Find the area of the shaded region.

Step-by-step explanation:

Hi there!

Please see the answer in the picture.

Hope it helps!

1. Approach

One is given a trigonometric equation with and one is asked to prove that it is true. Using the attached image, combined with the knowledge of trigonometry, one can evaluate each trigonometric function. Then one can simplify each ratio to solve. To yield the most accurate result, one has to each of the ratios in a fractional form, rather than simplifying it into a decimal form. Remember the right angle trigonometric ratios, these ratios describe the relationship between the sides and angles in a right triangle. Such ratios are as follows,

[tex]sin(\theta)=\frac{opposite}{hypotenuse}\\\\cos(\theta)=\frac{adjacent}{hypotenuse}\\\\tan(\theta)=\frac{opposite}{adjacent}\\\\csc(\theta)=\frac{hypotenuse}{opposite}\\\\sec(\theta)=\frac{hypotenuse}{adjacent}\\\\cot(\theta)=\frac{adjacent}{opposite}[/tex]

Please note that the terms (opposite) and (adjacent) are relative to the angle uses in the ratio, however the term (hypotenuse) refers to the side opposite the right angle, this side never changes its name. Use these ratios to evaluate the trigonometric functions. Then simplify to prove the identity.

2. Problem (9)

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

As per the attached image, the following statements regarding the value of each ratio can be made:

[tex]sin(60)=\frac{\sqrt{3}}{2}\\\\cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cos(60)=\frac{1}{2}[/tex]

Substitute,

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

Simplify,

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}[/tex]

Thus, this equation is true.

2. Problem (10)

Use a similar strategy to evaluate this equation,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

Use the attached image to evaluate the ratios.

[tex]cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cot(60)=\frac{1}{\sqrt{3}}[/tex]

Substitute,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

Simplify,

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}+1}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

Rationalize the denominator,

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}*\frac{\sqrt{3}-1}{\sqrt{3}-1}[/tex]

[tex]2-\sqrt{3}=\frac{(\sqrt{3}-1)^2}{3-1}[/tex]

[tex]2-\sqrt{3}=\frac{3-2\sqrt{3}+1}{2}[/tex]

[tex]2-\sqrt{3}=\frac{4-2\sqrt{3}}{2}[/tex]

[tex]2-\sqrt{3}=2-\sqrt{3}[/tex]

Therefore, this equation is also true.

Answer:

1≤f(x)≤5

Step-by-step explanation:

-1≤x≤1

-2≤2x≤2 (Multiplied by 2 both side)

-2+3≤2x+3≤2+3 (Adding three both sides)

1≤f(x)≤5

Answer:

A " (1,-2)

B " (4,0)

C " (6,-3)

Step-by-step explanation:

Hope it helped.

° ° °

Answer:

Step-by-step explanation:

so we know a few things about t his triangle,  it's a special one  :P  ,   it's called isosceles because  two of the legs are the same length, which also mean that the angle x  is the same at the other unknown angle sooo  there are  2 x's   if that makes sense?  and then we can solve this , b/c we also know that the interior angles of a triangle add up to 180°

180 = 32 + 2x

148 = 2x

74 =x

∠x = 74°

Answer:

x=61.71428 or 61 5/7

Step-by-step explanation:

We can use a ratio to solve

7 pieces              12 pieces

-----------------   = ---------------

36 seconds          x seconds

Using cross products

7x = 36*12

7x = 432

Divide by 7

7x/7 = 432/7

61 5/7

x=61.71428

Answer:

66

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

Let's simplify step-by-step.

−2z2+4z+2z2

Combine Like Terms:

=−2z2+4z+2z2

=(−2z2+2z2)+(4z)

=4z

Answer:

4z

Hope this helps <3 Need thanks?

Comment /hearthelp

[tex]\\ \sf\longmapsto -2z^2+4z+2z^2[/tex]

[tex]\\ \sf\longmapsto -2z^2+2z^2+4z[/tex]

[tex]\\ \sf\longmapsto (-2+2)z^2+4z[/tex]

[tex]\\ \sf\longmapsto 0z^2+4z[/tex]

[tex]\\ \sf\longmapsto 4z[/tex]

Answer:

O D. 3x2 + 2x-3

Answer:

A.7x2-5x questions 2of 20

Answer:

Step-by-step explanation:

In one hour pump1 can fill 1/8 of the tank and pump2 can fill 1/10 of the tank.

Two pumps can fill:

Time required to fill the tank:

Answer:

90°,60°&30°

Step-by-step explanation:

A 30-60-90 triangle is a right triangle with angle measures of 30º, 60º, and 90º (the right angle). Because the angles are always in that ratio, the sides are also always in the same ratio to each other.

Answer:

6.8 gallions i believe. im not quite sure

Answer:

A^7

Step-by-step explanation:

Answer:

240

Step-by-step explanation:

We are given a right triangle. Based on the leg rule, the following equation shows how the length of a leg in a right triangle relates with the segments connected to the hypotenuse:

Hypotenuse/leg = leg/part

Where,

Hypo = 400

Leg = x

Part = 144

Substitute

400/x = x/144

Cross multiply

400*144 = x*x

57,600 = x²

√57,600 = x

240 = x

x = 240

Answer:

-1,2 can be plotted together with these four points ...

