CPhill

If you want a "non-decimal" answer...

√20 = √4 * √5   = 2√ 5

Cosx=(2.4/3)

Using the inverse cosine function, we have

arcos (2.4/3) = about 36.87 degrees

Also, realize that this angle might also lie in the 2nd quadrant = 180 - 36.87 = `143.13 degrees

Using the identity   1 + tan²(θ )  = sec² (θ ), we have

1+tan^2(pi/2-x) = sec² (pi/2 - x ) = 1 / cos² (pi/2 - x) 

And since  cos(pi/2 -x) = sin(x)   we have

1/ (sin² (x) = csc²(x)

Well let's see...mutiply both sides by 3

3x^2 + 21x +159 = 11   Subtract 11 from both sides

3x^2 + 21x + 148 = 0

I can tell right away that this has no "real" solutions....Let's see what he "magic" calculator sez.......

$${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{148}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{1\,335}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}\right)}{{\mathtt{6}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{1\,335}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{21}}\right)}{{\mathtt{6}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6.089\: \!608\: \!635\: \!484\: \!241\: \!2}}{i}\right)\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{7}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6.089\: \!608\: \!635\: \!484\: \!241\: \!2}}{i}\\
\end{array} \right\}$$

Yep...we get two "complex" solutions...as expected....

OK....I just had to reduce my screen size....sorry!!

The other angle in the triangle is 90 - 17 = 73

So using the Law of Sines we have

10/sin17 = x / sin 73

x =10 (sin 73)/(sin 17) = 32.7 = the side opposite the 73 degree angle

Now just use the Pythagorean Theorem to find the hypoteneuse...

Sqrt (32.7^2  + 10^2)  =  hypoteneuse

I'll leave this for you to finish......just add all the sides for the answer!!

24

There are 4 ways to choose the first letter....3 ways to choose the second letter ......2 ways to choose the second letter.......and 1 way to choose the last letter

Thus......4 x 3 x 2 x 1 = 24

BTW......this is known as 4!    (4 factorial)

As to 10......(I assume this is a trig class??)

We have

sin 50 = 20 / L      

L = 20 / sin 50 ≈ 26.1 m

As to 11....the image is "off the page," so I can't tell what you've got....try sending another pic!!

OK

A = pi * r^2

So

314.2 = pi * r^2     Divide both sides by pi

314.2 / pi = r^2      Take the square root of both sides

√(314.2 / pi)   = r   = 10.00064829093329474 ft  or about 10 ft

We've just created a new math "identity"

1 question = "infinite" answers

Thus

1 = ∝

(And we have the "proof")

OK...good job, so far.....(we'll talk about one of your "steps" in a second)

So we have

3x + 3 = 2x + 8      Since I usually like all the "x's" on the left and everything else on the right, let's subtract 2x from both sides   We have

x + 3  = 8      Then subtract 3 from both sides    We get

x = 5  

And that's it  !!!

I don't know if this was a "mistype" or not, but in general

8 + 2x ≠ 8x + 2      (This only works if x = 1 !!)