Sin, Cos, Tan?

Question

Given cscx = (u^2+1)/(2u) and cot x < 0, find sinx, cosx, and tan x



pleeeasseee i'm so confused

Answer 1

By definition: sin x = 1/ csc x

sin x = 2u / (u²+1)

cos x = √(1-sin²x) = √( 1 - [4u² / (u²+1)²] ) = (u² - 1) / (u² + 1)

tan x = sin x / cos x = 2u / (u² -1)

Edit:
From sin x = 2u / (u²+1),
you can also consider sin x as O/ H (opposite over hypotenuse)
O = 2u , H = (u²+1)
Then A² = H² - O² = u⁴ + 2u² + 1 - 4u² = u⁴ - 2u² + 1
A = u² -1.

You can now apply SOHCAHTOA.
You still get the above answers.


d:

Answer 2

Lets assume you are dealing with a right triangle with and are looking at an angle other than the right angle. The stipulation that cot x < 0 means that this is the case. Our three sides are h (hypotenuse), a (adjacent), and o (opposite). The following trig identities hold true.

Sin x = o/h
Cos x = a/h
Tan x = o/a
Csc x = 1/Sin x = h/o
Sec x = 1/Cos x = h/a
Cot x = 1/Tan x = a/o

So from the information given

h = u^2 +1
o = 2u

We must use Pythagoreans theorem to find the third side a.

h^2 = o^2 + a^2

a^2 = h^2 - o^2

a^2 = (u^2 +1)^2 - (2u)^2

a^2 = u^4 + 2 u^2 + 1 - 4 u^2 = u^4 - 2 u^2 +1 = (u^2 - 1)^2

a = u^2 - 1

Now we can find tan x, sin x, and cos x.

sin x = o/h = (2 u) / (u^2 +1)
cos x = a/h = (u^2 -1) / (u^2 +1)
tan x = o/a = (2 u) / (u^2 - 1)

Answer 3

So am I. There is no constant for "u" and I'd have to work this out in BASIC to really see it.

It seems a linear equation, so it has to do with simplification of the u^2 and the 2u and then this becomes factoring with "u" for all the others.

This is gobbly **** algebra to me. Too abstract.

IF I were to put this into Visual BASIC and assign a value to "u" i would probably come up with something.

"u" has to equal a radian within the spectrum of something like -6 to +6

It's been a long time.

To get SINE, COSINE and TANGENT angle you have to deal in RADIANS or DEGREES and the "u" thing has to fit in that motif.

That's about all I can tell you.

If you're in degrees "u" has to be within a range of 0 to 360, but generally it's radian

That's what, -6.2 to + 6.2 or something like that.

There is a MAXIMUM and MINIMUM range of possible RADIANS in SINE, COSINE and TANGENT ANGLE

TRY assigning a value to "u" within the range of -6 and +6 and doing the math
(-2^2)/(-2*2) and the COTANGNET is less than ZERO

This is real world not abstract, but it might help

If you can't do the math, then you are in over you're head.

Not sure wash CSCX means

You have to define that one

OK

Let's UNDERSTAND what SINE COSINE and TANGENT are in the REAL WORLD

A surfer in a surf board

The WAVE

Tangent angle is a LINE drawn in the shallow of that WAVE where the SURF BOARD intersects (radian) at a point of SINE or COSINE

A boat is travelling in the ocean.

It's bottom intersects the SINE of one side of the wave and the COSINE of the opposite wave and the TANGENT is the line drawn on the outside of the curve where the bottom of the boat touches the SINE and COSINE

SINE and COSINE move accordingly and the boat RISES and FALLS inside that wave and the TANGENT is the straight line drawn against the CURVE of the WAVE at the point (sine or cosine) where the BOAT'S BOTTOM touches the CURVED WAVE.

Think of a SINE WAVE, a smooth circle, a SEMI CIRCLE

Think of a straight line going across that VALLEY from the down side of the wave on the left to the upside of the wave on the right.

That is the SINE (on one side) and COSINE

And these move as the BOAT rises with the CREST of the wave and the TANGENT is the line where the BOAT intersects the wave on the CURVE

Think of the Earth. The North Pole

We see a curved top.

The North Pole

We draw a straight line that touches the north pole

That line is the TANGENT ANGLE

If we move to the equator that is 90 degrees away to either side and we draw the same straight line and that is the TANGENT angle.

The TANGENT is a STRIGHT LINE drawn so it touches ONE SPOT (which can be defined as EITHER sine or COSINE) on the curve.

COSINE is the OPPOSITE of SINE

If you draw a line on the WESTERN side of the EARTH at the HORIZON at the EQUATOR and call that the TANGENT and the SINE then the COSINE is the EASTERN TANGENT

If you draw the TANGENT (straight line) at the NORTH POLE, and call this the SINE then the COSINE is the SOUTH POLE

An ocean wave

You find the right point of SINE, draw a TANGENT ANGLE and then postion a SURF BOARD at a 90 degree angle to the TANGET you get motion.

You get kicks

You get a sufer moving along under the power of that wave until he goofs up and misses the sine and then WIPES OUT

Because he loses that point, that TANGENT ANGLE that is OPTIMAL

This is the BEST I can do and I'm pretty weak on this stuff.

But this is the BEST I can do to explain the HARMONY of SINE, COSINE, TANGENT

Try some REAL WORLD figures between RADIANS -6 and +6 on this problem with U being in that range and SEE what you get when you DO the math

THEN turn it into an ABSTRACT

This is REVERSE engineering, which MATH people HATE

But it can work.

Answer 4

Construct a right angled triangle ∆ABC, with base CA, vertical side BC and angle ACB = 90°.

Now it is given that cscx = (u² + 1)/(2u) so, by definition, (u² + 1) = hypotenuse AB and 2u = BC. The base CA, by Pythagoras Theorem, will then be:

CA² = AB² - BC² = (u² + 1)² - (2u)² = (u² - 1)² or CA = u² - 1.

So that from ∆ABC: sinx = BC/AB = 2u/(u² + 1), cosx = CA/AB = (u² - 1)/(u² + 1) and tanx = BC/CA = 2u/(u² - 1).

Answer 5

cscx= 1/sinx
sinx = 1/cscx = (2u)/(u^2+1)
(cosx)^2 + (sinx)^2 = 1
cosx = (1-(sinx)^2)^.5 = u^4 - 4u^2 + 2u + 1
tanx = sinx/cosx = (2U) / ((u^2 + 1)(u^4 - 4u^2 + 2u + 1))
looks confusing

Answer 6

Could you put that in a calculator and press the sin, cos, and tan button? That was probably really stupid but yeah I'm not sure how to find that. Good luck.

Answer 7

sin x = 1/ cscx
sin x = 2u / (u²+1)
cos x = √(1-sin²x) =√( 1 - [4u² / (u²+1)²] ) = u² - 1 / u² + 1
tan x = sin x / cos x =2u / u² -1

man i hated that stuff hope that helps

Answer 8

The only help I can offer is, soc, cah, toa,

which is sin over cos, cos over something, and tan over something

I know this doesn't help much, but it's all I got.

Answer 9

sin(x)=(2u)/(u^2+1)
sin(x)/cot(x)=sin^2(x)/cos(x)
sin(x)/tan(x)=cos(x)