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Basic Math/trigonometric equations

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Expert: Josh - 1/21/2010


Question

trig
hey Josh,

sorry to keep bothering you

referrring to Q2  a,

i got 2 out of 3 readings, where does the 150 degrees final solution come from?
i have drawn the sin wave, to me it looked like this was as far as it could go, but the answers say different.

also,
find value of a stated second trigonmetric ratio without finding angle
how do i do this for
q1
0< x< 90 degrees, sin x=0.3, what is cos x? answer 0.954

q2
180 < x <270, sec x = -sqrt10, what is tan x? answer     3

answers to 3dcp

I,m not sure even where to start with this. i have a long list of these with answers, examples are  nil.

much appreciated,thanks for your time and patience
richard.

Answer
Part 1:

Remember, there are always 2 solutions for
sin(x)=const.
cos(x)=const.
tan(x)=const.
in the fundamental interval 0 <= x < 2*pi.

In yesterday's response, I referred to the "A.S.T.C" trick, which tells us the two quadrants in which sin(x) is positive. Please go over this. Because sin(x) is always positive in the 1st and 2nd quadrant (for angles x between 0 and 90, 90 and 180, respectively), we infer that sin(x) is always negative in the 3rd and 4th quadrant (for angles x between 180 and 270, 270 and 360, respectively).

Solve 2*[cos(x)]^2 + 3*sin(x) = 3   for 0 < x < pi.

Note: I think you got the sign wrong on line 3,
| intermediate steps, using 1-cos^2=sin^2
| 2*{1-[sin(x)]^2} + 3*sin(x) = 3
| 2-2*[sin(x)]^2 + 3*sin(x) = 3
| 2*[sin(x)]^2 - 3*sin(x) + 1 = 0
(2*sin(x)-1)(sin(x)-1) = 0

sin(x)=1 when x=pi/2 and (3/2)*pi

sin(x)= 1/2 when x = pi/6 (30 degr).

Using the rules given yesterday, where angles in the 1st through to the 4th quadrant are represented by:
x       if angle in 1st quadrant (0 <= x < pi/2) i.e., 0<=x<90,
pi-x    if angle in 2nd quadrant (pi/2 <= x < pi), i.e., 90<=x<180
pi+x    if angle in 3rd quadrant (pi <= x < 3*pi/2) i.e., 180<=x<270
2*pi-x  if angle in 4th quadrant (3*pi/2 <= x < 2*pi) i.e., 270<=x<360

The remaining solution comes from the 2nd quadrant, i.e., x= pi-pi/6 = (5/6)*pi or 150 degr.

Ans: x=pi/6, pi/2, (5/6)*pi, [(3/2)*pi is NOT an admissible solution, since it exceeds pi radian]

======
Part 2: Use fundamental definitions

Q1. Let angle x=^ACB in the following triangle.

A
|
|
|
B---------C

By definition, sin(x) = |AB|/|AC|. [side opposite angle/hypotenuse]
If we let length |AB|=0.3, |AC|=1, the ratio fits perfectly.

By definition, cos(x) = |BC|/|AC|. [side adjacent to angle/hypotenuse]
Using Pythagoras, |BC|=sqrt(|AC|^2-|AB|^2)=sqrt(1-0.3^2)=sqrt(0.91)=0.954.
Hence, cos(x)=|BC|/|AC|=0.954/1.

Q2. 180 < x <270, sec x = -sqrt(10), what is tan x?

Review "A.S.T.C." trick.

Recall sec(x)=1/cos(x).
Note:  cos(x) is negative in 3rd quadrant, so 1/cos(x) is negative as well.
      tan(x) is positive in 3rd quadrant, however.

Now, with the signs taken care off, we expect tan(x) to be positive.

For ease of analysis, consider a triangle in the 1st quadrant. I'll draw a mirror image because it would otherwise be impossible to line things up. Let angle x = ^ACB as before.

A
|
|
|
B--------C

By definition, cos(x)=|BC|/|AC|. So, sec(x)=|AC|/BC|.
If we let |AC|=sqrt(10), |BC|=1, then sec(x)=sqrt(10) as required.
Using Pythagoras, |AB|=sqrt(|AC|^2-|BC|^2)=sqrt(9)=3.

By definition, tan(x)=|AB|/|BC| ....(opposite/adjacent side)
So, tan(x)=3.

Note: Generally speaking, our job would still be incomplete at this stage. We still have the sign to take care of. If we just do the analysis in the first quadrant, without paying attention to the properties of "A.S.T.C.", we would only get the magnitude right (and possibly, the sign wrong). However, as we have already worked out that tan(x) is positive in the 3rd quadrant, and we are restricting x between 180 and 270 degrees, there is nothing more to be done.

Indeed, tan(x)=3.  

Josh

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When I work through problems, I like to emphasize concepts which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponential & Logarithms. Basic Calculus. Probability, Permutation and Combination. Mathematical Induction. Complex numbers. Physics problems.

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