CPhill

Just a calculator problem

arccos (0.21)  = 77.88°

log _{a + b} m  + log _{a - b} m   -  2log _{a+ b}  m* log _{a - b} m

Note  m^2  =  a^2 - b^2   = (a + b) (a - b)

And, using the change-of-base rule, we can write

log m / log(a + b) + log  m / log (a - b)  -  2 (  [ log m] [ log m]  /  [ log (a + b) * log (a - b) ] )

Get a common  denominator between the first two fractions and factor out log m 

[  log( m ) [ log (a + b) + log (a - b)]  ] /  [ log(a + b)* log(a - b)] -

2 (  [ log m] [ log m]  /  [ log (a + b) * log (a - b) ]  

{ log a + log b  =  log(a * b)  }

log (m) [ log [ (a + b)(a - b)]  / [ log(a + b)* log(a - b)] -

2 (  [ log m] [ log m]  /  [ log (a + b) * log (a - b) ]

log (m) [ log (m^2) ]  /  [ log(a + b)* log(a - b)] -

2 [ log m] [ log m]  /  [ log (a + b) * log (a - b) ]

{  log a^b  =  b log a }

log (m) * 2 log (m) /  [ log(a + b)* log(a - b)] -

2 (  [ log m] [ log m]  /  [ log (a + b) * log (a - b) ]

(2 [ log (m) }^2 ( / [ log(a + b)* log(a - b)] -   (2 [ log m] ^2 ( /  [ log (a + b) * log (a - b) ] =

0

Cosine  = x / r

Cosine in quadrant  II  is negative because x is negative and r is positive

Sine  =  y / r

Sine  in quadrant II  is negative because  y, r  are both positive 

5pi/ 6  is a 30°  reference angle in the 2nd quadrant....so....

sin 5pi/6  =   sin (30°)  =  1/2

Angle of elevation   =

arctan ( 35 / 50)  =   35°

We can use  the inverse tangent ( arctan)  function here

arctan (15/8)  =  x   =  61.9° 

We know an angle, the opposite side to the angle and we are looking for the hypotenuse.....the sine function relates these.....so we have

sin (65)  = 15 / d   rearrange as  

d  = 15 / sin (65)  ≈  16.6 ft

-32°  =  328°

Adding  360°  to this  gives us 688°

And  -32°  - 360°  =   -392°

arcsin (-1/2)   =  -pi/ 6