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Nastasia [14]
3 years ago
10

Prove that 1+cosθ+sinθ/1+cosθ-sinθ = 1+sinθ/cosθ

Mathematics
1 answer:
marusya05 [52]3 years ago
3 0

Step-by-step explanation:

(1 + cos θ + sin θ) / (1 + cos θ − sin θ)

Multiply by the reciprocal:

(1 + cos θ + sin θ) / (1 + cos θ − sin θ) × (1 + cos θ + sin θ) / (1 + cos θ + sin θ)

(1 + cos θ + sin θ)² / [ (1 + cos θ − sin θ) (1 + cos θ + sin θ) ]

(1 + cos θ + sin θ)² / [ (1 + cos θ)² − sin² θ) ]

Distribute and simplify:

(1 + cos θ + sin θ)² / (1 + 2 cos θ + cos² θ − sin² θ)

[ 1 + 2 (cos θ + sin θ) + (cos θ + sin θ)² ] / (1 + 2 cos θ + cos² θ − sin² θ)

(1 + 2 cos θ + 2 sin θ + cos² θ + 2 sin θ cos θ + sin² θ) / (1 + 2 cos θ + cos² θ − sin² θ)

Use Pythagorean identity:

(2 + 2 cos θ + 2 sin θ + 2 sin θ cos θ) / (sin² θ + cos² θ + 2 cos θ + cos² θ − sin² θ)

(2 + 2 cos θ + 2 sin θ + 2 sin θ cos θ) / (2 cos² θ + 2 cos θ)

(1 + cos θ + sin θ + sin θ cos θ) / (cos² θ + cos θ)

Factor:

(1 + cos θ + sin θ (1 + cos θ)) / (cos θ (1 + cos θ))

(1 + cos θ)(1 + sin θ) / (cos θ (1 + cos θ))

(1 + sin θ) / cos θ

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5. A community neighborhood wants to
Nastasia [14]

Answer:

51 trees

Step-by-step explanation:

Start at one end of the 50-foot line segment.

At position 0 ft, put one tree.

Then 1 ft from 0 ft, put tree number 2.

One more foot over, at position 2 ft from the start, put tree number 3.

Notice that each tree number is one more than the number of feet.

That means at 50 ft from the stat, you put tree number 51.

Answer: 51 trees

8 0
3 years ago
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficie
melamori03 [73]

Answer:

The complete solution is

\therefore y= Ae^{3x}+Be^{-\frac13 x}-\frac43

Step-by-step explanation:

Given differential equation is

3y"- 8y' - 3y =4

The trial solution is

y = e^{mx}

Differentiating with respect to x

y'= me^{mx}

Again differentiating with respect to x

y''= m ^2 e^{mx}

Putting the value of y, y' and y'' in left side of the differential equation

3m^2e^{mx}-8m e^{mx}- 3e^{mx}=0

\Rightarrow 3m^2-8m-3=0

The auxiliary equation is

3m^2-8m-3=0

\Rightarrow 3m^2 -9m+m-3m=0

\Rightarrow 3m(m-3)+1(m-3)=0

\Rightarrow (3m+1)(m-3)=0

\Rightarrow m = 3, -\frac13

The complementary function is

y= Ae^{3x}+Be^{-\frac13 x}

y''= D², y' = D

The given differential equation is

(3D²-8D-3D)y =4

⇒(3D+1)(D-3)y =4

Since the linear operation is

L(D) ≡ (3D+1)(D-3)    

For particular integral

y_p=\frac 1{(3D+1)(D-3)} .4

    =4.\frac 1{(3D+1)(D-3)} .e^{0.x}    [since e^{0.x}=1]

   =4\frac{1}{(3.0+1)(0-3)}      [ replace D by 0 , since L(0)≠0]

   =-\frac43

The complete solution is

y= C.F+P.I

\therefore y= Ae^{3x}+Be^{-\frac13 x}-\frac43

4 0
3 years ago
Please help will give brainliest
garri49 [273]

Answer:

b = 17

Step-by-step explanation:

For this triangle we have to

a=18.2\\B=62\°\\C=48\°

We want to find the length of b

We know that the sum of the internal angles of a triangle is 180 °

So

A + 62 +48=180\\\\A=180-62-48\\\\A=70\°

Now we use the sine theorem to find the length of b:

\frac{sin(A)}{a}=\frac{sin(B)}{b}=\frac{sin(C)}{c}

Then:

\frac{sin(A)}{a}=\frac{sin(B)}{b}\\\\b=\frac{sin(B)}{\frac{sin(A)}{a}}\\\\b=a*\frac{sin(B)}{sin(A)}\\\\b=(18.2)\frac{sin(62)}{sin(70)}\\\\b=17

5 0
3 years ago
Use the drop down menus to complete the statements to match the information shown<br> by the graph.
jenyasd209 [6]

Answer:

less than, slope, higher (the rest in not sure)

Step-by-step explanation:

just guessing, could you please show the drop downs??

8 0
3 years ago
Read 2 more answers
The perimeter of a rectangle is
kondor19780726 [428]
<h3 /><h3>verified \: answer  \sqrt}</h3>

Equation for Perimeter of a rectangle: Perimeter = 2W + 2L

<h3>Defining the variables, let</h3>

<h3>Width = x</h3><h3>Length = 2x+3 (3 more than twice the width)</h3>

<h3>Plugging everything into the equation</h3>

<h3>30= 2(x) + 2(2x+3) using the distributive property,</h3>

<h3>30=2x+4x+6 combining like terms</h3>

<h3>30=6x+6 subtracting 6 from both sides,</h3>

<h3>24=6x divide both sides by 6</h3>

<h3>4=x This means that the width is 4 m.</h3>

<h3>To get the length, use the expression L=2x+3 and plug in x = 4 that was already solved for</h3>

<h3>L=2(4)+3</h3>

<h3>L=8+3 = 11 m</h3>

<h3>So the dimensions of the rectangle are width is 4 m and length is 11 m.</h3>

8 0
3 years ago
Read 2 more answers
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