Answers
Answer: see proof below
Step-by-step explanation:
Given: A + B + C = π → A = π - (B + C)
→ B + C = π - A
Use the Pythagorean Identity: cos² A + sin² A = 1 → sin² A = 1 - cos² A
Use Double Angle Identities: cos 2A = 2 cos² A - 1 → cos² A = (cos 2A + 1)/2
→ cos A = 1 - 2 sin² (A/2)
Use Sum to Product Identity: cos A + cos B = 2 cos [(A + B)/2] · cos [(A - B)/2]
Use Cofunction Identities: cos (π/2 - A) = sin (A)
sin (π/2 - A) = cos A
cos (-A) = cos (A)
Proof LHS → RHS:
[tex]\text{LHS:}\qquad \qquad \sin^2\bigg(\dfrac{B}{2}\bigg)+\sin^2 \bigg(\dfrac{C}{2}\bigg)-\sin^2\bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Pythagorean:}\qquad 1-\cos^2 \bigg(\dfrac{B}{2}\bigg)+1-\cos^2 \bigg(\dfrac{C}{2}\bigg)-\bigg[1-\cos^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =1-\cos^2 \bigg(\dfrac{B}{2}\bigg)-\cos^2 \bigg(\dfrac{C}{2}\bigg)+\cos^2 \bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Double Angle:}\quad 1-\bigg(\dfrac{\cos(2\cdot \frac{B}{2})+1}{2}\bigg)-\bigg(\dfrac{\cos (2\cdot \frac{C}{2})+1}{2}\bigg)+\bigg(\dfrac{\cos (2\cdot \frac{A}{2})+1}{2}\bigg)\\\\\\.\qquad \qquad \qquad =1-\dfrac{\cos B}{2}-\dfrac{1}{2}-\dfrac{\cos C}{2}-\dfrac{1}{2}+\dfrac{\cos A}{2}+\dfrac{1}{2}\\\\\\.\qquad \qquad \qquad =\dfrac{1}{2}[1-(\cos B+\cos C)+\cos A][/tex]
[tex]\text{Sum to Product:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\cos \bigg(\dfrac{B+C}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Given:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\cos \bigg(\dfrac{\pi -A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Cofunction:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Double Angle:}\qquad \dfrac{1}{2}\bigg[1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)+1-2\sin^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =\dfrac{1}{2}\bigg[2-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)-2\sin^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =1-\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin^2 \bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Factor:}\qquad \qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin \bigg(\dfrac{A}{2}\bigg)\bigg][/tex]
[tex]\text{Given:}\qquad \qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin \bigg(\dfrac{\pi -(B+C)}{2}\bigg)\bigg][/tex]
[tex]\text{Cofunction:}\qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)+\cos \bigg(\dfrac{B+C}{2}\bigg)\bigg][/tex]
[tex]\text{Sum to Product:}\ 1-\sin \bigg(\dfrac{A}{2}\bigg)\cdot 2 \cos \bigg(\dfrac{(B-C)+(B-C)}{2\cdot 2}\bigg)\cdot \cos \bigg(\dfrac{(B-C)-(B+C)}{2\cdot 2}\bigg)\\\\\\.\qquad \qquad \qquad =1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(-\dfrac{C}{2}\bigg)[/tex][tex]\text{Cofunction:}\qquad =1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{LHS = RHS:}\quad \checkmark\\\\\quad 1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)=1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)\quad[/tex]
Related Questions
what is (10y+6) (8y-6) ?
Answers
Answer:
y = 3/4 or y = -3/5
Step-by-step explanation:
Solve for y:
(8 y - 6) (10 y + 6) = 0
Hint: | Find the roots of each term in the product separately.
Split into two equations:
8 y - 6 = 0 or 10 y + 6 = 0
Hint: | Look at the first equation: Factor the left hand side.
