We've mastered using the unit circle with the standard angles: and the axes. Angular functions which can be described as ratios of the sides of a right triangle to each other.It's almost graduation time here at Unit Circle U. ![]() A trigonometric function which represents the ratio of the opposite side of right triangle to its adjacent side. A trigonometric function which represents the ratio of the opposite side of a right triangle to its hypotenuse, or the y coordinate of a point on a unit circle. Triangles which have congruent angles and proportional sides. ![]() A triangle which contains a 90° or right angle Similar triangles One complete revolution is equal to 2π radians. A unit of angular measurement that relates the radius of a circle to the amount of rotation of an angle. It is used to find the distance between two points. An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. A value at which a periodic function begins to repeat. The side of a right triangle which is opposite the angle in question. A relationship between the sine of an angle of a triangle and its side which can be used to determine the dimensions of a triangle. A relationship between the cosine of an angle of a triangle and its sides which can be used to determine the dimensions of a triangle. The longest side of a right triangle which is opposite the right angle. A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol °. A trigonometric function that relates the ratio of the adjacent side of a right triangle to its hypotenuse, or the x coordinate of a point on a unit circle. A geometric figure created by two lines drawn from the same point. A characteristic of a periodic graph represented by half the distance between its maximum and minimum. The side of a right triangle which forms one side of the angle in question. Also, they can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound. For example, the times of sunsets, sunrises, and comets can all be calculated thanks to trigonometric functions. Mathematicians and scientists are now able to describe many types of natural phenomena which reoccur periodically with trigonometric functions. The periodicity of trigonometric functions is more important to modern trigonometry than the ratios they represent. Since the tangent is equal to y/x, its range is - ∞ to ∞ and its amplitude is ∞. Like the sine and cosine graphs, the tangent function is periodic, but it has a period of 180° or π radians. Of these, the most important is the graph of the tangent function. Graphs of the other trigonometric functions are possible. They also have an amplitude of one which is defined as half the difference between the maximum (1) and minimum (-1) values. The sine and cosine graphs are periodic because they repeat their values, or have a period, every 360° or 2π radians. (Angles which are greater than 360° or 2π radians represent an angle with more than one revolution of rotation). The magnitude of an angle can be any real number, so the domain of the graphs is all real numbers. Since the value for x and y can never be greater than one on a unit circle, the range for the sine and cosine graphs is between 1 and -1. With the trigonometric functions defined as such, a graph of each can be developed by plotting its value versus the magnitude of the angle it represents. The trigonometric functions could then be represented by the following equations. For example, coordinates of point P(x,y) can be used to define a right triangle with a hypotenuse of length r. If we consider the circle to represent the rotation of a side of an angle, then the trigonometric functions can be defined by the x and y coordinates of the point of rotation. A unit circle is one which has a radius of one unit which means x 2 + y 2 = 1.
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