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3 years ago

If (-10,8) is a point on the terminal side f a angle 0 in standard position what is the value of tan0?

Mathematics

1 answer:

fiasKO [112]3 years ago

5 0

Answer:

The value of the tangent is $-\frac{4}{5}$ .

Step-by-step explanation:

Let be $A(x,y) = (-10, 8)$ on the terminal side of an angle in standard position and let be $O(x,y)$ the origin of the angle and the origin of the Cartesian plane. By definition of tangent, we make use of the following expression:

$\tan \theta = \frac{y_{A}-y_{O}}{x_{A}-x_{O}}$ (1)

Where:

$x_{A}, x_{O}$ - x-Coordinates of point A and point O.

$y_{A}, y_{O}$ - y-Coordinates of point A and point O.

If we know that $x_{A} = -10, x_{O} = 0$ , $y_{A} = 8$ and $y_{O} = 0$ , then the value of the tangent is:

$\tan \theta = \frac{y_{A}-y_{O}}{x_{A}-x_{O}}$

$\tan \theta = \frac{8-0}{-10-0}$

$\tan \theta = -\frac{4}{5}$

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I am having trouble with this relative minimum of this equation.<br>

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Answer:

So the approximate relative minimum is (0.4,-58.5).

Step-by-step explanation:

Ok this is a calculus approach. You have to let me know if you want this done another way.

Here are some rules I'm going to use:

$(f+g)'=f'+g'$ (Sum rule)

$(cf)'=c(f)'$ (Constant multiple rule)

$(x^n)'=nx^{n-1}$ (Power rule)

$(c)'=0$ (Constant rule)

$(x)'=1$ (Slope of y=x is 1)

$y=4x^3+13x^2-12x-56$

$y'=(4x^3+13x^2-12x-56)'$

$y'=(4x^3)'+(13x^2)'-(12x)'-(56)'$

$y'=4(x^3)'+13(x^2)'-12(x)'-0$

$y'=4(3x^2)+13(2x^1)-12(1)$

$y'=12x^2+26x-12$

Now we set y' equal to 0 and solve for the critical numbers.

$12x^2+26x-12=0$

Divide both sides by 2:

$6x^2+13x-6=0$

Compaer $6x^2+13x-6=0$ to $ax^2+bx+c=0$ to determine the values for $a=6,b=13,c=-6$ .

$a=6$

$b=13$

$c=-6$

We are going to use the quadratic formula to solve for our critical numbers, x.

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x=\frac{-13 \pm \sqrt{13^2-4(6)(-6)}}{2(6)}$

$x=\frac{-13 \pm \sqrt{169+144}}{12}$

$x=\frac{-13 \pm \sqrt{313}}{12}$

Let's separate the choices:

$x=\frac{-13+\sqrt{313}}{12} \text{ or } \frac{-13-\sqrt{313}}{12}$

Let's approximate both of these:

$x=0.3909838 \text{ or } -2.5576505$ .

This is a cubic function with leading coefficient 4 and 4 is positive so we know the left and right behavior of the function. The left hand side goes to negative infinity while the right hand side goes to positive infinity. So the maximum is going to occur at the earlier x while the minimum will occur at the later x.

The relative maximum is at approximately -2.5576505.

So the relative minimum is at approximate 0.3909838.

We could also verify this with more calculus of course.

Let's find the second derivative.

$f(x)=4x^3+13x^2-12x-56$

$f'(x)=12x^2+26x-12$

$f''(x)=24x+26$

So if f''(a) is positive then we have a minimum at x=a.

If f''(a) is negative then we have a maximum at x=a.

Rounding to nearest tenths here: x=-2.6 and x=.4

Let's see what f'' gives us at both of these x's.

$24(-2.6)+25$

$-37.5$

So we have a maximum at x=-2.6.

$24(.4)+25$

$9.6+25$

$34.6$

So we have a minimum at x=.4.

Now let's find the corresponding y-value for our relative minimum point since that would complete your question.

We are going to use the equation that relates x and y.

I'm going to use 0.3909838 instead of .4 just so we can be closer to the correct y value.

$y=4(0.3909838)^3+13(0.3909838)^2-12(0.3909838)-56$

I'm shoving this into a calculator:

$y=-58.4654411$

So the approximate relative minimum is (0.4,-58.5).

If you graph $y=4x^3+13x^2-12x-56$ you should see the graph taking a dip at this point.

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Step-by-step explanation:

Here we will show you how to calculate 8 divided by 1/6. We will give you the answer in fraction form and in decimal form.

Here is 8 divided by 1/6 displayed mathematically in colors:

The numbers in 8 divided by 1/6 are labeled below:

8 = whole number

1 = numerator

6 = denominator

To make it a fraction form answer, you multiply the whole number by the denominator and make the result the new numerator. The old numerator becomes the new denominator:

8 x 6

Thus, the answer to 8 divided by 1/6 in fraction form is:

To make the answer to 8 divided by 1/6 in decimal form, you simply divide the numerator by the denominator from the fraction answer above:

48 / 1 = 48

The answer is rounded to the nearest two decimal points if necessary.

48/1 is an improper fraction and should be written as 48.

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