Factor constant terms from the left hand side:
2 (4 y - 3) = 0 or 10 y + 6 = 0
Hint: | Divide both sides by a constant to simplify the equation.
Divide both sides by 2:
4 y - 3 = 0 or 10 y + 6 = 0
Hint: | Isolate terms with y to the left hand side.
Add 3 to both sides:
4 y = 3 or 10 y + 6 = 0
Hint: | Solve for y.
Divide both sides by 4:
y = 3/4 or 10 y + 6 = 0
Hint: | Look at the second equation: Factor the left hand side.
Factor constant terms from the left hand side:
y = 3/4 or 2 (5 y + 3) = 0
Hint: | Divide both sides by a constant to simplify the equation.
Divide both sides by 2:
y = 3/4 or 5 y + 3 = 0
Hint: | Isolate terms with y to the left hand side.
Subtract 3 from both sides:
y = 3/4 or 5 y = -3
Hint: | Solve for y.
Divide both sides by 5:
Answer: y = 3/4 or y = -3/5
Use the given endpoint Y and the midpoint M of YZ (there is a line over YZ) to find the coordinates of the other endpoint Z
Answers
The x-coordinate of our midpoint = 3
The x-coordinate of Y = 0
k = x-coordinate of Z
3 = (0 + k) / 2
Multiply both sides by 2
6 = k + 0
Simplify
6 = k
The y-coordinate of our midpoint = 3
the y-coordinate of Y = 5
w = y-coordinate of Z
3 = (w + 5) / 2
Multiply both sides by 2
6 = w + 5
Subtract 5 from both sides
1 = w
For problem #17, Z(6, 1)
Let's do problem 18 now.
The x-coordinate of our midpoint = 5
The x-coordinate of Y = -1
k = x-coordinate of Z
5 = (k + (-1)) / 2
Multiply both sides by 2
10 = k + (-1)
Simplify the right side
10 = k - 1
Add 1 to both sides
11 = k
The y-coordinate of our midpoint = 9
The y-coordinate of Y = -3
w = y-coordinate of Z
9 = (w + (-3)) / 2
Multiply both sides by 2
18 = w + (-3)
Simplify the right side
18 = w - 3
Add 3 to both sides
21 = w
For problem #18, Z(11, 21)
x -0.63 = 1.16 I couldn't find the answer can someone please help me?
Answers
Answer:
x= 1.79
Step-by-step explanation:
x-0.63=1.16 add 0.63 to both sides
x= 1.79
Answer:
x =1.79
Step-by-step explanation:
x -0.63 = 1.16
Add .63 to each side
x -0.63+.63 = 1.16+.63
x =1.79
Write the fraction 12/28 in the simplest form
Answers
Answer: 3/7
Step-by-step explanation:
Answer:
12/28 is 3 /7 is your answer
The price of a tv is rs3000.if the discount is rs1560,find the percent of discount.
Answers
Answer:
price of tv = 3000
discount =1560
discount % = 1560/3000 * 100% = 52%
Step-by-step explanation:
Answer:
52
Step-by-step explanation:
Formula =discount ×100÷original price
1560×100÷3000=52%
Evaluate the expression for x = -3 and y = 4
3x2 – 2xy
Answers
Answer: -36 is your answer
Step-by-step explanation:
about what is 4/5+8/9
Answers
Answer:
1.688 the 8 will keep repeating
Step-by-step explanation:
Answer:
76/45 or 1.69 (rounded)
Step-by-step explanation:
First, you want to find the LCM (Lowest Common Multiple) of the two fractions.
Here, the LCM would be 45 (5*9).
So the fraction would become:
[tex]\frac{4*9}{45} + \frac{8*5}{45} \\\\\frac{36+40}{45} \\\\\frac{76}{45}[/tex]
milo runs 4.5 miles around his house he has to run 6.7 miles how many miles does he have to run left
Answers
Answer:
2.2 miles
Step-by-step explanation:
6.7 - 4.5 = 2.2
Answer:
6.7 miles
Step-by-step explanation:
The sentence about 4.5 miles is extra info. you don't need. In the question it says he needs to run 6.7 miles, then continues to ask how many miles he needs to run.
