1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
nikklg [1K]
3 years ago
14

Trig. help. How does

29%20%3D%20sinx" id="TexFormula1" title="2sin (\frac{1}{2} x)cos( \frac{1}{2} x) = sinx" alt="2sin (\frac{1}{2} x)cos( \frac{1}{2} x) = sinx" align="absmiddle" class="latex-formula"> ?
Mathematics
1 answer:
kotegsom [21]3 years ago
5 0
Recall that 2sin(x) cos(x) is actually equal to sin(2x).

We can prove this by expanding sin(2x) to sin(x + x).
sin(x + x) = sin(x) cos(x) + cos(x) sin(x) = 2sinxcosx

Thus, 2sin(x/2)cos(x/2) can be rewritten in the form:
sin(2x/2), and this simplifies down to sinx.
You might be interested in
How do you graph 3(y-3)= -2x using the slope and intercept?
stira [4]
Are you sure you wrote that equation right
6 0
3 years ago
Consider the quadratic equation x? = 4x - 5. How many solutions does the equation have? A The equation has one real solution. Th
anzhelika [568]

Answer:

{-1, 5} (two real solutions).

Step-by-step explanation:

On the left side we want x^2, not x?

We need to rewrite x? = 4x - 5 in standard form, that is, as a quadratic with x terms in decreasing powers of x:

x^2 - 4x + 5 = 0

This factors easily to (x - 5)(x + 1) = 0.

We let each factor equal zero (0) separately and solve for x:  

{-1, 5} (two real solutions).

4 0
3 years ago
Find an example for each of vectors x, y ∈ V in R.
rjkz [21]

(a) Both conditions are satisfied with <em>x</em> = (1, 0) for \mathbb R^2 and <em>x</em> = (1, 0, 0) for \mathbb R^3:

||(1, 0)|| = √(1² + 0²) = 1

max{1, 0} = 1

||(1, 0, 0)|| = √(1² + 0² + 0²) = 1

max{1, 0, 0} = 1

(b) This is the well-known triangle inequality. Equality holds if one of <em>x</em> or <em>y</em> is the zero vector, or if <em>x</em> = <em>y</em>. For example, in \mathbb R^2, take <em>x</em> = (0, 0) and <em>y</em> = (1, 1). Then

||<em>x</em> + <em>y</em>|| = ||(0, 0) + (1, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2

||<em>x</em>|| + ||<em>y</em>|| = ||(0, 0)|| + ||(1, 1)|| = √(0² + 0²) + √(1² + 1²) = √2

The left side is strictly smaller if both vectors are non-zero and not equal. For example, if <em>x</em> = (1, 0) and <em>y</em> = (0, 1), then

||<em>x</em> + <em>y</em>|| = ||(1, 0) + (0, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2

||<em>x</em>|| + ||<em>y</em>|| = ||(1, 0)|| + ||(0, 1)|| = √(1² + 0²) + √(0² + 1²) = 2

and of course √2 < 2.

Similarly, in \mathbb R^3 you can use <em>x</em> = (0, 0, 0) and <em>y</em> = (1, 1, 1) for the equality, and <em>x</em> = (1, 0, 0) and <em>y</em> = (0, 1, 0) for the inequality.

(c) Recall the dot product identity,

<em>x</em> • <em>y</em> = ||<em>x</em>|| ||<em>y</em>|| cos(<em>θ</em>),

where <em>θ</em> is the angle between the vectors <em>x</em> and <em>y</em>. Both sides are scalar, so taking the norm gives

||<em>x</em> • <em>y</em>|| = ||(||<em>x</em>|| ||<em>y</em>|| cos(<em>θ</em>)|| = ||<em>x</em>|| ||<em>y</em>|| |cos(<em>θ</em>)|

Suppose <em>x</em> = (0, 0) and <em>y</em> = (1, 1). Then

||<em>x</em> • <em>y</em>|| = |(0, 0) • (1, 1)| = 0

||<em>x</em>|| • ||<em>y</em>|| = ||(0, 0)|| • ||(1, 1)|| = 0 • √2 = 0

For the inequality, recall that cos(<em>θ</em>) is bounded between -1 and 1, so 0 ≤ |cos(<em>θ</em>)| ≤ 1, with |cos(<em>θ</em>)| = 0 if <em>x</em> and <em>y</em> are perpendicular to one another, and |cos(<em>θ</em>)| = 1 if <em>x</em> and <em>y</em> are (anti-)parallel. You get everything in between for any acute angle <em>θ</em>. So take <em>x</em> = (1, 0) and <em>y</em> = (1, 1). Then

||<em>x</em> • <em>y</em>|| = |(1, 0) • (1, 1)| = |1| = 1

||<em>x</em>|| • ||<em>y</em>|| = ||(1, 0)|| • ||(1, 1)|| = 1 • √2 = √2

In \mathbb R^3, you can use the vectors <em>x</em> = (1, 0, 0) and <em>y</em> = (1, 1, 1).

8 0
3 years ago
How many quarters in $100 ​
Y_Kistochka [10]
400 quarters

$1 = 4q

4q • 100 = 400

400q=$100
7 0
3 years ago
Read 2 more answers
There are 7 people on a cycling team. Their coach mush pick two of them to be Co - Captain. How many different pairs can the cou
garik1379 [7]
21, because 7 choose 2 = 21
4 0
3 years ago
Other questions:
  • How many tablespoon are equivalent to 3 pints?
    8·2 answers
  • Simplify the expression 5(n^2+n)-3n(2n^2+4n-2)
    5·1 answer
  • An amount of $18,000 is borrowed for 10 years at 8.25% interest, compounded annually. If the loan is paid in full at the end of
    15·1 answer
  • Look at the table. Make a conjecture about the sum of the first 25 positive even numbers.
    10·2 answers
  • . A sports store sells a 4 pack of tennis balls for $11.49, a 6 pack of tennis balls for $16.70, and a 9 pack for $22.99. Which
    10·2 answers
  • 15 PTS!!!!!!!!!!
    5·1 answer
  • In a shopping mall, two elevators pass each other moving in opposite directions. Each elevator is moving at the same speed. The
    15·2 answers
  • 1 3/7 plus 1 5/7 is how much in all
    10·1 answer
  • Solve for y.<br> O 10<br> O 12<br> O 15<br> O 18
    14·2 answers
  • Object B has a mass of 10kg. Object b collides with another object. If the momentum of object b after collision is 18kg•m/s18kg.
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!