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3 years ago

Sin(a+b) + sin (a-b) = 2 sin a cos b verify the identitiy

Mathematics

1 answer:

Bas_tet [7]3 years ago

6 0

Write out the sum formula for sin
sin(x + y) = sinxcos + sinycosx 

Then expand sin(a + b) + sin(a - b) 
sinacosb + sinbcosa + sinacosb - sinbcosa 
The 2nd and 4th terms cancel and you get 
2sinacosb

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Find the volume of the wedge with vertices at points (0,0,0), (1,0,0), (0,1,0), (0,0,1) by integrating the area of cross-section

Angelina_Jolie [31]

Answer:

V = 1/6 cubic units

Step-by-step explanation:

Applying the concept of integrals for volume calculation:

$V = \int\limits^b_a {S(x)} \, dx$ (1)

V = volume of the solid bounded by x = a and x = b

S(x) = cross section area of the solid, perpendicular to the x axis

From the figure we have that S is the area of a triangle that has base Z and height Y

Area of the triangle = $S(x)=\frac{y(x)*z(x)}{2}$ (2)

Calculation of y(x) and z(x)

We apply the equation of the point-slope line (plane xy):

Slope = $m = \frac{y_{2} - y_{1} }{x_{2} - x_{1}}$ (3)

Equation of the line = $y - y_{1} =m(x-x_{1} )$ (4)

Replacing the points (1,0) and (0,1) in (3):

$m=\frac{1-0}{0-1} =-1$

Replacing the point (1,0) and m = -1 in (4):

$y-0=(-1)(x-1)$

y(x) = -x + 1 (Line A-B) (5)

We apply the equation of the point-slope line (plane xz):

Slope = $m = \frac{z_{2} - z_{1} }{x_{2} - x_{1}}$ (6)

Equation of the line = $z - z_{1} =m(x-x_{1} )$ (7)

Replacing the points (1,0) and (0,1) in (6):

$m=\frac{1-0}{0-1} =-1$

Replacing the point (1,0) and m = -1 in (7):

$z-0=(-1)(x-1)$

z(x) = -x + 1 (Line A-C) (8)

Replacing (5) and (8) in (2)

$S(x) = \frac{(-x + 1) * (-x + 1)}{2} =\frac{(-x + 1)^{2} }{2}$ (9)

Replacing (9) in (1) and knowing that a = 0 and b = 1:

$V = \int\limits^1_0 {\frac{(-x + 1)^{2} }{2}} \, dx = \int\limits^1_0 {\frac{x^{2}-2x+1 }{2}} \, dx$

$V =\frac{1}{2} (\frac{x^{3} }{3} -2\frac{x^{2} }{2} +x)$ evaluated from x=0 to x=1

$V= \frac{1}{2} (\frac{1}{3} -1 +1) = \frac{1}{6}$

3 0

3 years ago

What is an example of proportional relationship

Dennis_Churaev [7]

Example: Oranges are sold in a bag of 5 for $2. The ratio of roanges to cost is 5:2 if i had 20 oranges for ? cost it would look like: 5/2=20/?

3 0

3 years ago

Find all solutions for the following equations. Write your answers in radians using exact values. (Enter your answers as a comma

sasho [114]

Answer:

Step-by-step explanation:

Given that:

sin2xcos x + cos2xsin x = -1

Recall that;

SInAcosB + cosASinB = SIn (A+B)

SIn (2x +x) = -1

Sin (3x) = -1

$x = \bigg [\dfrac{Sin ^{-1} (-1)}{3} \bigg ]$

$x = \dfrac{\pi}{2} + \dfrac{2 \pi \ k}{3}$

$x = \dfrac{\pi}{2}, \dfrac{7 \pi}{6}, \dfrac{11 \pi}{6}$

5 0

3 years ago

Use the Law of Sines to find the length of AC

ddd [48]

<h3>Answer: 30.78181 meters</h3>

The value is approximate. Round that however you need to.

========================================================

Explanation:

lowercase a = side opposite angle uppercase A
lowercase b = side opposite angle uppercase B
b = AC

Using the law of sines, we can say:

a/sin(A) = b/sin(B)

45/sin(30) = b/sin(20)

b/sin(20) = 45/sin(30)

b = sin(20)*45/sin(30)

b = 30.78181 approximately

You'll need to make sure your calculator is in degree mode.

3 0

2 years ago

Find the value of m .

kondor19780726 [428]

Step-by-step explanation:

if the 2 matrices are inverse, then their product must be the identity matrix

1 0

0 1

so,

m×3 + 2×-7 = 1

7×3 + 3×-7 = 0

m×-2 + 2×m = 0

7×-2 + 3×m = 1

that means we have to solve only

3m - 14 = 1

3m = 15

m = 5

7 0

2 years ago