=-x+5 and y=x-1, then -x+5=x-1.

-x+5=x-1

Add x on both sides:

   5=2x-1

Add 1 on both sides:

   6=2x

Divide 2 on both sides:

  6/2=x

Simplify

   3=x

You should see them cross when x is 3.

The y there should by 3-1=2.

They should cross at the ordered pair (3,2).

Answer:

3

Step-by-step explanation:

If a graph shows a system of equations: y = -x + 5 and y = x - 1, the x-coordinate of the solution to the system of equations is 3.

y=-x+5

y=x-1

-x+5=x-1

The system of equations provided consists of a circle and a line:

1. [tex]\( x^2 + y^2 = 49 \)[/tex] (This represents a circle with a radius of 7, centered at the origin.)

2. [tex]\( y = -x - 7 \)[/tex] (This is a linear equation.)

The first equation has already been graphed, showing a circle with a radius of 7. To find the intersection points of the circle and the line, which are the solutions of the system, we can substitute the expression for y from the second equation into the first one:

[tex]\[ x^2 + (-x - 7)^2 = 49 \][/tex]

[tex]\[ x^2 + x^2 + 14x + 49 = 49 \][/tex]

[tex]\[ 2x^2 + 14x + 49 - 49 = 0 \][/tex]

[tex]\[ 2x^2 + 14x = 0 \][/tex]

[tex]\[ x(2x + 14) = 0 \][/tex]

This gives us two solutions for x:

[tex]\[ x = 0 \quad \text{or} \quad 2x + 14 = 0 \][/tex]

[tex]\[ x = 0 \quad \text{or} \quad x = -7 \][/tex]

Now we can substitute these x-values into the second equation to find the corresponding y-values:

For [tex]\( x = 0 \)[/tex]:

[tex]\[ y = -0 - 7 \][/tex]

[tex]\[ y = -7 \][/tex]

For [tex]\( x = -7 \)[/tex]:

[tex]\[ y = -(-7) - 7 \][/tex]

[tex]\[ y = 7 - 7 \][/tex]

[tex]\[ y = 0 \][/tex]

Therefore, the system of equations has two solutions where the line intersects the circle:

[tex]\[ (0, -7) \quad \text{and} \quad (-7, 0) \][/tex]

These calculations provide us with the step-by-step solution to the system of equations. The graphical solution would show the line [tex]\( y = -x - 7 \)[/tex] intersecting the circle [tex]\( x^2 + y^2 = 49 \)[/tex] at these two points.

Here is the graph showing the system of equations:

- The circle represented by [tex]\( x^2 + y^2 = 49 \)[/tex].

- The line represented by [tex]\( y = -x - 7 \).[/tex]

The red points indicate where the line intersects the circle, which are the solutions to the system of equations. These points are at (0, -7) and (-7, 0).

Answer:

The area of the circle is A = 12.56 m²

Step-by-step explanation:

* Lets explain how to solve the problem

- The area of any circle is A = π r² , where r is the radius of the circle

- In any circle the length of the radius is half the length of its diameter

* Lets solve the problem

- The diameter of the circle is 4 meters

∵ The radius of the circle = 1/2 diameter

∵ The diameter = 4 meters

∴ The radius = 1/2 × 4 = 2 meters

- The area of the circle is A = π r²

∵ The value of π = 3.14

∵ The length of r = 2 meters

- Substitute the value of r in the rule of the area

∵ A = π r²

∴ A = 3.14 × (2)²

∴ A = 3.14 × 4 = 12.56 meters²

* The area of the circle is A = 12.56 m²

Answer:

72.5°

Step-by-step explanation:

This is a right triangle problem.  The ladder is the hypotenuse, and we have a length for that of 50 feet.  We also have the distance along the ground that the ladder's base is from the wall against which it is leaning.  This is the side adjacent to the angle for which we are seeking, so it the side adjacent to the angle, and it measures 15 feet.

We need a trig ratio that relates the side adjacent to the hypotenuse.  That will be the cosine.  Setting up:

[tex]cos\theta =\frac{adjacent}{hypotenuse}[/tex] and then filling in:

[tex]cos\theta=\frac{15}{50}[/tex]

Use the 2nd button on your calculator, and then press cos and you will get a display on your screen that looks like this:

[tex]cos^{-1}([/tex]

After that open parenthesis enter the fraction 15/50 and hit enter and you'll get the angle of 72.5 degrees, rounded.

just an addition to that superb reply above by @Luv2Teach

to the risk of sounding redundant.

Check the picture below.

make sure your calculator is in Degree mode.

2222 = 2220 + 2 = 555 * 4 + 2

2333 = 2332 + 1 = 583 * 4 + 1

Then

2222 + 2333 = (555 + 583) * 4 + 3

leaving a remainder of 3.

