Solve algebraically: √2 sin(x) - 1 = 0?

Question

The first part is easy, you end up with x = 45 degrees. However, it is the domain the question gives ( -2pi <= x <= 2pi) that is giving me problems. I always forget how to rotate and get all the rest of the answers for the given domain. Thanks for the help!

Answer 1

√2 sin(x) - 1 = 0, -2π ≤ x ≤ 2π

So sin(x) = 1/√2

In the domain 0 ≤ x ≤ 2π x is in the first or second quadrant and so x = π/4, π - π/4

ie x = π/4, 3π/4

However this domain has been enlarged by an extra rotation of -2π to -2π ≤ x ≤ 2π

Thus two additional solutions arise by subtracting 2π (ie one extra revolution) from the solutions above

So as x is in the first or second quadrant then, in this domain,

x = π/4 - 2π, 3π/4 - 2π, π/4, 3π/4
= -7π/4, -5π/4, π/4, 3π/4

Answer 2

√2 sin (x) - 1 = 0

Add 1 to both sides:

√2 sin (x) = 1

Divide by √2:

sin (x) = 1/√2

One solution, as you have indicated, is 45 degrees. Remember that on the unit circle, the point forming an angle theta with the positive x-axis has coordinates (cos theta, sin theta). The other point on the unit circle with the same y-coordinate as 45 degrees is 135 degrees. So sin (135) = 1/√2 also. Thus, 45 degrees and 135 degrees work, or pi/4 or 3pi/4.

At this point, we know x = pi/4 or x = 3pi/4 work. Since the sine function is periodic with period 2pi, then (pi/4) - 2 * pi and (3pi/4) - 2 * pi also work.

Solution: x = -7pi/4, -5pi/4, pi/4, 3pi/4.

Answer 3

let x = sin x 6x^2 - x - 2 =0 factorise (3x-2)(2x+1) = 0 x = 2/3 or x = -1/2 which means sin x = 2/3 or sin x = -1/2 x = arc sin 2/3 or x = arc sin -1/2 x = 41.81° or 30° the possible answers 41.81° , 180°+30° or 360° - 30° if the angle is between 0 to 360°

Answer 4

ok so u cant have an answer of 45 degress when the problems in radians so its not 45 its sqroot2/2.

look: sinx= 1/sqroot2 (then rationalize)
Then you have to look on ur unit circle and see all the places that sin is 2/sqroot 2. Those are ALL of ur answers. he problem includes the info ( -2pi <= x <= 2pi) because its saying just to give the answers in 2pi (or 360 degress), or else the answers would go on forever because you could ust keep goin around the school

holla if u need more help love

[e-mail]

Answer 5

Sin x = 1/sqrt(2) at two points in -2pi< x <2pi

at x = pi/4 and x = pi/4+pi/2 = 3pi/4 or 45+90 = 135 deg. That's it.

However, you can also get to pi/4
by going -2pi+pi/4 = -7pi/4
and
-2pi + 3pi/4 = -5pi/4 = -180 -45 = -225 deg

See, your domain overlaps.

Answer 6

Here is a little graphical mnemonic I learned as a young lad in England, that really helps figure out what the signs of sine, cosine, and tan (theta) are in the four domains formed by the x- and y-axes.

(I tried typing so that it would appear properly on the Answer page, but the spacings don't work out. So I'll just describe it, instead.)

Put a big 'C' in the 4th quadrant, big 'A' in the 1st, big 'S' in the 2nd, and big 'T' in the 3rd quadrant. Reading anti-clockwise from the 'C', these letters spell the word CAST. What good is that, you may well ask?

Well, first of all, I've remembered that word "CAST" for over 50 years, without needing to use it very much (but it's been there, lurking in my mind, when I did); so there must be something memorable about it!

Secondly, it tells you what you want to know:

'C' means: only Cosine is positive here.
'A' means: All are positive here.
'S' means: only Sines are positive here; and
'T' means: only Tangents are positive here.

So: these four letters C, A, S, T tell you which of (or in the 'A' case that all of) the trigonometric functions sine, cosine and tangent is (or are) positive in that quadrant.

I hope this has helped.

Live long and prosper.

Answer 7

√2sin(x) = 1
sin(x) = 1/√2 = √2/2
x = arcsin(√2/2) = 45° = π/4
Since sin has even periodicity around nπ/2 (where n is an odd integer), then the 'other' answer is π/4 + π/2 = 3π/4 (or 135°). And, of course, since sin has a total period of 2π, adding 2π to either of those answers will produce another answer. This is why you're 'limited' to
-2π <= x <= 2π.


Doug

Answer 8

root2sinx=1
sinx=1/root2
x=45
in the first and second quadrants sin is +ve
x=135

now add 90 or simply to say write it as 45+2npi where n can take values from 0,1,2............

now see what satisfies..

45,135,-225(135+90 in third quad)and -315 (in 4th quad,225+90)
in last two,sin is -ve,so - sign..

Answer 9

rt2sinx=1
sinx=1/rt2
x=pi/4
sin is +ve in the first and second quadrant
x=3pi/4
also you can add any number of2pi or subtract any no of 2pi
so sin -7pi/4=pi/4 and sin-5pi/4=3pi/4
so the four angles for which it is true will be
pi/4,3pi/4,-5pi/4 and -7pi/4

Answer 10

Ya know i hate to break it to you, but teachers *do* check these.