All categories

3 years ago

Can anyone please show me how to find sin 18 in fraction form? plzzzzzzz $_$

Mathematics

1 answer:

Inessa [10]3 years ago

7 0

Answer:

(-1 + √5) / 4

Step-by-step explanation:

18° is 1/5 of 90°. So if we say x = 18:

5x = 90

You probably won't find an identity for sin(5x) in your textbook. But you will find double angle and triple angle formulas. So if we split 5x into two terms:

2x + 3x = 90

Rearrange:

2x = 90 − 3x

Take sine of both sides:

sin(2x) = sin(90 − 3x)

Use phase shift identity:

sin(2x) = cos(3x)

Apply double and triple angle formulas:

2 sin x cos x = 4 cos³ x − 3 cos x

Simplify:

0 = 4 cos³ x − 3 cos x − 2 sin x cos x

0 = cos x (4 cos² x − 3 − 2 sin x)

We know cos 18° isn't 0, so we can divide it out:

0 = 4 cos² x − 3 − 2 sin x

Using Pythagorean identity and simplifying:

0 = 4 (1 − sin² x) − 3 − 2 sin x

0 = 4 − 4 sin² x − 3 − 2 sin x

0 = 1 − 4 sin² x − 2 sin x

0 = 4 sin² x + 2 sin x − 1

Solve with quadratic formula (or you can complete the square):

sin x = [ -2 ± √(2² − 4(4)(-1)) ] / 2(4)

sin x = [ -2 ± √(4 + 16) ] / 8

sin x = (-2 ± √20) / 8

sin x = (-2 ± 2√5) / 8

sin x = (-1 ± √5) / 4

18° is in the first quadrant, so we know sin x > 0. Therefore:

sin x = (-1 + √5) / 4

You might be interested in

Which of the following are progressions?

professor190 [17]

In mathematics: Arithmetic progression, sequence of numbers such that the difference of any two successive members of the sequence is a constant. Geometric progression, sequence of numbers such that the quotient of any two successive members of the sequence is a constant. so i chose b)Sequences

7 0

3 years ago

Read 2 more answers

Regular hexagon ABCDEF has vertices at A(4, 4!3), B(8, 4!3), C(10, 2!3), D(8, 0), E(4, 0) and F(2, 2!3).

marissa [1.9K]

Since the given hexagon is a regular hexagon all it's sides will be of equal length. Now, we know that the Area of any regular hexagon is given by:

$A=\frac{3\sqrt{3}}{2} a^2$

Where $A$ is the area of the regular hexagon

$a$ is the side length of the regular hexagon

Also, it's Perimeter is given by:

$P=6a$

Thus, all that we need to do is to find the side length of any one of the sides and to do that let us have a look at at the data of vertices points given and find out which points are definitely adjacent to each other and are also easy to calculate.

A quick search will yield that D(8, 0) and E(4, 0) are definitely adjacent to each other.

Please check the attached file here for a better understanding of the diagram of the original regular hexagon. Points D and E indeed are adjacent to each other.

Let us now find the distance between the points D and E using the distance formula which is as:

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

Where $d$ is the distance.

$(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of points D and E respectively. (please note that interchanging the values of the coordinates will not alter the distance $d$ )

Applying the above formula we get:

$d=\sqrt{(8-4)^2+(0-0)^2} =\sqrt{4^2}=4$

$\therefore d=4$

We know that this distance is the side length of the given regular hexagon.

$\therefore d=a=4$

Now, if the sides of the given regular polygon are reduced by 40%, then the new length of the sides will be:

$a_{small}=4-\frac{40}{100}\times 4=2.4$

Thus, the area of the smaller hexagon will be:

$A_{small}=\frac{3\sqrt{3}}{2} a_{small}^2=\frac{3\sqrt{3}}{2} (2.4)^2\approx14.96$ unit squared

and the new smaller perimeter will be:

$P_{small}=6a_{small}=6\times 2.4=14.4$ unit

Which are the required answers.

5 0

3 years ago

Mariza é costureira e, para determinar o comprimento de elástico, em cm, que utiliza na confecção de determinado modelo de calça

tresset_1 [31]

X = 40, Y = 15
40(2) + 15(2) = 80 + 30 = 110 :)

8 0

3 years ago

how many liters is 70% alcohol solution must be added to 50 L of 40% alcohol solution to produce a 50% alcohol solution

Anettt [7]

Answer:

50%

Step-by-step explanation:

This question is solved by C1V1 + C2V2 ÷ V1 + V2 = final concentration

Let x = volume of 70% solution

let's express the concentrations as decimals, so 70% = 0.7, 50% = 0.5, and 40% = 0.4

then 0.7X + 50L.4 ÷ X + 50L = 0.5

this equation becomes 0.5(X + 50) = 0.7X + 20

which yields 0.5X + 25 = 0.7X +20

combining, we get 5 = 0.2X, and 25 = X

when we mix the two volumes, we get 50L x 40% + 25L x 70%,

which yields 20L of 100% alcohol, and 17.5L of 100% alcohol, for a total of 37.5L of alcohol in 75L of solution

which is a final concentration of 50%

3 0

3 years ago

WILL MARK BRAINLIEST! PLEASE HELP

zmey [24]

Part 1) In triangle PQR, mP= 43, PQ=7.5 and PR= 8.4. What is mR to the nearest degree?

we know that
the law of cosines establishes that
c²=a²+b²-2*a*b*cos C

in this problem

c=QR
a=PQ=7.5
b=PR=8.4
C=angle P=43°
c²=a²+b²-2*a*b*cos C----> c²=7.5²+8.4²-2*7.5*8.4*cos 43°
c²=34.66--------> c=5.89
QR=5.9

applying the law of sines
7.5/sin R=5.9/sin 43-----> 7.5*sin 43=5.9*sin R---> sin R=7.5*sin 43/5.9
sin R=0.8669---------> R=arc sin (0.8669)----> R=60.11°----> R=60°

the answer Part 1) is the option
D. 60

Part 2) In triangle ABC mA=43, mB=62 and BC=22 in. What is AB to the nearest tenth of an inch?

we know that
∠C=180-(62+43)-----> ∠C=75°

Applying the law of sines
22/sin 43=AB/sin 75------> AB=22*sin 75/ sin 43-----> AB=31.16 in
AB=31.2 in

the answer Part 2) is
D.31.2

4 0

3 years ago

Can anyone please show me how to find sin 18 in fraction form? plzzzzzzz $_$​

Can anyone please show me how to find sin 18 in fraction form? plzzzzzzz $_$