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Vlad1618 [11]
2 years ago
7

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.

Mathematics
1 answer:
Luba_88 [7]2 years ago
4 0

Answer:

here's the answer to your question!

Step-by-step explanation:

Work shown in the attachment,

Refer to the attachment!

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Find the exact value of the expression.<br> tan( sin−1 (2/3)− cos−1(1/7))
Sonja [21]

Answer:

\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}

Step-by-step explanation:

I'm going to use the following identity to help with the difference inside the tangent function there:

\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}

Let a=\sin^{-1}(\frac{2}{3}).

With some restriction on a this means:

\sin(a)=\frac{2}{3}

We need to find \tan(a).

\sin^2(a)+\cos^2(a)=1 is a Pythagorean Identity I will use to find the cosine value and then I will use that the tangent function is the ratio of sine to cosine.

(\frac{2}{3})^2+\cos^2(a)=1

\frac{4}{9}+\cos^2(a)=1

Subtract 4/9 on both sides:

\cos^2(a)=\frac{5}{9}

Take the square root of both sides:

\cos(a)=\pm \sqrt{\frac{5}{9}}

\cos(a)=\pm \frac{\sqrt{5}}{3}

The cosine value is positive because a is a number between -\frac{\pi}{2} and \frac{\pi}{2} because that is the restriction on sine inverse.

So we have \cos(a)=\frac{\sqrt{5}}{3}.

This means that \tan(a)=\frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}}.

Multiplying numerator and denominator by 3 gives us:

\tan(a)=\frac{2}{\sqrt{5}}

Rationalizing the denominator by multiplying top and bottom by square root of 5 gives us:

\tan(a)=\frac{2\sqrt{5}}{5}

Let's continue on to letting b=\cos^{-1}(\frac{1}{7}).

Let's go ahead and say what the restrictions on b are.

b is a number in between 0 and \pi.

So anyways b=\cos^{-1}(\frac{1}{7}) implies \cos(b)=\frac{1}{7}.

Let's use the Pythagorean Identity again I mentioned from before to find the sine value of b.

\cos^2(b)+\sin^2(b)=1

(\frac{1}{7})^2+\sin^2(b)=1

\frac{1}{49}+\sin^2(b)=1

Subtract 1/49 on both sides:

\sin^2(b)=\frac{48}{49}

Take the square root of both sides:

\sin(b)=\pm \sqrt{\frac{48}{49}

\sin(b)=\pm \frac{\sqrt{48}}{7}

\sin(b)=\pm \frac{\sqrt{16}\sqrt{3}}{7}

\sin(b)=\pm \frac{4\sqrt{3}}{7}

So since b is a number between 0 and \pi, then sine of this value is positive.

This implies:

\sin(b)=\frac{4\sqrt{3}}{7}

So \tan(b)=\frac{\sin(b)}{\cos(b)}=\frac{\frac{4\sqrt{3}}{7}}{\frac{1}{7}}.

Multiplying both top and bottom by 7 gives:

\frac{4\sqrt{3}}{1}= 4\sqrt{3}.

Let's put everything back into the first mentioned identity.

\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}

\tan(a-b)=\frac{\frac{2\sqrt{5}}{5}-4\sqrt{3}}{1+\frac{2\sqrt{5}}{5}\cdot 4\sqrt{3}}

Let's clear the mini-fractions by multiply top and bottom by the least common multiple of the denominators of these mini-fractions. That is, we are multiplying top and bottom by 5:

\tan(a-b)=\frac{2 \sqrt{5}-20\sqrt{3}}{5+2\sqrt{5}\cdot 4\sqrt{3}}

\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}

4 0
3 years ago
What is another way to say 230 mm?
MArishka [77]

Answer:23 cm

Step-by-step explanation:

Because it would equal the same amount because there are 10 mm in a cm

7 0
3 years ago
Read 2 more answers
Pizzazz HELP MEEEE
Art [367]

Answer:

8x less than or equal to $50

Step-by-step explanation:

bc the x represents the amount of books you can get for 8 dollars per book and can be less than or equal to, but not greater than 50, the max amount you may spend

hope this helped :)

4 0
3 years ago
The polynomial function f(x) is a fourth degree polynomial. Which of the following could be the complete list of the roots of f(
cluponka [151]

Answer:

the answer is A

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
I need help on listing the four points that have a distance of 8 from B, thank you!
dolphi86 [110]

Answers:

  • (12, 3)
  • (-4, 3)
  • (4, 11)
  • (4, -5)

A diagram is shown below.

===========================================================

Explanation:

First we locate point B at (4,3). The '4' in the x position means we've gone 4 units to the right of the origin, and the 3 in the y coordinate means we've gone 3 units up.

Now we need to find four points that are 8 units away from this central location. We'll ignore point A.

What we can do is add 8 to the x coordinate of point B to get x+8 = 4+8 = 12. Keeping the y coordinate the same, the point (12, 3) is exactly 8 units away from (4,3). Think of drawing a number line through those two points. Then we can note there are 8 spaces between (4,3) and (12,3). Or we can say "there are 8 spaces between 4 and 12" since that '3' doesn't really play much of a role in this case.

So (12,3) is one answer.

---------------------------------

Move back to point B(4,3)

Instead of adding 8 to the x coordinate, let's subtract 8 from it

x-8 = 4-8 = -4

Again the y coordinate is kept the same.

The point (-4, 3) is 8 units away from (4,3). The number line distance from -4 to 4 is 8 units.

So (-4,3) is another answer.

----------------------------------

Move back to point B

From this starting point, we can move 8 units up. Doing this means we add 8 to the y coordinate, while the x coordinate is kept intact.

(4,3) turns into (4,11) after doing so.

(4,11) is another answer.

---------------------------------

Move back to point B

From this point, we can move down 8 units to arrive at (4,-5). note how y-8 = 3-8 = -5. The x coordinate is kept the same.

(4,-5) is another answer.

----------------------------------

The four points we found were:

  • (12, 3)
  • (-4, 3)
  • (4, 11)
  • (4, -5)

which are four possible answers

Those points are found by going 8 units right, 8 units left, 8 units up and 8 units down in that order. Each time we started at point B as our center of operation. The use of the word "center" is done because we can form a circle of radius 8, and the center of the circle is at B(4,3). Any point on the circle's edge is exactly 8 units away from the center point B.

Refer to the diagram below. The four answers are the red points P,Q,R,S. The point T is another point that could be an answer, but it's not as easy to find compared to the four other points.

3 0
3 years ago
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