How do you calculate the tangent of an angle if you only have its sine?

Question

How do you calculate all of the angle's trigonometric functions using only one? There are formulas for this, does anybody have them?

Answer 1

www.google.com - try trig identities

Answer 2

If you have a right triangle with an angle of interest a, and

x = adjacent side (the side adjacent to the angle of interest)
y = opposite side (the side opposite the angle of interest)
r = hypotenuse

then

sin a = y / r
cos a = x / r
tan a = y / x = (y / r) / (x / r) = sin a / cos a
cot a = x / y = 1 / tan a
sec a = r / x = 1 / cos a
cosec a = r / x = 1 / sin a

The Pythagorean theorem says

x^2 + y^2 = r^2

but x = r cos a
and y = r sin a,

so making the appropriate substitutions and dividing through by r^2 gives the trigonometric identity

(cos a)^2 + (sin a)^2 = 1

from which you can get

cos a = sqrt (1 - (sin a)^2)

Therefore,

tan a = sin a / sqrt (1 - (sin a)^2)

You can write any trigonometric function in terms of any other trig function by a similar process.

Answer 3

let sin A=a
1. make a right angled triangle PQR with a right angle at Q andone angle at R=A. Call the length of the side opposite to A i.e PQas'' a and hipotenuse=1. By pythagoras theorem, we can find the side QR=sqrt(1-a^2). then the tangent of the angle A is given by
tanA=PQ/QR or -PQ/QR. the signof tanA is determined by the quadrant in which the angle A lies
2.if sin A=a, cosA=sqrt(1-a^2) and
tanA=+/-a/sqrt(1-a^2). the sign of tanA is found againas explained above

Answer 4

cos^2(x) = 1 - sin^2(x) ..........(1)
is the one you want here.
tan(x) = sin(x) / cos(x)
From (1) cos(x) = +/- sqrt(1 - sin^2(x))
You need to determine the quadrant of the angle to ensure you get the right sign on the square root. For angles in the range 0 to 90deg, the + is the right one.

Other forrmulae derived from cos^2(x) = 1 - sin^2(x) are:
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = cosec^2(x).

Answer 5

If you have the sin of an angle and want a strictly calculation approach all you have to do to get the tan is hit inverse sin to get the angle in degrees. From there you can hit tan, sin, cos, whatever you want to get those properties.

Cheers

Answer 6

Suppose you know sin x = .631= 631/1000
Then 1000 represents the hypotenus and 631= side opposite. So find 3rd side = sqrt(1000^2-631^2)= 775.78.
So tan x = 631/775.78= .813375.

Answer 7

Let x be your angle. You know what sin(x) is. Two ways:

1) arcsin(sin(x)) = x; so, tan(arcsin(sin(x)))=tan(x)
2) cos(x) = square root of (1 - sin(x)); tan(x) = sin(x)/cos(x) = sin(x) / (1 - sin(x)) ^ (1/2)

Either way, double check the signs! tan(x) > 0 for 0 < x < 90° with period 180°.

Answer 8

If you have sin(θ), you can find cos(θ) using the identity
sin^2 (θ) + cos^2 (θ) = 1.

From there you can easily find the other four using the sine and cosine values, and by keeping the following in mind:
tan(θ) = sin(θ) / cos(θ)
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1/tan(θ), or cos(θ) / sin(θ)