What is Trigonometric functions: Definition and 163 Discussions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

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  1. O

    Derivative of inverse hyperbolic trigonometric functions

    I'm working on a pre-freshman year math packet for college, and at one point it asks for the derivative of sinh-1(x), followed up by the derivative of ln( x + sqrt(1+x2) ). In high school, we never really covered hyperbolic trigonometry, but I have previously derived that the inverse of sinh is...
  2. S

    Differentiation of Inverse Trigonometric Functions

    Hey guys, I have to know how to Differentiate Inverse Trigonometric Functions in my next exam and need somewhere to study up on them. Do you know of any web sites I could read? Can't find anything on Karl's Calculus. Thanks
  3. T

    Trigonometric functions like sin(2x)=2sin(x)cos(x)

    Hi guyz, as we know we have some known relations in the trigonometric functions like sin(2x)=2sin(x)cos(x) and sin(x/2)=1/2-1/2 cos2x My question is are there similar formulas for arcsin and arccos? I know those only ! arcsin x =ln(ix-sqrt(1-x^2)) arccos x =ln(-ix-sqrt(1-x^2))...
  4. F

    Applying trigonometric functions to some real life situations

    I'm having some trouble with applying trigonometric functions to some real life situations, particularly this one problem in my homework. Andrea, a local gymnast, is doing timed bounces on a trampoline. The trampoline mat is 1 meter above ground level. When she bounces up, her feet reach a...
  5. L

    Chapter Summary: Trigonometric Functions

    The problem reads: Find \sin\theta and \cos\theta Part a gives me the coordinates \left(-1,1\right) The triangle I got had the x-length as -1, while the y-length was 1. The hypotenuse I got was \sqrt{2} Since \sin is \frac{opposite}{hypotenuse} I got \sin\theta=\frac{1}{\sqrt{2}}...
  6. D

    Graphing Trigonometric Functions Question

    Ok I can graph sin(x) , cos(x) , and tan(x) pretty easily, but I'm having a hard time graphing the csc, sec, and cot ones. For the first three I just found the values of pi/2 pi 3pi/2 and 2pi. So for example pi/2 for sin(x) would be sin 90 or sin pi/2 which is equal to 1. I then just put a...
  7. M

    Periods of Trigonometric Functions

    Here are two general questions How would you find the period of: sin(2Pi*t)+sin(4Pi*t) or cos(3t)sin(2t) Thanks
  8. M

    Trigonometric Functions - Confusing question

    What equation in the form y=sin(theta)+c best models the dtat in the chart below... ____________________________________ (theta)radians : 1 : pi/2 : pi : 3pi/2 : 2pi ______________________________________ y-------------: 2 :--3--:-2-:-1----:--2 __________________________________ I am...
  9. V

    Adding Trigonometric Functions: Get the Answer Here!

    Hi, wat would you get if you add these: 3cos(600*pi*t) + 5sin(1000*pi*t - 45degrees) + 5sin(1200*pi*t)? thanks in advance...
  10. K

    Inverse Trigonometric Functions / Rates of change with 2 points of references

    Hi, A questions regarding trigonometric (or inverse) functions. I just can't get it right. Q1: A picture 2 m high is hung on the wall with its bottom 6 m above the observer's eye level. How far should the viewer stand for the picture to subtend the largest possible vertical angle with the...
  11. Jeebus

    Integrating Trigonometric Functions: Need Help!

    I need help with these problems: (1) STan^-1(y)dy and (2) Ssin^-1(x)dx (S=integral) I can't seem to figure out the trigonometric formula for integration. A little help or hint would be nice. Thanks!
  12. M

    Trigonometric Functions Graphing

    Ok I have hit a little bumb in the road on my trigonometric function learning. Ok i need to graph the function y=3cos2(x-120degrees)-3, for x is less than or equal to 360degrees but is greater than or equal to -360degrees. Ok so i know the: verticle translation is 3 down the phase shift is...
  13. M

    Trigonometric Functions- Grade 11

    Ok I have two questions both i have done but not 100% sure I have done them correct. I am good with math but not so well at Trigonemetric stuff. Ok here it is. 1) Convert the following angles to radian measure, leave in simpilest rational form. 350 degrees. Ok what i have done is this...
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