hectictar

AHHA! I knew I had seen this..

took some digging, but here it is:    

1.

In all 30-60-90 triangles:

the shortest side—across from the 30º angle = n

the middle length side—across from the 60º angle = √(3)n

the longest side/hypotenuse—across from the 90º angle = 2n

So..... if the shortest side = n = 8

The hypotenuse = 2n = 2(8) = 16 meters

2.

In all 45-45-90 triangles:

both legs—across from the 45º angles = n

the hypotenuse—across from the 90º angle = √(2)n

So...if a leg = n = 5

The hypotenuse = √(2)n = √(2)(5) = 5√(2) ≈ 7.071 inches

3.

x is one of the legs of a 45-45-90 triangle.

The other leg is the hypotenuse of a 30-60-90 triangle.

Both legs are the same length in a 45-45-90 triangle.

So..

Find the hypotenuse of the 30-60-90 triangle when the middle length side = 2√(3)

and that will be the same length as x.

Can finish this one on your own? It is now very similar to problem number 1.

4.

The sin(45º) = the length of a leg in a 45-45-90 triangle when the hypotenuse is 1.

So..

Find the length of a leg in a 45-45-90 triangle when the hypotenuse = 1

and that will be sin(45º)

Can you finish it on your own? It is very similar to problem number 2.

5.

Here is an illustration:

area of hexagon = 12 * area of third triangle.

area of third triangle = (1/2) * base * height

area of third triangle = (1/2) * 6 * orange line

area of third triangle = 3 * orange line

area of third triangle = 3 * √(3)(6)

area of third triangle = 18√3

area of hexagon = 12 * 18√3

area of hexagon = 216√3 ≈ 374.123 sq. cm.

AB will be 6 when x = ± 3, that is when y = 9...

I don't have time to do any more but maybe this helps get you started.

Here's a rough drawing (Not to scale):

We don't know what the right side is.

-17b - 22 - 15b + 9 = n

Combine like terms.

-32b - 13 = n

\(6^{-10}=\frac{1}{6^{10}}=\frac{1}{60,466,176}\approx 0.0000000165381717\approx1.654\times10^{-8}\)

If you just want to put it into the calculator, you can type it like this:

6^(-10)

Here is a drawing of what cos(x)= 1/2 really means:

We can find sin(x) just using the Pythagorean Theorem.

\((\frac12)^2+(\sin (x))^2=1^2 \\~\\ \mathbf{sin(x)}=\sqrt{1-\frac14}=\sqrt{\frac34}\mathbf{=\frac{\sqrt3}{2}}\)

tan = sin / cos, so...

tan(x) = sin(x) / cos(x) = \(\frac{\sqrt3}{2}\div\frac{1}{2}=\frac{\sqrt3}{2}\cdot\frac{2}{1}\mathbf{=\sqrt3}\)

Q: What is condensation on grass?

A: dew

Q: What makes the water cycle continue? (hint: gives heat at summer)

A: the sun (and other things too, like gravity... but I think the sun is it)

Those are the only ones I really know...

5(-3x - 2) - (x - 3) = -4(4x + 5) + 13

Distribute.

-15x - 10 - x + 3 = -16x - 20 + 13

Combine like terms.

-16x - 7 = -16x - 7

Both sides say the exact same thing.

So yes, it is true for all values of x.       :)

Is it like this?

(The position of the red square inside the blue square doesn't matter.)

To find the area of the shaded region, all we need to do is find the area of the bigger, blue square, and subtract from it the area of the smaller, red square.

shaded area = blue square area - red square area

blue square area = side² = (x + 5)²

red square area = side² = (x - 2)²

shaded area = (x + 5)² - (x - 2)²

shaded area = (x + 5)(x + 5) - (x - 2)(x - 2)

shaded area = [x² + 10x + 25] - [x² - 4x + 4]

shaded area = x² + 10x + 25 - x² + 4x - 4

shaded area = 14x + 21 sq. units.