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Leni [432]
3 years ago
8

Rationalize the denominator: Please give step by step solution

Mathematics
1 answer:
MrMuchimi3 years ago
5 0

Answer:

1 + i

Step-by-step explanation:

To rationalize multiply numerator and denominator by (2 + i )

\frac{3+i}{2-i}= \frac{3+i}{2-i} \times \frac{2+i}{2+i} = \frac{(3+i)(2+i)}{(2-i)(2+i)} = \frac{6+3i+2i-1}{4-(-1)} = \frac{5+5i}{5} = 1 + i

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How to use trig to find missing side or angle of right triangles.
andrew11 [14]
Explanation

Right triangles are triangles with one right angle i.e. one angle with a measure of 90° so when we deal with this type of figures we know at least one of its internal angles. The opposite side to the right angle is known as the hypotenuse whereas the other two sides are called legs.

Trigonometric functions in right triangles give us the following identities:

\begin{gathered} \tan x=\frac{\text{opposite side}}{\text{adjacent side}} \\ \cos x=\frac{\text{adjacent side}}{\text{hypotenuse}} \\ \sin x=\frac{\text{opposite side}}{\text{hypotenuse}} \end{gathered}

We also have the Pythagorean theorem that states that the square of the hypotenuse is the sum of the squares of the legs.

Let's imagine that we have a right triangle like the following:

If we want to find the missing angles A and/or B we need to know at least two sides of the triangle.

For example, if we want to find B and we have b and a we can use the tangent of B and its inverse the arctangent (tan^(-1)):

\tan B=\frac{b}{a}\rightarrow B=\tan^{-1}(\frac{b}{a})

If we know a and c we can find B using its cosine:

\cos B=\frac{a}{c}\rightarrow B=\cos^{-1}(\frac{a}{c})

If we know b and c then we can use the sine:

\sin B=\frac{b}{c}\rightarrow B=\sin^{-1}(\frac{b}{c})

We can also do the same for angle A, we just need to use its corresponding opposite side and adjacent side. And that's how you find a missing angle using trigonometry in a right triangle.

If we are looking for a missing side we can also use trigonometry. In order to find a missing side we need two other sides or another side and an angle. For example, let's assume that we want to find side a.

If we have b and c then we just need to use the Pythagorean theorem:

c^2=a^2+b^2\rightarrow a=\sqrt{c^2-b^2}

If we have b and A then we can use the tangent of A to built an equation for a:

\tan A=\frac{a}{b}\rightarrow a=b\cdot\tan A

If we have b and B we can use the tangent of B:

\tan B=\frac{b}{a}\rightarrow a=\frac{b}{\tan B}

If we have A and c we can use the sine of A:

\sin A=\frac{a}{c}\rightarrow a=c\cdot\sin A

If we have B and c we can use the cosine of B:

\cos B=\frac{a}{c}\rightarrow a=c\cdot\cos B

And that's how you find a missing side using trigonometry.

8 0
9 months ago
What is 16 over 32 simplified
Vinvika [58]
Alright, so to simplify we would take the largest number that would go into both 16 and 32 and divide both numbers by that. Do you know what that would be?
7 0
3 years ago
Read 2 more answers
One large bubble separates into four small bubbles so that the total area of the small bubbles is equal to the area of the large
anzhelika [568]

Answer:

10 cm.

Step-by-step explanation:

We'll begin by calculating the area of the small bubble. This can be obtained as follow:

Radius (r) = 5 cm

Area (A) =?

Since the bubble is circular in nature, we shall use the formula for area of circle to determine the area of the bubble. This is illustrated below:

A = πr²

A = π × 5²

A = 25π cm²

Next, we shall determine the total area of the small bubbles. This can be obtained as follow:

Area of 1 bubble = 25π cm²

Therefore,

Area of 4 bubbles = 4 × 25π cm²

Area of 4 bubbles = 100π cm²

Finally, we shall determine the radius of the large bubble. This can be obtained as follow:

Area of large bubble = total area of small bubbles = 100π cm²

Radius (r) =?

A = πr²

100π = πr²

100 = r²

Take the square root of both side

r = √100

r = 10 cm

Thus, the radius of the large bubble is 10 cm

5 0
2 years ago
(03.05 LC) The graph of a function f(x) is shown below: graph of line segment going through ordered pairs negative 3, negative 4
Aleks [24]

Answer:

−3 < x ≤ 1

Step-by-step explanation:

The domain of a function is the set of x-values.

In this graph, the open circle at (-3, -4) means the segment goes back up to this point but this point is not part of the segment itself.

The closed circle at (1, 2) means this is the endpoint and part of the segment.

This means the x-values range from almost -3 up to and including 1; this gives us the inequality

−3 < x ≤ 1

4 0
2 years ago
Read 2 more answers
Solve for x . <br> X+9 .. 2x-3 .
nadezda [96]
X + 9 = 2x - 3
<u>-x         -x        </u>
     9 = x - 3
<u>   +3       +3</u>
   12 = x
6 0
3 years ago
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