Answers
Answer:
see explanation
Step-by-step explanation:
Using the identity
cos2Θ = 1 - 2sin²Θ, then
1 - 2sin²([tex]\frac{\pi }{4}[/tex] - [tex]\frac{0}{2}[/tex] )
= cos [2([tex]\frac{\pi }{4}[/tex] - [tex]\frac{0}{2}[/tex] )]
= sos([tex]\frac{\pi }{2}[/tex] - Θ )
= cos[tex]\frac{\pi }{2}[/tex]cosΘ + sin
= 0 × cosΘ + 1 × sinΘ
= 0 + sinΘ
= sinΘ = right side
Answer: see proof below
Step-by-step explanation:
Use the Difference Identity: sin (A + B) = sin A cos B - cos A sin B
Use the following Half-Angle Identities:
[tex]\sin\bigg(\dfrac{A}{2}\bigg)=\sqrt{\dfrac{1-\cos A}{2}}\\\\\cos\bigg(\dfrac{A}{2}\bigg)=\sqrt{\dfrac{1+\cos A}{2}}[/tex]
Use the Pythagorean Identity: cos²A + sin²A = 1 --> sin²A = 1 - cos²A
Use the Unit Circle to evaluate: [tex]\cos\dfrac{\pi}{4}=\sin\dfrac{\pi}{4}=\dfrac{1}{\sqrt2}[/tex]
Proof LHS → RHS
[tex]\text{Given:}\qquad \qquad \qquad 1-2\sin^2\bigg(\dfrac{\pi}{4}-\dfrac{\theta}{2}\bigg)\\\\\text{Difference Identity:}\quad 1-2\bigg(\sin\dfrac{\pi}{4}\cdot \cos \dfrac{\theta}{2}-\cos \dfrac{\pi}{4}\cdot \sin\dfrac{\theta}{2}\bigg)^2\\\\\text{Unit Circle:}\qquad \qquad 1-2\bigg(\dfrac{1}{\sqrt2}\cos \dfrac{\theta}{2}-\dfrac{1}{\sqrt2}\sin \dfrac{\theta}{2}\bigg)^2\\\\\\\text{Half-Angle Identity:}\quad 1-2\bigg(\dfrac{\sqrt{1+\cos A}}{2}-\dfrac{\sqrt{1-\cos A}}{2}\bigg)^2[/tex]
[tex]\text{Expand Binomial:}\quad 1-2\bigg(\dfrac{1+\cos A}{4}-\dfrac{2\sqrt{1-\cos^2 A}}{4}+\dfrac{1-\cos A}{4}\bigg)\\\\\text{Simplify:}\qquad \qquad \quad 1-2\bigg(\dfrac{2-2\sqrt{1-\cos^2 A}}{4}\bigg)\\\\\text{Pythagorean Identity:}\quad 1-\dfrac{1}{2}\bigg(2-2\sqrt{\sin^2 A}\bigg)\\\\\text{Simplify:}\qquad \qquad \qquad 1-\dfrac{1}{2}(2-2\sin A)\\\\\text{Distribute:}\qquad \qquad \qquad 1-(1-\sin A)\\\\.\qquad \qquad \qquad \qquad \quad =1-1+\sin A\\\\\text{Simplify:}\qquad \qquad \qquad \sin A[/tex]
RHS = LHS: sin A = sin A [tex]\checkmark[/tex]
Related Questions
Rectangle A has length 12 and width 8. Rectangle B has length 15 and width 10. Rectangle C has length 30 and width 15.Explain how you know that Rectangle C is not a scaled copy of Rectangle B.
Answers
Step-by-step explanation:
Because the ratio of length:width is..
15:10 for b
and
30:15 for c.
Now we will divide..
3:2 for b
and
2:1 for c.
Those are NOT the same things!
A set of points forming a closed figure, every point of which is a fixed distance from a given center best defines which shape? *
Rectangle
Square
Rhombus
Circle
Answers
Answer: circle
Step-by-step explanation:
because each point is the same from the center
please someone help me..!!!!