I hope this answers your question and you understand! Have a great rest of your day! :)
Problem
3x + 5 + 2x = 12 + 4x
Answers
Answer:
x=7
Step-by-step explanation:
because u add the like terms so
3x+2x
which would be
5× so now its
5x+5=12+4x
-4. -4
x+5=12
-5. -5
x=7
a seven sided number cube with sides numbered 1,2,3,4,5,6,7 is rolled. what is the probability that a 4 or odd number is rolled ? which fraction ? 5/7 2/7 3/7 4/49
Answers
The probablity of rolling a 4 = 1/7
There are 4 odd numbers, so the probablity of rolling an odd number = 4/7
4/7 + 1/7 = 5/7
What type of information is being presented on this graph?
Answers
Describe the transformation.
Answers
Answer:
C. reflection across the x-axis
Answer:
C
Step-by-step explanation:
This is a reflection across the x-axis because it is the same thing but reflected.
Hope this helps.
6(m+2)+6m=-5(-7m-7) Solve for m.
Answers
Answer:
m = -1
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
m = -1
☽------------❀-------------☾
Hi there!
~
Let's solve your equation step-by-step.
[tex]6(m+2)+6m=-5(-7m-7)[/tex]
Step 1: Simplify both sides of the equation.
[tex]6(m+2)+6m=-5(-7m-7)[/tex]
[tex](6)^{(m)} + (6)(2)+6m= (-5)(-7m)+(-5)(-7)[/tex] (Distribute)
[tex]6m+12+6m=35m+35[/tex]
[tex](6m + 6m)+(12)=35m + 35[/tex] (Combine Like Terms)
[tex]12m + 12 = 35m + 35[/tex]
[tex]12m + 12 = 35m + 35[/tex]
Step 2: Subtract 35m from both sides.
[tex]12m + 12 - 35m = 35m + 35 - 35m[/tex]
[tex]-23m + 12 = 35[/tex]
Step 3: Subtract 12 from both sides.
[tex]-23m + 12 - 12= 35 - 12[/tex]
[tex]-23m = 23[/tex]
Step 4: Divide both sides by -23.
[tex]\frac{-23m}{-23} = \frac{23}{-23}[/tex]
[tex]m = 1[/tex]
❀Hope this helped you!❀
☽------------❀-------------☾
A basketball player averages 12.5 points per game. There are 24 games in a season. At this rate, how many points would the player score in an entire season?
Answers
Answer:
300
Step-by-step explanation:
hope this helps
The two rectangles in the sketch below are congruent. Which sequence exhibits their congruence? A rotation 90° clockwise about the origin followed by a reflection across the x-axis A rotation 90° counterclockwise about the origin followed by a reflection across the y-axis A rotation 90° counterclockwise about the origin followed by a reflection across the x-axis A rotation 180° counterclockwise about the origin followed by a reflection across the y-axis
Answers
Answer:
A rotation 90° counterclockwise about the origin followed by a reflection across the x-axis
Step-by-step explanation:
it's rotated about the origin counterclockwise by 90 degrees and then reflected over the X axis in order to find the shape there.
Simplify 0.05x + 1.02x.
Answers
Answer:
107x-
-------
100
Step-by-step explanation:
7¹² ÷ 7⁴ is equal to ___. A. 7³ B. 7⁸ C. 3 D. 1³
Answers
Answer:
B. 7⁸
Step-by-step explanation:
The four factors of 7 in the denominator cancel four of the twelve factors of 7 in the numerator, leaving 8. When a product consists of 8 factors of 7, we write it as ...