Answer:

the answer is option 1

Step-by-step explanation:

the negative angle => a quarter round angle (clockwise) = ¼ x ( -360°) = -90°

and

the positive angle => a full round angle + 270° = (360°)+270°

= 630°

Answer:

correct answer is -90 and 630

Step-by-step explanation:

Answer:

apoco la propiedad asociativa en los siguiente ejercicio 25x11x18=

Answer:

BD = 10 √3

Step-by-step explanation:

ABC ∆ ,

BD ÷ 20 = sin 60

BD = 20 sin 60

BD = (20 √3) / 2

= 10 √3

Answer:

k = 14

Step-by-step explanation:

Prime factorize 32

32 = 2 * 2 * 2 * 2 * 2  = 2⁵

[tex]\frac{2^{12}}{2^{\frac{k}{2}}}= 32\\\\\frac{2^{12}}{2^{\frac{k}{2}}}=2^{5}\\\\2^{12-\frac{k}{2}}=2^{5}[/tex]

Both sides base are same.So, compare exponents

[tex]12-\frac{k}{2}=5\\[/tex]

Subtract 12 from both side

[tex]-\frac{k}{2}=5-12\\\\-\frac{k}{2}=-7\\[/tex]

Multiply both sides by (-2),

[tex](-2)*(-\frac{k}{2})=-7*(-2)\\\\k = 14[/tex]

Answer:

Tb bight by icing I jb Yves j by by by navy by iffy i by by by

Answer:

1/4a -1/6b + 1/10c

Step-by-step explanation:

1/2 a  -1/3 b + 1/5c   + -1/4a + 1/6 b - 1/10c

Combine like terms

1/2a - 1/4a   -1/3b + 1/6b    +1/5c - 1/10 c

2/4a -1/4a    -2/6b + 1/6b   +2/10c -1/10c

1/4a -1/6b + 1/10c

Answer:

(-1,-6)

Step-by-step explanation:

(13 + x)/2 = 6

13+x= 12

x = -1

~~~~~~~~~~~~~~~

(-2 + y ) / 2 = -4

-2 + y = -8

y = -6

The coordinates of the other endpoint will be (-1,-6). The correct option is C.

Divide the measurement of the distance between the two end locations by 2. The middle of that line is located at this separation from either end.

A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis.

Given that the midpoint of a segment is (6,−4) and one endpoint is (13,−2).

The x- coordinate will be calculated as:-

(13 + x)/2 = 6

13+x= 12

x = -1

The y-coordinate will be calculated as:-

(-2 + y ) / 2 = -4

-2 + y = -8

y = -6

Therefore, the coordinates of the other endpoint will be (-1,-6). The correct option is C.

To know more about midpoints of the line follow

https://brainly.com/question/24431553

#SPJ2

Answer:

x = ± 10

Step-by-step explanation:

To solve the equation let y = 0 , that is

0 = 100 - x²( add x² to both sides )

x² = 100 (I take the square root of both sides )

x = ± [tex]\sqrt{100}[/tex] = ± 10

Then solution is x = - 10, x = 10

Answer:

10

Step-by-step explanation:

the guy on top put -10 he's right but it's just 10

Answer:

The area of the circle is 100 square units.

Step-by-step explanation:

We are given that the circumference of a circle is 20π, and we want to determine its area.

Recall that the circumference of a circle is given by the formula:

[tex]\displaystyle C = 2\pi r[/tex]

Substitute:

[tex]20 \pi = 2 \pi r[/tex]

Solve for the radius:

[tex]\displaystyle r = \frac{20\pi}{2\pi} = 10[/tex]

The area of a circle is given by:

[tex]\displaystyle A = \pi r^2[/tex]

Since the radius is 10 units:

[tex]\displaystyle A = \pi (10)^2[/tex]

Evaluate:

[tex]\displaystyle A = 100\pi\text{ units}^2[/tex]

In conclusion, the area of the circle is 100 square units.

Answer:

i use logical reasoning

Answer:

It's a combination, BUT...

Step-by-step explanation:

Definition: A combination is a grouping of outcomes in which the order does not matter. That's the difference between combination and permutation.

Combination Formula: nCr = (n!)/[r!(n-r)!)] where n stands for number of things and r the pair chosen

Applied to this case: We have n=5 (12345) and r=5 for the 5-digit lock combination.

5C5 = (5!)/[5!(5-5)] = 1

But!! As you can see, we cannot have 1 possible combinations so.. We have to assume that this is a permutation!!! Conceptually, this looks pretty much like a combination, but being the same range as number of items, we conclude that it's a permutation.

Permutation Formula: nPr = (n!)/(n-r)!

nPr = (5!)/(5-5)! = 120

Well, this looks way more accurate now, right?

Final Answer: The scenario involves a permutation, and there's 120 possible combinations of those 5 numbers in a 5-digit lock.

Answer:

on my screen I cant see anything sorry!

Step-by-step explanation:

Answer:

Solution given:

x^5=225

we have

x=[tex] \sqrt[5]{225} [/tex]

x=2.9541

Hello!

[tex] \bf {x}^{5} = 225[/tex]

[tex] \bf x = \sqrt[5]{225} [/tex]

[tex] \bf x ≈2.95418[/tex]

Answer: x ≈ 2,95418

Good luck! :)