A direct variation always goes through the origin of the coordinate plane. TrueFalse

Prove by mathematical induction that

postnew [5]

For $n=1$ , on the left we have $\cos\theta$ , and on the right,

$\dfrac{\sin2\theta}{2\sin\theta}=\dfrac{2\sin\theta\cos\theta}{2\sin\theta}=\cos\theta$

(where we use the double angle identity: $\sin2\theta=2\sin\theta\cos\theta$ )

Suppose the relation holds for $n=k$ :

$\displaystyle\sum_{n=1}^k\cos(2n-1)\theta=\dfrac{\sin2k\theta}{2\sin\theta}$

Then for $n=k+1$ , the left side is

$\displaystyle\sum_{n=1}^{k+1}\cos(2n-1)\theta=\sum_{n=1}^k\cos(2n-1)\theta+\cos(2k+1)\theta=\dfrac{\sin2k\theta}{2\sin\theta}+\cos(2k+1)\theta$

So we want to show that

$\dfrac{\sin2k\theta}{2\sin\theta}+\cos(2k+1)\theta=\dfrac{\sin(2k+2)\theta}{2\sin\theta}$

On the left side, we can combine the fractions:

$\dfrac{\sin2k\theta+2\sin\theta\cos(2k+1)\theta}{2\sin\theta}$

Recall that

$\cos(x+y)=\cos x\cos y-\sin x\sin y$

so that we can write

$\dfrac{\sin2k\theta+2\sin\theta(\cos2k\theta\cos\theta-\sin2k\theta\sin\theta)}{2\sin\theta}$

$=\dfrac{\sin2k\theta+\sin2\theta\cos2k\theta-2\sin2k\theta\sin^2\theta}{2\sin\theta}$

$=\dfrac{\sin2k\theta(1-2\sin^2\theta)+\sin2\theta\cos2k\theta}{2\sin\theta}$

$=\dfrac{\sin2k\theta\cos2\theta+\sin2\theta\cos2k\theta}{2\sin\theta}$

(another double angle identity: $\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta$ )

Then recall that

$\sin(x+y)=\sin x\cos y+\sin y\cos x$

which lets us consolidate the numerator to get what we wanted:

$=\dfrac{\sin(2k+2)\theta}{2\sin\theta}$

and the identity is established.

8 0

2 years ago

Which of the following is the square of a binomial

Ronch [10]

Answer:

Step-by-step explanation:

(a+b)² = a² + 2ab + b²

The only choice that has the form a² + 2ab + b² is D.

5 0

2 years ago

3. Consider the expressions sq root of 15, sq root of 97 and sq root of 40 Locate the approximate value of each on a number line

Crank

Answer:

Square root of 15

√ 15 = 3.9 (Rounding to the nearest tenth)

Square root of 97

√97 = 9.9 (Rounding to the nearest tenth)

Square root of 40

√40 = 6.3 (Rounding to the nearest tenth)

Step-by-step explanation:

Square root of 15

√ 15 = 3.9 (Rounding to the nearest tenth)

Square root of 97

√97 = 9.9 (Rounding to the nearest tenth)

Square root of 40

√40 = 6.3 (Rounding to the nearest tenth)

Location on a number line

√ 15, between + 3 and + 4, very close to +4

√40, between + 6 and + 7, close to +6

√97, between + 9 and + 10, very close to +10

5 0

2 years ago

Tell which set or sets the number below belongs to: natural numbers, whole numbers, integers, rational numbers, irrational

aalyn [17]

Answer:

if what you're asking if 0 belongs to anyone of the sets then zero belongs to whole numbers.

whole numbers is 0,1,2,3,4 etc

Natural numbers are 1,2,3,4 etc

integers are -3,-2,-1,0,1,2,3 etc

rational numbers are like fractions etc

irrational numbers are when you don't necessarily get a exact or whole numbers for example like π/pie

3 0

2 years ago

To the nearest hundredth of a second, how much time does it take a car to travel 50 ft from a stop at a constant acceleration of

Genrish500 [490]

Answer:

Turkey

Step-by-step explanation:

6 0

2 years ago