2222 = 2220 + 2 = 370 * 6 + 2

2333 = 2328 + 5 = 388 * 6 + 5

Then

2222 + 2333 = (370 + 388) * 6 + 7 = (370 + 388 + 1) * 6 + 1

leaving a remainder of 1.

2222 = 2223 - 1 = 741 * 3 - 1

2333 = 388 * 6 + 5 = (388 * 2) * 3 + 5 = (388 * 2 + 1) * 3 + 2

Then

2222 + 2333 = (741 + 388 * 2 + 1) * 3 + 1

leaving a remainder of 1.

2222 = 555 * 4 + 2 = 185 * 3 * 4 + 2 = 185 * 12 + 2

2333 = 2400 - 67 = 2400 - 60 - 7 = (200 - 5) * 12 - 7

Then

2222 + 2333 = (185 + 200 - 5) * 12 - 5

leaving a remainder of -5, or 7. (because 12 - 5 = 7)

Answer:

Step-by-step explanation:

Let the other leg = x

x^2 + 4^2 = (3x)^2

x^2 + 4^2 = 9x^2

4^2 = 9x^2 - x^2

16= 8x^2

16/8 = x^2

x^2 = 2

x = sqrt(2)

The lengths of the sides

x = sqrt(2)      

other side =4

hypotenuse = 3*sqrt(2)

x = 1.4

other side= 4

hypotenuse = 3*1.4142

hypotenuse = 4.2

Answer:

4 feet, 1.4 feet, 4.2 feet

Step-by-step explanation:

We are looking for the lengths of the three sides of a right triangle. We are given that one leg has length 4ft. Let x be the length of the other leg. Since the hypotenuse of the right triangle is three times the length of this leg, we can represent the hypotenuse as 3x. This is a right triangle, so we can use the Pythagorean Theorem to find x.

42+x216+x2=(3x)2=9x2

Subtracting x2 from both sides, then dividing by 8 to isolate the x, we have

8x2x2x=16=2=±2–√

Considering only the positive value for x, the lengths of the three sides of the triangle are approximately 4 feet, 2–√≈1.4 feet, and 32–√≈4.2 feet.

Answer:

She used 49 of type B

Step-by-step explanation:

Step-by-step explanation:

Same steps, different problem.

x - pounds of type A coffee for current month

y - pounds of type B coffee for current month

x+y=145

4.2x+5.3y=659.9

x+y=145 -> multiply both sides by 4.2

4.2x+5.3y=659.9

4.2x+4.2y=609

1.1y=50.9

Answer: binomial probability

Step-by-step explanation:

A binomial probability indicates to the probability of having exactly x successes on n repeated trials in an particular binomial experiment which has only two possible outcomes.

If the probability of success on an single trial is b (which does not change) , then , the probability of failure will be (1-b)  .

The binomial probability for success in x trials out of n trials is given by :-

[tex]^nC_x\ b\ (1-b)^{n-x}[/tex]

Final answer:

The distribution called for a fixed number of independent trials with a constant success probability is the binomial probability distribution, defined by the equation P(X = x).

Explanation:

The probability distribution that shows the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a binomial probability distribution. The key characteristics of a binomial distribution include a fixed number of independent trials with two possible outcomes (success or failure) and a constant probability of success in each trial.

Mathematically, the binomial distribution is defined using the equation P(X = x) = (n choose x) * px * qn-x, where p is the probability of success, q = 1 - p is the probability of failure, and (n choose x) is the combination of n taken x at a time.

Answer:

the expected value is if your sum is odd because half of the values you could roll are odd witch means you have 50% chance to get odd

because their are not high odds of getting something good

Answer:

The triangles are similar

Step-by-step explanation:

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional

step 1

In the right triangle FED

Find the length of side FD

Applying the Pythagoras Theorem

[tex]FD^{2}=FE^{2}+DE^{2}[/tex]

substitute the given values

[tex]FD^{2}=3^{2}+4^{2}[/tex]

[tex]FD^{2}=25[/tex]

[tex]FD^{2}=5\ units[/tex]

step 2

In the right triangle BUG

Find the length of side GU

Applying the Pythagoras Theorem

[tex]BG^{2}=BU^{2}+GU^{2}[/tex]

substitute the given values

[tex]10^{2}=6^{2}+GU^{2}[/tex]

[tex]GU^{2}=100-36[/tex]

[tex]GU^{2}=8\ units[/tex]

step 3

Find the ratio of its corresponding sides

If the triangles are similar

[tex]\frac{FD}{BG}=\frac{FE}{BU}=\frac{DE}{GU}[/tex]

substitute the given values

[tex]\frac{5}{10}=\frac{3}{6}=\frac{4}{8}[/tex]

[tex0.5=0.5=0.5[/tex] -----> is true

therefore

The triangles are similar

Answer:

1/root 3

Step-by-step explanation:

pi =180degrees

11×180/6

11×30

330

tan 330=tan (360-30)

=-tan30

=1/root3

i have answered ur question

Answer:

Base of the original triangle is 12 cm.

