The sin, cos and tan functions

Background: In what follows we assume that you are familiar with trigonometry. The sin, cos and tan functions are important in trigonometry and many other areas of mathematics. Here is how they are derived. Consider the vector (the red arrow) in the picture to the right. It has its tail at the origin, has length r and is oriented at angle θ.

Let (x, y) denote the coordinates of the head of the vector (i.e. let x and y be the movements in the x and then in the y direction required to get from the tail to the head of the vector.) The three arrows form a triangle in standard position.


Now imagine changing the angle θ. The vector will point in another direction but its head will still be somewhere on the dotted circle (because its length r is unchanged).

The values of x and y will change. For example in the picture to the right the values of x and y are both negative.



Definitions: The sine (or sin), cosine (or cos) and tangent (or tan) functions are defined as returning the following ratios:
These ratios are functions of θ because x and y change with θ.



Graph of the sin function: The picture on the left shows the red vector pointing at various angles θ and the graph on the right shows the resulting function sin (θ):



Graph of the cos function: The next picture on the left again shows the red vector pointing at various angles θ and the graph on the right shows the resulting function cos (θ):



Graph of the tan function: The next graph shows the function tan (θ). The dotted vertical lines are asymptotes (lines that the function approaches but never touches):



In the above three graphs the angle θ is measured in radians. If you want θ to be measured in degrees then simply change the horiontal scale so that θ runs from 0 to 360° instead of from 0 to 2π radians; the shapes of the graphs are otherwise unchanged. The sin, cos and tan functions are said to be periodic. This means that they repeat themselves in the horizontal direction after a certain interval called a period. The sin and cos functions have a period of 2π radians and the tan function has a period of π radians.


Domain and range: From the graphs above we see that for both the sin and cos functions the domain is all real numbers and the range is all reals from −1 to +1 inclusive. For the tan function the domain is all real numbers except ±π/2, ±3π/2, ±5π/2, …, (or in degrees: ±90°, ±270°, ±450°, …), where the tan function is undefined. The range of the tan function is all real numbers.

The definitions of sin, cos and tan can be extended to the complex numbers by defining the functions by their Taylor series instead of by the ratio of two lengths. In that case, the domain and range of the sin and cos functions is all complex numbers, and the domain of the tan function is all complex numbers except ±π/2, ±3π/2, ±5π/2, …, where the tan function is undefined, and the range is all complex numbers.


Special values: For the two triangles shown below, Pythagoras' theorem gives simple, exact values for the lengths of the sides and hence for the values of the sin, cos and tan functions. The following table gives these values as well as those for angles of 0° and 90° :




Algorithms for calculating sin, cos and tan: Have you ever wondered how calculators and computers are able to calculate functions like sin, cos and tan? The answer is that they make use of formulas like these:


These formulas are called polynomial approximations and are based on Taylor's series. To use them x must be in radians. They are very accurate when x is close to 0 but lose accuracy as x gets bigger. When x = π/4 radians (i.e. 45°) the sin formula is only accurate to within ±0.00004, cos to within ±0.000004 and tan to within ±0.004.

If x is greater than π/4 these formulas are too inaccurate to be used directly. Instead cofunctions and symmetries of the sin, cos and tan functions are exploited to reduce the angle x and improve the accuracy. For example, to calculate sin(440°), use is made of the fact that this is the same as sin(80°), which is the same as cos(10°) which is the same as cos(0.174533 radians), which is then computed using the cos formula. Click here to see algorithms that computers use for calculating the sin function, the cos function and the tan function.


How to use the sin, cos and tan functions in the Algebra Coach