Trig functions Definition and 12 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. pairofstrings

    B Cosine of 1 degree and cosine of 60 degrees?

    Why is cos (1)° = 0.9998? cos(60)° = ½? Thanks.
  2. R

    Calculating the half maximum point of a function

    This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
  3. C

    What is conceptual physics?

    I have been wondering, what is conceptual physics? I remember taking a class in high school that was physics oriented, for example two trains leave a station at different speeds, and arrive at a central point, where do they overlap. Also there were trig functions on how to find the height of a...
  4. Jeviah

    Answer to cos^2x^2

    Homework Statement hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
  5. Amlan mihir

    Find the value of the trigonometric sum

    Homework Statement i have no idea how to proceed
  6. OldWorldBlues

    B Using trig to find distance?

    Hi there! I haven't yet taken a trigonometry course (I'm in High-school), but I have an amateur interest in surveying. Recently I began thinking about how I could calculate the height of a point relative to me, or the distance of the object from me. Naturally, I immediately thought of the...
  7. DeathbyGreen

    I Infinite series of trigonometric terms

    I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...
  8. G

    B Add two functions, same frequency to produce one greater?

    Is their any way to add two wave functions like sin or cos in such a way that you could double the frequency or at least increase it?
  9. F

    Need help solving this trig equation

    Homework Statement what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114 Homework Equations (sinθ)^2 + (cosθ)^2 = 1 The Attempt at a Solution I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
  10. SSGD

    I Solve 2*C*sin(W)+P*cos(N*W)=P for W

    Is there a way to solve for W in the below equation. There has to be multiple solution for W, but I am at a loss as to how to solve this. 2*C*sin(W)+P*cos(N*W)=P or 2*C/P*sin(W)+cos(N*W)=1 C and P are constants N is an integer
  11. Erenjaeger

    Which of these relations are functions of x on R

    Mentor note: moved to homework section y = sin(x) y = cos(x) y = tan(x) y = csc(x) y = sec(x) y = cot(x) (a) 0 (b) 4 (c) 6 (d) 2 I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2
  12. I

    High School Level Physics Homework

    Homework Statement A highway is to be built between two towns, one of which lies 35.0km south and 72.0km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed with respect to due west. Homework...