Answers
Answer: see proof below
Step-by-step explanation:
Use the Triple Angle Identity: sin 3A = 3sinA - 4sin³A
[tex]\text{Given:}\quad \sin\bigg(\dfrac{\theta}{3}\bigg)=\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)[/tex]
Proof LHS → RHS
LHS: sin Ф
Let A = Ф/3 sin 3A
Triple Angle Identity: 3sinA - 4sin³A
Substitute Ф/3 = A: 3sin(Ф/3) - 4sin³(Ф/3)
Substitute sin(Ф/3): [tex]3\bigg(\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)\bigg)-4\bigg(\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)\bigg)^3[/tex]
[tex]\text{Simplify:}\qquad \qquad \dfrac{3m}{2}+\dfrac{3}{2m}-4\bigg(\dfrac{1}{8}\bigg(m^3+3m+\dfrac{3}{m}+\dfrac{1}{m^3}\bigg)\bigg)\\\\.\qquad \qquad \qquad =\dfrac{3m}{2}+\dfrac{3}{2m}-\dfrac{m^3}{2}-\dfrac{3m}{2}-\dfrac{3}{2m}-\dfrac{1}{2m^3}\\\\.\qquad \qquad \qquad =-\dfrac{m^3}{2}-\dfrac{1}{2m^3}\\\\\text{Factor:}\qquad \qquad -\dfrac{1}{2}\bigg(m^3+\dfrac{1}{m^3}\bigg)[/tex]
[tex]\text{LHS = RHS:}\quad -\dfrac{1}{2}\bigg(m^3+\dfrac{1}{m^3}\bigg)=-\dfrac{1}{2}\bigg(m^3+\dfrac{1}{m^3}\bigg)\quad \checkmark\\[/tex]
Evaluate 6 - 2(-1) +1|-5|= 1
a.9
b.1
c.13
Answers
Answer:
c
Step-by-step explanation:
Which set of side length is a Pythagorean triple? A. 2,3,13 b. 5,7,12 C. 10, 24, 29 D. 11,60,61
Answers
Answer:
B. 5,7,12 is what I got.
In the equation 2y=18, determine which property of equality will be used to isolate the variable. Choose the best answer. A. Multiplication Property of Equality B, Addition Property of Equality C. Division Property of Equality D. Subtraction Property of Equality
Answers
Answer:
(C) Division Property of Equality
Step-by-step explanation:
Looking at the equation [tex]2y=18[/tex], we see that if we want to isolate y, we need to divide it by 2 - because we're multiplying it by 2, and 2 times a number divided by 2 is just the number.
The Division Property of Equality states that if we have an equation, [tex]a=b[/tex], then [tex]a\div c = b\div c[/tex].
So, that means [tex]2y\div 2[/tex] will be equal to [tex]18 \div 2[/tex], which is 9.
Hope this helped!
Answer:
C. Division Property of Equality
Step-by-step explanation:
The equation is
2y = 18
The answer is of the form
y = a number
Now, we have 2y. We want y alone. 2y means 2 multiplied by y. Since we want y alone, to undo the multiplication of 2 and y, we need to use division. We need to divide 2y by 2. Then we also need to divide the right side by 2. Since we are using division, the answer is:
C. Division Property of Equality
I don't understand this problem...like how to combine the k terms or how to get the answer.
Answers
Answer:
-6 k +5
Step-by-step explanation:
-2k -4k +5
Think of k is apples
-2 apples -4 apples
-6 apples
-6 k +5
Answer:
Hey there!
It's totally fine if you are having trouble with these kinds of questions, especially if you're new to them.
We can simplify the problem to be -2k-4k+5. Treat k as some number, so we have -6k+5.
Let me know if this helps :)
Solve x+y+z=10 −2x−2y−z=−32 3x−2y+2z=−8 Enter your answer, in the form (x,y,z), in the boxes in simplest terms. x= y= z=
Answers
Answer:
(12, 10, -12)
x = 12
y = 10
z = -12
Step-by-step explanation:
Let's start by picking a variable to eliminate from the three equations, since "z" is at the end, let's go with "z".
