Find the exact value of sin x if tan x = 5 and 0 ≤ x ≤ π/2?

Answer 1

tanx=sinx/cosx and cosx=sqrt(1-sin^2(x))
==> sinx/sqrt(1-sin^2x)=5
sin^2(x)/(1-sin^2x)=25
sin^2(x)=25-25sin^2x
26sin^2(x)=25
sin^2(x)=25/26
sinx=+-5/sqrt(26)
0= ==> sinx=+5/sqrt(26)

Answer 2

merely split the crucial at aspect t. F(x) = a million/[4 sin x - 3 cos x] t ? F(x) dx + -?/2 ?/2 ? F(x) dx = t (After comparing the bounds from the left and authentic) (-2/5) ln(2) - ?i/5 The imaginary area comes from the indefinite crucial being: (a million/5) * (ln[ (3 * tan(x/2) - a million) / (tan(x/2) + 3)]) ignore concerning the scaling ingredient for the on the spot and look at each and every time period: @ x = ?/2 ? -ln(2) @ x = -?/2 ? ln(2) + i? Now: -ln(2) - (ln(2) + i? ) = -2ln(2) - ?i there is merely one time period that has the ln of a detrimental form. hi Dr. D. truly Maple would not like the crucial any better than Mathematica (apparently) does. i'm integrating from -?/2 to arctan(3/4), then including to that the crucial from arctan(3/4) to ?/2 by using a dummy variable then taking the reduce. it truly is merely the second one period that promises the imaginary area. It merely so takes position that Maple will locate the indefinite crucial, which even as evaluated on the endpoints supplies the same consequences because the previous technique. I gained't have better time to post till later, yet when the question continues to be open i visit teach the outcomes for the intermediate calculations. Later.... O.ok. So I went lower back and regarded back at what I had executed. Dr. D is sweet, I neglected a demonstration. the incredible answer is as he reported: (-2/5) ln(2) thanks Dr. D for assisting me locate that mistakes! Later nonetheless.... I found out the thanks to rigidity Maple to guage this crucial, i ask your self if Mathematica has the same selection. do this (once you've Maple) int(a million/(4*sin(x)-3*cos(x)), x = -(a million/2)*Pi .. (a million/2)*Pi, 'CauchyPrincipalValue') without the command to locate the central fee, Maple says the crucial is undefined. it truly is why I at the start changed into doing it by hand, which carry about my mistakes above. i'm particular happy for this discovery, that is going to likely be quite sensible contained in the destiny. i'd guess Mathematica has the same selection. thanks.

Answer 3

Two approaches:

1. Use scientific calculator to find
Sin( InvTan(5 radians) ) = 0.98058 radians.

2. Use basic principles
Tan(x) = Sin(x) / Cos(x) = 5
Sin(x) * Sin(x) = 25 * Cos(x) * Cos(x) = 25 * ( 1 - (Cos(x) * Cos(x)) )
Sin(x) * Sin(x) = 25 - (25 * (Cos(x) * Cos(x)) )
25 = 26 * Sin(x) * Sin(x)
Sin(x) = SquareRoot(25/26) = 0.98058
(Taking the square root is easiest with a scientific calculator but could be done with a table of logarithms if necessary.)

Answer 4

Sin = .9805 Tan = 5 Cos = .1961

Answer 5

we can write tanx as sinx/cosx

write cosx as sin(90 - x)

according to u x can take the value 90 (pie by 2) which means sin0 and hence sinx = infinity

ther can never be one specific value for sinx under the given conditions as the range sinx is from -1 to 1

Answer 6

the length of the hyp is sqrt(25+1)=sqrt(26)

sinx=5/sqrt(26)

sqrt(26)=5.0990195

sinx=0.9805806

Answer 7

it might be approximately 0.981

Answer 8

xyz