How to find the tangent if we know the cosine

The concept of a tangent is one of the basic in trigonometry. It represents some trigonometric function which is periodic, but not continuous in the area of definitions of sine and cosine. And has discontinuities at the points (+,-)PI*n+PI/2, where n is the period of the function. In Russia he is referred to as tg(x). It can be represented using any trigonometric function, as they are all closely interrelated.

You will need

• Textbook on trigonometry.

Instruction

1. In order to Express the tangent of an angle using a sine, you need to recall a geometric definition of tangent. So, the tangent of an acute angle in a right triangle is the ratio opposite over adjacent.
2. On the other hand, consider a Cartesian coordinate system where the unit circle is drawn with radius R=1 and with center in the origin. Take counterclockwise rotation as positive and the opposite direction negative.
3. Mark a certain point M on the circle. It will drop a perpendicular on the axis Oh, call it point N. get the triangle OMN, in which the angle ONM is a direct.
4. Now consider acute angle MON, by definition of sine and cosineand an acute angle in a right triangle
   sin(MON) = MN/OM cos(MON) = ON/OM. Then MN= sin(MON)*OM and ON = cos(MON)*OM.
5. Returning to the geometric definition of the tangent (tg(MON) = MN/ON), substitute the above expression. Then:
   tg(MON) = sin(MON)*OM/cos(MON)*OM cut OM, then tg(MON) = sin(MON)/cos(MON).
   
6. From the basic trigonometric identities (sin^2(x)+cos^2(x)=1) Express the cosine, using the sine: cos(x)=(1-sin^2(x))^0,5 Substitute the expression obtained in step 5. Then tg(MON) = sin(MON)/(1-sin^2(MON))^0,5.
7. Sometimes there is a need to calculate the tangent double-angle and half-hearted. Here, too, relations are derived:tg(x/2) = (1-cos(x))/sin(x) = (1-(1-sin^2(x))^0,5)/sin(x);tg(2x) = 2*tg(x)/(1-tg^2(x)) = 2*sin(x)/(1-sin^2(x))^0,5/(1-sin(x)/(1-sin^2(x))^0,5)^2) =
   = 2*sin(x)/(1-sin^2(x))^0,5/(1-sin^2(x)/(1-sin^2(x)).
8. It is also possible to Express the square of the tangent of a double angle cosineand or sine. tg^2(x) = (1-cos(2x))/(1+cos(2x)) = (1-1+2*sin^2(x))/(1+1-2*sin^2(x)) = (sin^2(x))/(1-sin^2(x)).