7⁸
Answer:
B
Step-by-step explanation:
7^12 / 7^4 = 7^8
Round 8,542,310 to the nearest million
Answers
Answer:
9,000,000
Step-by-step explanation:
Since the 5 next to the 8 is a number with a value of 5 or higher, you can round the 8 up to a 9 thus rounding to the nearest million.
please help me with these two questions ( show work plz )( it's ok if you only now one)
Answers
Answer:
1.8.09
2.-6
Step-by-step explanation:
1. 11b=89
b=89/11
8.09
2. 13+k/3=11
k/3=11-13
k/3=-2
k=-2*3
=-6
Answer:
Number 2 is - 6
Step-by-step explanation:
2. 13 + k/3 = 11
We can start off my subtracting 13 in both sides.
-13 + 13 + k/3 = 11 - 13
k/3 = -2. now we have to multiply 3 in both sides
3 x k/3 = -2 x 3
k = -6
Hope this helps!!
The temperature falls from 0 degrees to -12.4 degrees in 3 hours. Which expression finds the change in temperature
per hour?
Answers
Answer: should be 4.13
Step-by-step explanation: -12.4 divided by 3 is -4.13333333333 and goes on
Please emergency!!!
Please
Answers
Answer:
plane Q
Step-by-step explanation:
Please marke as a brainlist...
13.) The ratio of the weight of an object on Mars to the weight of that object on Earth is 4:10. If a rocket weighs 15 tons on earth, how many tons would it weigh on Mars? (Need help quick please)
Answers
Answer: 6 tons
Step-by-step explanation: 4:10 is also 2/5 so all we need to do is 2/5 of 15 which is 6. [15x(2/5)]
Hope this helps!
Please mark Brainliest!
I give Brainliest to whom says this question is stupid. What is -2x-8
Answers
Answer:
No solution it IS STUPID.
Step-by-step explanation:
Answer:
that question is stupid.
how does math play a role in the activities of your everyday life activities
Answers
Answer:
-Managing money $$$
-Balancing the checkbook.
-Shopping for the best price.
-Preparing food.
-Figuring out distance, time and cost for travel.
-Understanding loans for cars, trucks, homes, schooling or other purposes.
-Understanding sports (being a player and team statistics)
-Playing music.
These are things we do everyday that use Math.
Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills.
Hi can someone tell me how is the answer 113?? how do i type it in the calculator?? im confused can someone please explain ASAP!!!!!!!!
THANK YOU
Answers
You shouldn't need a calculator for this. The quadratic equation is written in standard form.
ax^2 + bx + c = 0
You substitute a, b, and c into the discriminant formula and solve as shown in the example in the picture you provided.
(-7)^2 - 4(2)(-8)
49 - 4(-16)
49 - (-64)
49 + 64
113
Best of Luck!
Find all points on the curve x=4cos(t),y=4sin(t) that have the slope of 12.
Answers
Answer:
[tex]\left (-\dfrac{4}{\sqrt{5}},\dfrac{8}{\sqrt{5}}\right )\text{ and }\left (\dfrac{4}{\sqrt{5}},-\dfrac{8}{\sqrt{5}}\right )[/tex].
Step-by-step explanation:
We need to find all the points on the curve x=4cos(t),y=4sin(t) that have the slope of 1/2.
[tex]x=4cos (t)[/tex]
[tex]\dfrac{dx}{dt}=-4sin (t)[/tex]
[tex]y=4sin (t)[/tex]
[tex]\dfrac{dy}{dt}=4cos (t)[/tex]
Now,
[tex]\dfrac{dy}{dx}=\dfrac{dy}{dt}\times \dfrac{dt}{dx}[/tex]
[tex]\dfrac{dy}{dx}=4cos (t)\times \dfrac{1}{-4sin (t)}[/tex]
[tex]\dfrac{dy}{dx}=-\cot t[/tex]
So, slope of the curve is [tex]-\cot t[/tex].
