Expert: Paul Klarreich - 5/8/2009

Question
QUESTION: solve for x to the nearest tenth of a degree on the interval 0 less then or equal to x less then or equal to 360.
sin squared x = 4sinx + 5 ?

ANSWER: Questioner:   danielle
Country:  United States
Category:  Advanced Math
Private:  Yes  << Don't mark this.
 
Subject:  algebra and trigonometry
Question:  solve for x to the nearest tenth of a degree on the interval 0 less then or equal to x less then or equal to 360.
sin squared x = 4sinx + 5 ?
..........................
Hi, Danielle,

If  sin^2(x) = 4 sin x + 5, this is quadratic equation like:

s^2 = 4s + 5

s^2 - 4s - 5 = 0  << standardize

(s - 5)(s + 1) = 0  << factor

sin x = 5,  sin x = -1    << solve and re-expand
 
sin x = 5 does not exist, but

sin x = -1  give  x = 270 degrees  [check your standard graph].


---------- FOLLOW-UP ----------

QUESTION: part a) from the top of a building 30 feet tall, the angle of depression to the foot of a building across the street is 60 degrees and the angle of elevation to the top of the same building is 70 degrees. How tall is the building?

part b) if cos x is greater then 0 and sinx = -12/13, then find cosx, tanx, secx, cscx, and cotx.

Answer
Questioner:   danielle
Country:  United States
Category:  Advanced Math
Private:  No

---------- FOLLOW-UP ----------

QUESTION: part a) from the top of a building 30 feet tall, the angle of depression to the foot of a building across the street is 60 degrees and the angle of elevation to the top of the same building is 70 degrees. How tall is the building?

part b) if cos x is greater then 0 and sinx = -12/13, then find cosx, tanx, secx, cscx, and cotx.
........................................
Hi, Danielle,

Angle of depression means 'angle looking down'.  See diagram.  

Step 1: Use the lower triangle to find x, using:

tan 60 = 30/x

x = 30/tan 60

You can use your calculator for that.

Step 2:  Use the upper to find y, using:

tan 70 = y/x, where you know x from step 1.

y = x tan 70, and use your calculator again.

Step 3:  The height is 30 + y.

...................................

b) is a standard thing --  from any one trig function of an angle, and one other piece of info, you can get all the other functions by suitable detective work.  Do the following:

Step 1: Rewrite the badly written problem:

part b) if cos t is greater then 0 and sin t = -12/13, then find cos t, tan t, sec t, csc t, and cot t.

(NEVER use x for the angle, we need it for other stuff.)

Step 2: The definitions are:

sin t = y/r

cos t = x/r

.....

And remember  r^2 = x^2 + y^2, and r is never negative.

Step 3:  Make assignations:

If sin t = -12/13 = y/r,  ASSIGN  r = 13,  y = -12.

Now the Pythagorean Theorem says x = +- 5.  Which?  Does

cos t = 5/13  or

cos t = -5/13 ?

Aha! The great detective (that's you) notes the sentence "cos t is greater than 0" and concludes x = +5, not -5.

Now you can just fill in all the functions.