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KLscilevothma
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sin hx, cos hx, tan h x... What are they? What does the "h" mean here?
Any explanations or websites would be appreciated.
Any explanations or websites would be appreciated.
Originally posted by Lonewolf
Also:
sinh x = -i*sin(i*z)
cosh x = cos(i*z)
Where i is sqrt(-1)
sinh x = -i*sin(i*z) ...(1)
sinh x = (ex-e-x)/2...(2)
Originally posted by KL Kam
I guess z is a complex number, right?
Differentiate sin hx = ex (interesting )
but I don't think a function involving complex number, -i*sin(i*z), is differentiable, or am I wrong?
Why -i*sin(i*z) = ex-e-x)/2 = sin hx ? [/B]
#Lonewolf meant to use x on both sides, I think.
oops, I swollowed the "-" sign when I differentiated e^(-x)d(sinh x)/dx=(1/2)(ex+e-x)
Is it because we can represent a function with complex variables by using an Argand(sp?) Diagram, just as we use a Cartesian plane to represent functions with real variables?Yes, you can differentiate functions with a complex variable, in almost the same way as a real function.
Originally posted by KL Kam
Is it because we can represent a function with complex variables by using an Argand(sp?) Diagram, just as we use a Cartesian plane to represent functions with real variables?
(1/2)(ex+e-x)=cosh x
Is it the defination of cos hx ?
I've heard of Euler's formula eix = cos x + i sin x
are the hyperbolic functions somehow related to Euler's formula?
X = A sin h θ
how can we write θ in terms of X and A ?
Sin hx, cos hx, and tan hx are all trigonometric functions that are used to calculate the relationship between the sides and angles of a right triangle. The main difference between these functions is the ratio that they represent. Sin hx is the ratio of the opposite side to the hypotenuse, cos hx is the ratio of the adjacent side to the hypotenuse, and tan hx is the ratio of the opposite side to the adjacent side.
Trigonometric functions like sin hx, cos hx, and tan hx are used in various scientific fields such as physics, engineering, and astronomy. They are used to calculate distances, angles, and forces in different systems. For example, in physics, these functions are used to calculate the trajectory of a projectile or the amplitude of a wave.
The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system. It is used to understand the values of sin hx, cos hx, and tan hx at different angles. The x-coordinate of a point on the unit circle represents cos hx, and the y-coordinate represents sin hx. The tangent of an angle can be calculated by dividing the y-coordinate by the x-coordinate at that angle.
Some important properties of trigonometric functions include periodicity, symmetry, and monotonicity. Sin hx and cos hx are periodic functions with a period of 360 degrees or 2π radians. They also exhibit symmetry about the origin. Tan hx is a monotonic function, meaning it continuously increases or decreases without repeating any values.
The inverse trigonometric functions, such as sin^-1 hx, cos^-1 hx, and tan^-1 hx, can be used to find the angle given the ratio of sides. For example, if we know the ratio of the opposite side to the adjacent side is 0.5, we can use the inverse tangent function to find the angle. tan^-1 (0.5) = 26.57 degrees or 0.463 radians.