[tex]-\cot t=\dfrac{1}{2}[/tex]
[tex]-\tan t=2[/tex]
[tex]\tan t=-2[/tex] ...(1)
Using [tex]\sec^2t=1+\tan^2t[/tex], we get
[tex]\sec^2t=1+(-2)^2[/tex]
[tex]\sec^2t=1+4[/tex]
[tex]\sec t=\pm \sqrt{5}[/tex]
[tex]\cos t=\pm \dfrac{1}{\sqrt{5}}[/tex]
Now,
[tex]\sin^2t=1-cos^2t[/tex]
[tex]\sin t=\pm \sqrt{1-\dfrac{1}{5}}[/tex]
[tex]\sin t=\pm \sqrt{\dfrac{4}{5}}[/tex]
[tex]\sin t=\pm \dfrac{2}{\sqrt{5}}[/tex]
It equation (1), tan(t) is negative. So, sin and cos have different signs.
If [tex]\sin t= \dfrac{2}{\sqrt{5}}[/tex], then [tex]\cos t=- \dfrac{1}{\sqrt{5}}[/tex].
[tex]x=4cos (t)=-\dfrac{4}{\sqrt{5}}[/tex]
[tex]y=4sin (t)=\dfrac{8}{\sqrt{5}}[/tex]
If [tex]\sin t=- \dfrac{2}{\sqrt{5}}[/tex], then [tex]\cos t= \dfrac{1}{\sqrt{5}}[/tex].
[tex]x=4cos (t)=\dfrac{4}{\sqrt{5}}[/tex]
[tex]y=4sin (t)=-\dfrac{8}{\sqrt{5}}[/tex]
Therefore, the two points are [tex]\left (-\dfrac{4}{\sqrt{5}},\dfrac{8}{\sqrt{5}}\right )\text{ and }\left (\dfrac{4}{\sqrt{5}},-\dfrac{8}{\sqrt{5}}\right )[/tex].
Read the headline below and complete the instruction that follows. “Sloppy Play Sends Tigers to Final Loss” Determine which word indicates bias. Sloppy Tigers Final Loss
Answers
Answer:
Sloppy
_________
Answer:
sloppy
Step-by-step explanation:
it basically give a description that not everyone would think about it. if i described texas A&M as Trash not all people think that therefore making the statement an opinion and bias.
Which polynomial is in standard form?
A)
2x - 3x2 - 4x} + 6x4 + 8
B)
2x4 - 7x² - 6x² + 3x + 9
5 + 7x2 + 6x + 3x5 + x
D)
5 + 7x3 - 6x² + 3x + 9x
Answers
Answer:
A
Step-by-step explanation:
A. B. 3(x+2) x+2 =18 =6 1) How can we get Equation BBB from Equation AAA? Choose 1 answer: Choose 1 answer: (Choice A) A Add/subtract the same quantity to/from both sides (Choice B) B Add/subtract a quantity to/from only one side (Choice C) C Multiply/divide both sides by the same non-zero constant (Choice D, Checked) D Multiply/divide only one side by a non-zero constant
Answers
Answer:
C. Multiply/divide both sides by the same non-zero constant
A. Yes
Just Did It On Khan Academy
Step-by-step explanation:
To get equation A from equation B multiply/divide only one side by a non-zero constant option (C) Multiply/divide only one side by a non-zero constant is correct.
What is a linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
It is given that:
Two linear equations:
A. 3(x + 2) = 18
B. x + 2 = 18
In equation B multiply by 3 on one side:
3(x + 2) = 18
Thus, to get equation A from equation B multiply/divide only one side by a non-zero constant option (C) Multiply/divide only one side by a non-zero constant is correct.
Learn more about the linear equation here:
brainly.com/question/11897796
#SPJ2
2/7(x+4)=-6 what is the value of x?
Answers
Answer:
2/7 as decimal is 0.28
2/7(x+4)=-6
2/7x+ 1 1/4= -6
-6 - 1 1/4 = -7.25
0.28x = -7.25
x = -25.89