Next we'll eliminate z from two of the equations by adding them together so:
(x+y+z=10) + (-2x-2y-z=-32)
you'll get: -x-y=-22
Next we'll eliminate z from the remaining equation in the same way so:
(-2x-2y-z=-32) + (3x-2y+2z=-8)
you'll get: -x-6y=-72
We now have two new equations:
-x-y=-22
-x-6y=-72
From these let's eliminate another variable, let's go with "x" and since both "x" are negative we'll multiply one of the equations by the negative sign of the other equation so we'll end up with one positive x and one negative x, which looks like this:
-x-y=-22
x+6y=72
and when we add these two equations we get:
5y=50
Finally we can solve for y:
y=10
With the known value of y we can use y to plug into the other equations and solve for the x and z variables to get x=12, y=10, z=-12
Answer it answer it answer it
Answers
Answer:
It sooo D
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
How to find the value of :-
[tex] \sqrt{35} [/tex]
[tex] \sqrt{2} [/tex]
Answers
Q1. Find y, if √12 - √147 + y√3 = 0
Q2. Without using tables or calculators, find the value of Sin45° - Cos30°
Hints: Sin45° = √2/2 Cos 30° = √3/2.
Answers
Answer:
Q1. y = 5 Q2. (√2 - √3)/2
Step-by-step explanation:
Q1
√12 - √147 + y√3 = 0
Taking -√147 and √12 to the right hand side, we have
y√3 = √147 - √12
y√3 = √(3 × 49) - √(3 × 4)
y√3 = √3 × √49 - √3 × √4
y√3 = √3 × 7 - √3 × 2
y√3 = 7√3 - 2√3
y√3 = 5√3
Dividing both sides by √3, we have
y√3/√3 = 5√3/√3
y = 5
Q2
Sin45° - Cos30°
Since Sin45° = √2/2 and Cos 30° = √3/2
Substituting these values into the equation, we have
Sin45° - Cos30° = √2/2 - √3/2
Taking L.C.M of both factors, we have
Sin45° - Cos30° = (√2 - √3)/2
Given the area, find the dimension of a rectangle
The width of a rectangle is 30 feet less than twice its length. its area is 68 square feet. Find the dimensions of the rectangle.
Answers
Answer:
Length=17
Width=4
Step-by-step explanation:
Length=x
Width=2x-30
x(2x-30)=68
2x²-30x=68
2x²-30x-68=0
2(x²-15x-34)=0
2(x+2)(x-17)=0
So x equals either 17 or -2. Since it can’t be -2, x=17
Length=x=17
Width=2x-30=2*17-30=34-30=4
Answer:
Length L = 17 ft.
Width W = 4 ft
Step-by-step explanation:
Area of a rectangle = L * W
Area = 68 ft²
W = 2L - 30
Area = L * W
68 = L * (2L - 30)
68 = 2L² - 30L
2L² - 30L -68 = 0 ( solve L by using quadratic equation)
-(-30) ± √ -30² - 4 * 2 * (-68)
L = -------------------------------------------
2 * 2
L = 17 ; L= - 2 therefore use L = 17 ft.
solve for W:
W = 2L - 30
W = 2(17) - 30
W = 34 - 30
W = 4 ft
check:
Area = L * W
68 = 17 * 4
68 = 68 ----- OK
Pls halp due today >_<. Thank you
Answers
Answer:
5√8, 14.025, 40/3, √170
Step-by-step explanation:
Step 1: Evaluate √170
√170 = 13.0384
Step 2: Evaluate 5√8
5√8 = 14.1421
Step 3: Evaluate 40/3
40/3 = 13.333
Step 4: Align greatest to least
14.1421 > 14.025 > 13.333 > 13.0384
Step 5: Revert
5√8, 14.025, 40/3, √170
Write the number name and expanded form for the number of driver ants that could be in two colonies. Up to 22,000,000 driver of ants can live in a single colony
Answers
Answer: number name = 22 million
Expanded form: 22,000,000 = 20,000,000+2,000,000
Step-by-step explanation:
To write the number name and expanded form for the number of driver ants that could be in two colonies.
Given: Up to 22,000,000 driver of ants can live in a single colony
Since 1, 000, 000 =1 million
So 22,000,000= 22 million
Expanded form: 22,000,000 = 20,000,000+2,000,000
Hence, number name = 22 million
Expanded form: 22,000,000 = 20,000,000+2,000,000
-7•(-9)
need help on this one with full work including
Answers
Answer:
-7×-9
=63
Step-by-step explanation:
If u multiple - with -, I'll get + sign
Answer:
63
Step-by-step explanation:
-7*(-9)
7*9
63
Best of Luck!
find the area of square of one of its side is (3x+4y)cm.If x=4 and y=1,find its actual area.
