Properties of Trigonometric Functions

Trigonometric Functions on the Unit Circle

Given a point, P(x, y), on the unit circle we can create a triangle by drawing the straight lines from the point to the x-axis and from P to the origin. This creates a triangle with vertices P, (x, 0), and (0, 0). This also creates an angle starting at the x-axis and ending at the line segment from P to the origin. This allows us to define the six trigonometric(trig) functions based on the coordinates of P. All of the trigonometric functions take the angle created by the mentioned line segment, when defined.

The sine function outputs the y coordinate of P.

The cosine function outputs the x coordinate of P.

The tangent function outputs the ratio of the x-coordinate of P to the y-coordinate of P, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {y}{x}}}

The cosecant function outputs the reciprocal of the sine function output, when defined. So when y is nonzero, the cosecant function outputs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{y}}}

The secant function outputs the reciprocal of the cosine functions output, when defined.

The cotangent function outputs the reciprocal of the tangent functions output, when defined.

Basic Properties

To each angle we can associate one point on the unit circle. Thus, we can compute the Sine and Cosine of any angle. This means the domain of both functions is all reall numbers.

As for the tangent function, on the unit circle it outputs the ratio of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {y}{x}}} for the point (x, y) associated to any angle. Thus, the tangent function is defined as long as the angle does not coincide with the y-axis, so 90$({\frac {\pi }{2}})$ or 180Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\pi)} degrees (radians), when restricing the looking at the angle when measured between 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} .

Since the secant function is defined as the radius of the circle through the point (x, y) divided by the x-coordinate, so it is undefined when the angle corresponds with the y-axis.

Similarly the cotangent and cosecant functions are defined for all angles not corresponding with the x-axis.

We can make the observation that the radius is larger than either coordinate of the point associated to an angle, unless the angle corresponds to an axis, in which case the radius is equal to the absolute value of the non-zero coordinate. Thus we have the range of both Sine and Cosine is [-1, 1].

By a similar argument we can conclude that the range of both Secant and Cosecant is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1] \cup [1, \infty)}

It may be a bit harder to understand, but we can also conclude that the range of both the tangent and cotangent functions is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, \infty)}

Periods of trigonometric functions

We can notice that after going all the way around the circle the values for the trig functions we just repeat, since the angle coincides with a angle from the first revolution around the circle.

We can observe that Sine, Cosine, Secant, and Cosecant will repeat the same patterns each revolution around the circle. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin(\theta + 2\pi) = sin(\theta)} . We could replace sin with any of the other three mentioned functions and the statement would still hold.

For the last two functions, tangent and cotangent, we only need to rotate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} radians before the values begin repeating. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tan(\theta + \pi) = tan(\theta)} and the same holds for cotangent.

Even/Odd Properties

We can observe that sine, cosecant, tangent, and cotangent are odd functions, while cosine and secant are even.

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