Step-by-step explanation:

Let base of triangle be x  

two equal legs  of triangles by y

therefore perimeter of triangle

                       x+y+y = 50 or x+2y =50

according to the question

 if base is x+3 and leg is y-4

then both are equal

   that is x+3 = y-4

            y-x = 7 or y =x+7

   x+2y =50

   x+2(x+7) = 50

    x+2x +14 =50

     3x +14 =50

     3x = 50 -14

      3x = 36

      x = 12

therefore base of the original  triangle is 12 cm                    

Answer:

  A.  24  hm

Step-by-step explanation:

The formula for the circumference of a circle is ...

  C = 2πr

Fill in the given values and solve.

  150.7968 hm = 2×3.1416×r

  (150.7968 hm)/6.2832 = r = 24 hm . . . . . divide by 2π

The radius of the circle is 24 hm.

Answer:  (a)  (602.95,705.37)

(b) 33

Step-by-step explanation:

(a) Given : Sample size : [tex]n=40[/tex]

Sample mean : [tex]\overline{x}=654.16[/tex]

Standard deviation : [tex]\sigma= 165.23[/tex]

Significance level :[tex]\alpha=1-0.95=0.05[/tex]

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

The confidence interval for population mean is given by :-

[tex]\mu\ \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]

[tex]=654.16\pm(1.96)\dfrac{165.23}{\sqrt{40}}\\\\\approx654.16\pm51.21\\\\=(654.16-51.21,\ 654.16+51.21)=(602.95,705.37)[/tex]

Hence, the 95% (two-sided) confidence interval for true average [tex]CO_2[/tex] level in the population of all homes from which the sample was selected.

(b) Given : Standard deviation : [tex]s= 167\text{ ppm}[/tex]

Margin of error : [tex]E=\pm57\text{ ppm}[/tex]

Significance level :[tex]\alpha=1-0.95=0.05[/tex]

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

The formula to calculate the sample size is given by :-

[tex]n=(\dfrac{z_{\alpha/2}s}{E})^2\\\\\Rightarrow\ n=(\dfrac{(1.96)(167)}{57})^2=32.9758025239\approx33[/tex]

Hence, the minimum required sample size would be 33.

Answer: 0.2140

Step-by-step explanation:

Given : A company produces steel rods. The lengths of the steel rods are normally distributed with

[tex]\mu=92.6 \text{ cm}[/tex]

[tex]\sigma=2\text{ cm}[/tex]

Sample size : [tex]n=12[/tex]

Let x be the length of randomly selected item.

z-score : [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x=92 cm

[tex]z=\dfrac{92-92.6}{\dfrac{2}{\sqrt{12}}}\approx-1.04[/tex]

For x=92.4 cm

[tex]z=\dfrac{92.4-92.6}{\dfrac{2}{\sqrt{12}}}\approx-0.35[/tex]

The probability that the average length of a randomly selected bundle of steel rods is between 92-cm and 92.4-cm by using the standard normal distribution table

= [tex]P(92<x<92.4)=P(-1.04<z<-0.35)=P(z<-0.35)-P(z<-1.04)[/tex]

[tex]= 0.3631693-0.14917=0.2139993\approx0.2140[/tex]

Hence, the probability that the average length of a randomly selected bundle of steel rods is between 92-cm and 92.4-cm is 0.2140.

Answer:

  -4432.5 m

Step-by-step explanation:

Displacement is measured in meters, so will be the product of velocity in m/s and time in s.

  (-98.5 m/s)×(45.0 s) = -4432.5 m

___

If you're concerned with significant figures, you can round this to -4430 m, which has the required 3 significant figures.

Answer:

-4432.5 m

Step-by-step explanation:

distance (or displacement) = rate times time.

Here, the displacement is

                                             (-98.5 m/s)(45.0 s) = -4432.5 m

Answer:

Step-by-step explanation:

For the equation ...

  y = a(x -h)² +k

you can find the x-intercepts by setting y=0 and solving for x.

  0 = a(x -h)² +k

  -k = a(x -h)² . . . . . . subtract k

  -k/a = (x -h)² . . . . . divide by a

  ±√(-k/a) = x -h . . . . take the square root

  h ± √(-k/a) = x . . . . add h . . . . this is the general solution

__

So, for each of your problems, fill in the corresponding numbers and do the arithmetic. If (-k/a) is a negative number, the square root gives imaginary values, so there is "no solution".