Answers
Answer:
Step-by-step explanation:
hello :
the area is : (3x+4y)²cm² when :x=4 and y=1, the area is : (3(4)+4(1))²cm² =16²cm2=256cm²
Which expression is equivalent 4/5 - 1/3?
Answers
Answer:
7/15
Step-by-step explanation:
Well 4/5 - 1/3 = 12 / 15 - 5 / 15 which is equal to 7/15, but there aren't any expressions in your question.
Answer:
7/15
Step-by-step explanation:
4/5 - 1/3 = 12 / 15 - 5 / 15 = 7/15
Nancy bought 8 amusement park tickets for $82. About how much did each ticket cost?
Answers
Answer:
10.25
Step-by-step explanation:
Answer:
the answer would be 10.25
PLEASE HELP! I will mark brainliest!
Answers
Answer:
D
Step-by-step explanation:
Instead of going through each of the table, let's create one ourselves.
So, we have the piecewise function:
[tex]f(x)=-x+8,\text{ if } x<4\\f(x)=x, \text{ if } x\geq 4[/tex]
So, let's create a table of values starting with 2 and skipping the odd numbers until 12.
2:
For 2, since 2 is less than one, plug it into the first equation. Thus:
[tex]f(2)=-(2)+8=6[/tex]
So, the first value is (2,6).
For 4, since 4 is not less than 4 but rather equal to 4, use the second equation. Thus:
[tex]f(4)=(4)=4[/tex]
The second value is (4,4).
For 6, the same thing. 6 is greater than 4 so use the second equation:
[tex]f(6)=(6)=6[/tex]
So the third value is (6,6).
And this pattern will repeat. Therefore, our table of equations is:
x | y
2 | 6
4 | 4
6 | 6
8 | 8
10 | 10
12 | 12
The choice that represents this is D. D is the correct answer.
Chloe George
If m_ABD = 83° and m_ABC = (x + 10) ° and mZDBC = (4x - 12), what is the measure of each angle?
Answers
Answer:
ABC = 27
DBC = 56
Step-by-step explanation:
ABC + CBD = ABD
x+10 + 4x-12 = 83
Combine like terms
5x -2 = 83
Add 2 to each side
5x-2+2 = 83+2
5x = 85
Divide each side by 5
5x/5 = 85/5
x =17
ABC = x+10 = 17+10 = 27
DBC = 4x-12 = 4*17 -12 = 68-12 = 56
Answer:
Angle ABC = 27°
Angle DBC = 56°
Step-by-step explanation:
So we have to set an equation first.
x + 10 + 4x - 12 = 83
This simplified = 5x - 2 = 83
--> 5x = 85
so x = 17
Plug it in to the formula for ABC and DBC
ABC = x + 10
17 + 10 = 27
ABC = 27
DBC = 4x - 12
68 - 12
DBC = 56
Thus, we have our 2 answers.
Hope this helps!
area is calculated by:
Answers
Answer:
multiplying the length and the width.
Step-by-step explanation:
To find the area of a rectangle, multiply its height by its width.
For a square you only need to find the length of one of the sides then multiply this by itself to find the area.
What is the product of (negative 6) (negative 7) (negative 1)?
Answers
Answer:
-42
Step-by-step explanation:
negative * negative = positive
positive * negative = negative
negative * positive = negative
(-6)(-7)(-1) =
= 42(-1)
= -42
Answer:
-42
Step-by-step explanation:
negative * negative = positive
positive * negative = negative
negative * positive = negative
(-6)(-7)(-1) =
= 42(-1)
= -42
Simplify the expression.
