Chapter 3: Parallel and Perpendicular Lines
... d. consecutive interior 4. Drag point C or F so that the measure of any of the angles is 90. a. What do you notice about the measures of the other angles? b. Make a conjecture about a transversal that is perpendicular to one of two parallel lines. Explore 3-2 Geometry Software Lab: Angles and Parall ...
... d. consecutive interior 4. Drag point C or F so that the measure of any of the angles is 90. a. What do you notice about the measures of the other angles? b. Make a conjecture about a transversal that is perpendicular to one of two parallel lines. Explore 3-2 Geometry Software Lab: Angles and Parall ...
IMO problems from Kalva `s Web
... is colored red or blue, so that if (h, k) is red and h' ≤ h, k' ≤ k, then (h', k') is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 sub ...
... is colored red or blue, so that if (h, k) is red and h' ≤ h, k' ≤ k, then (h', k') is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 sub ...
File
... How many three-digit numbers more than 600 can be formed by using the digits 2,3,4,6,7. How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4,5, 6, 7 and 8. ...
... How many three-digit numbers more than 600 can be formed by using the digits 2,3,4,6,7. How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4,5, 6, 7 and 8. ...
Contemporary Arguments For A Geometry of Visual Experience
... from the eye will bear the same relations to the spherical figures that are relied upon by Yaffe to establish the equivalency (see Belot 2003: 583). Up to this point, the contemporary argument has just involved picking out a set of points, or lines, in physical space and stipulating that they are th ...
... from the eye will bear the same relations to the spherical figures that are relied upon by Yaffe to establish the equivalency (see Belot 2003: 583). Up to this point, the contemporary argument has just involved picking out a set of points, or lines, in physical space and stipulating that they are th ...
Exam 1 Sol
... (c)(5 pts) If s(t) = 2t2 − 13t + 5 for t ≥ 0 describes the position of an object (in feet) at time t, find the average velocity of the object from t = 1 second to t = 2 seconds. (d)(5 pts) Note that if g(x) = x2 , then g 0 (x) = 2x. Now, suppose L is the tangent line to y = g(x) at the point (1, 1). ...
... (c)(5 pts) If s(t) = 2t2 − 13t + 5 for t ≥ 0 describes the position of an object (in feet) at time t, find the average velocity of the object from t = 1 second to t = 2 seconds. (d)(5 pts) Note that if g(x) = x2 , then g 0 (x) = 2x. Now, suppose L is the tangent line to y = g(x) at the point (1, 1). ...
Trigonometric functions
In mathematics, the trigonometric functions (also called the circular functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.