1.  x = 5 ± √9 = {5 -3, 5 +3} = {2, 8} . . . . the x-intercepts are 2 and 8

2.  x = -3 ± √(-2) . . . . . . no solution; the roots are complex

3.  x = 5 ± √(-8/4) . . . . . no solution; the roots are complex

Answer:

Step-by-step explanation:

These are all done the exact same way.  I'll do the first one in its entirety, and you can do the rest, following my example.

Finding x-intercepts means that you find the places in the polynomial where the graph of the function goes through the x-axis.  Here, the y-coordinates will be 0.  To find these x-intercepts, you have to set y equal to 0 and then factor.  First, though, we need to know exactly what the polynomial looks like in standard form.  The ones you have are all in vertex form.  We find the standard form by first expanding the binomial, like this:

[tex]0=(x-5)(x-5)-9[/tex]

FOIL those out to get

[tex]x^2-10x+25-9=0[/tex]

Combine like terms to get

[tex]0=x^2-10x+16[/tex]

Now we have to factor that.  I'll use regular old factoring, although the quadratic formula will work also.

In our quadratic, a = 1, b = -10 and c = 16

The product of a * c = 16.  The factors of 16 are:

1, 16

2, 8

4, 4

Some combination of those factors will give us a -10, the b term.  2 and 8 will work, as long as they are both negative.  -2 + -8 = -10.  Fit them into the polynomial with the absolute value of the largest number named first:

[tex]x^2-8x-2x+16=0[/tex]

Now we group them by 2's without ever changing their order:

[tex](x^2-8)-(2x+16)=0[/tex]

and then factor out the common thing in each set of parenthesis.  The common thing in the first set of parenthesis is an x; the common thing in the second set is a 2:

[tex]x(x-8)-2(x-8)=0[/tex]

Now the common thing is (x - 8), so we factor that out and group together in a separate set of parenthesis what's left over:

[tex](x-8)(x-2)=0[/tex]

By the Zero Product Property, either x - 8 = 0 or x - 2 = 0.  Solving the first one for x:

x - 8 = 0 so x = 8

Solving the second one for x:

x - 2 = 0 so x = 2

The 2 solutions are x = 2 and x = 8, choices a and d.

The correct option is C. Never The product of (a - b)(a - b) is always equal to a² - 2ab + b².

The product of (a - b)(a - b) can be expanded using the distributive property:

(a - b)(a - b) = a(a) - a(b) - b(a) + b(b)

Simplify by multiplying the terms:

= a² - ab - ab + b²

Combine like terms:

= a² - 2ab + b²

As you can see, the product of (a - b)(a - b) is a² - 2ab + b², not a² - b².

However, there is a well-known algebraic identity called the difference of squares, which states that a² - b² can be factored as (a + b)(a - b). So, the correct statement is:

(a - b)(a - b) is equivalent to a² - 2ab + b², not a² - b².

Therefore, the answer is C. Never the product of (a - b)(a - b) is always equal to a² - 2ab + b².

To know more about product click here :

https://brainly.com/question/20297047

#SPJ2

Answer:

The correct option is B.

Step-by-step explanation:

It is given that Susan drove an average speed of 30 miles per hour for the first 30 miles of a trip & then at a average speed of 60 miles/hr for the remaining 30 miles of the trip.

[tex]Time=\frac{Distance}{Speed}[/tex]

Time taken by Susan in first 30 miles is

[tex]T_1=\frac{30}{30}=1[/tex]

Time taken by Susan in remaining 30 miles is

[tex]T_2=\frac{30}{60}=0.5[/tex]

Total distance covered by Susan is

[tex]D=30+30=60[/tex]

Total time taken by susan to complete 60 miles trip is

[tex]T=T_1+T_2[/tex]

[tex]T=1+0.5=1.5[/tex]

Susan's avg speed in miles/hr for the entire trip is

[tex]S=\frac{60}{1.5}=40[/tex]

The average speed of susan for entire trip is 40. Therefore the correct option is B.

Explanation:

1.  Sine and Cosine are never greater than 1 because they are the ratio of the leg of a right triangle to the hypotenuse. The hypotenuse is never shorter than one of the legs of a right triangle, so the ratio is at most 1.

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2. Secant and cosecant are the reciprocals of the cosine and sine (respectively), so are the reciprocals of numbers that are at most 1. Therefore, they can never be less than 1 (in magnitude).

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3. Tangent and cotangent (and secant and cosecant) are sometimes undefined because they are the ratios of triangle side lengths. The side length used in the denominator may be zero, causing the ratio to be undefined.