-b + 2 + 9b
Answers
Answer:
8b+2
Step-by-step explanation:
Combine Like Terms:
=−b+2+9b
=(−b+9b)+(2)
=8b+2
{ (-2, 4), (0, 2), (-1, 3), (4, -2) } The set of ordered pairs above: A mapping diagram has 2 boxes. Box 1 is labeled domain with entries negative 3, 4, negative 1, 5. Box 2 is labeled range with entries 3, 7, negative 2. Arrows show which domain value match onto which range value. An arrow goes from negative 3 to 3, 4 to 7, and arrows go from both negative 1 and 5 to negative 2. The mapping diagram above:
Answers
Answer:
Function
Step-by-step explanation:
Answer:
A - Both answers are functions
Step-by-step explanation:
Choose the correct expressions using the fewest number of bases possible that are equivalent to the current expression. Select All that apply. (Note: There are 2 correct choices) Question 4 options: 23−13 210 213−3 2−10
Answers
The rule we use here is [tex]\frac{a^b}{a^c} = a^{b-c}[/tex]
If the font size is too small, then it says (a^b)/(a^c) = a^(b-c)
We subtract the exponents when dividing exponentials like this. The bases must be the same.
So this means the exponents 13 and 3 subtract to get 13-3 = 10
We either have an exponent of "13-3" or an exponent of "10" as an equivalent expression. The base stays at 2 the entire time.
If $-3\le x+5 \le 8$ and $x$ is an integer, what is the sum of all the possible solutions?
Answers
Answer:
-30
Step-by-step explanation:
The integers are the whole numbers and the opposite of the whole numbers.
That is {...,-3,-2,-1,0,1,2,3,...} is the set of integers.
If it is a number you count with or the opposite of a number you count with it or 0, then it is an integer.
So we need to solve the inequality [tex]-3 \le x+5 \le 8[/tex] first, and then just say all the integers included in that compound inequality. Last step is to add since it is asking for the sum of those integers.
Let's do this.
[tex]-3 \le x+5 \le 8[/tex]
Subtract 5 on all sides:
[tex]-3-5 \le x+5-5 \le 8-5[/tex]
Simplify all sides:
[tex]-8 \le x+0 \le 3[/tex]
[tex]-8 \le x \le 3[/tex]
This inequality includes all numbers between -8 and 3, and both include -8 and 3 because of the equal part on both of the inequality parts.
So we have -8,-7,-6,-5,-4,-3,-2,-1,0,1,2, and 3 are the integer solutions of this inequality.
Let's add them. I'm going to put opposites together because I know they will zero out.
So I have this:
-8+-7+-6+-5+-4+-3+-2+-1+0+1+2+3
-8+-7+-6+-5+-4+(-3+3)+(-2+2)+(-1+1)+0
-8+-7+-6+-5+-4+0 +0 +0 +0
-8+-7+-6+-5+-4
-(8+7+6+5+4)
-([8+7]+[6+5]+4)
-([15]+[11]+4)
-(15+11+4)
-([15+11]+4)
-([26]+4)
-(26+4)
-(30)
-30
Which of the following correctly uses exponents to write "6 factors of 2"? A.62 B.26 C.6 ∙ 2 D.12
Answers
Answer:
A. 6 to the Second Power.
Step-by-step explanation:
Hope it Helped!
is -10 a rational number
Answers
Answer: no -10 isn't a rational number
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Hi my lil bunny!
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Is -10 a rational number?
No -10 is not a rational number, because A rational number is, as the name implies, any number that can be expressed as a ratio, or fraction. The number 6 is rational number because it can be expressed as 6/1, though this would be unusual. 4.5 is a rational number, as it can be represented as 9/2.
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Have a great day/night!
❀*May*❀
Simplify the expression: 4a - 3(4 - a)
Answers
Answer:
7a-12Step-by-step explanation:
[tex]4a-3\left(4-a\right)\\\\\mathrm{Expand}\:-3\left(4-a\right):\quad -12+3a\\\\=4a-12+3a\\\\\mathrm{Simplify}\:4a-12+3a:\quad 7a-12\\\\\mathrm{Simplify}\:4a-12+3a:\quad 7a-12\\\\=7a-12[/tex]
Simplify the expression below:
2(4x + 5y)
Answers
Answer:
8x + 10y
Hope this helps!
Answer: 8x+10y
Step-by-step explanation: you to distribute the 2 in the parentheses and the you will have 8x+10y but you can't combine uncommon symbols