About Trigonometry. Triangles

Transcription

1 About Trigonometry

2 TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion... 3 Trigonometry Rules... 3 Converting Between Radians and Degrees... 4 Functions... 4 Trigonometric Functions... 4 Graphing... 6 Graphing Trigonometric Functions... 6 Glossary... 8 References... 10

3 What is TRIGONOMETRY? About Trigonometry Trigonometry explores relationships between the sides and angles of a triangle. Combining arithmetic, algebra, and geometry, trigonometry is used in disciplines such as architecture and physics. Background Triangles Deriving its name from the Greek word for triangle, trigonometry relates the measure of the sides of a triangle to the measure of its angles. With this stated, it is important to note that triangles do not hold a monopoly over trigonometric functions, as these functions can also be defined using circles. Trigonometry with Triangles A right triangle is a triangle with a right angle (90 degrees or π/2 radians). The Pythagorean Theorem is central to trigonometry. The theorem states that for a right triangle with side lengths of a and b, and a hypotenuse of length c, the following relationship always holds: a 2 + b 2 = c 2 1

4 Trigonometric functions relate to angle measurement. The Greek letter Ө is often used to represent the angle, as shown in the figure below. The six trigonometric functions are defined as follows: sin(ө) = a/c = the side opposite the angle divided by the hypotenuse cos(ө) = b/c = the side adjacent to the angle divided by the hypotenuse tan(ө) = a/b = the side opposite the angle divided by the side adjacent to the angle sec(ө) = c/b = the reciprocal of cos(ө) csc(ө) = c/a = the reciprocal of sin(ө) cot(ө) = b/a = the reciprocal of tan(ө) The standard abbreviations for the functions were used above, but the full names are as follows: sine, cosine, tangent, secant, cosecant, and cotangent. Trigonometry with Circles Circles The trigonometric functions can be defined using a unit circle with centre (0, 0). A unit circle is a circle with a radius of 1. For any point (x, y) on the circle, the angle Ө is defined as the angle between the positive x-axis and the line segment joining (0, 0) to that point. 2

5 The trigonometric functions, as related to circles, are as follows: sin(ө) = y cos(ө) = x tan(ө) = y/x sec(ө) = 1/x csc(ө) = 1/y cot(ө) = x/y To check that these are correct, simply draw a right triangle inside the circle with sides x and y and hypotenuse 1. Trigonometry Rules Rules/Conversion Recall that the Cartesian Coordinate plane contains 4 quadrants. The CAST rule uses these quadrants to illustrate some helpful properties, as follows: Quadrant 4: Cos function positive Quadrant 1: All trigonometric functions positive Quadrant 2: Sin function positive Quadrant 3: Tan function positive 3

6 Another essential tool in remembering the properties of trigonometric functions is that of the SOHCAHTOA rule, illustrated below. Note that when you take out the first letter of each term in the equations, you get the phrase "SOHCAHTOA". Converting Between Radians and Degrees The convention is that positive angles are measured counterclockwise from the positive x-axis, and negative angles are measured clockwise from the positive x-axis. Note: π radians = 180 degrees Trigonometric Functions Functions As it is geometrically impossible for a right triangle to contain any obtuse angles, we turn to the circle definition, as it works for any angle. To apply this, draw a right triangle within the circle, using the ray that defines the angle and using the x-axis. In the first quadrant, this is simply the right triangle containing the angle Ө. In the second quadrant, it is the right triangle containing the supplementary angle (180 - Ө or π - Ө), as illustrated below. 4

7 In the third quadrant, the right triangle contains the angle minus the straight angle (Ө or Ө - π), as illustrated in the figure to the right. In the fourth quadrant, the right triangle contains the conjugate angle (360 - Ө or 2π - Ө), as illustrated in the figure to the right. The point connecting the circle to the ray is of the form (x, y), where x = cos(ө) and y = sin(ө). Consider the figure below, relating it back to the circle. Provided below, for reference, is a chart containing the important angles used in relation to trigonometric functions. 5

8 Graphing Graphing Trigonometric Functions Graphs applicable to the trigonometric functions previously considered, are provided below. Sin Cos Tan Csc Sec Cot Consider the following: - sin and cos are defined everywhere, while all other trigonometric functions have asymptotes. - The portion of the graph from -2π to 0 is the same shape as the portion of the graph from 0 to 2π for sin, cos, sec, and csc (e.g. cos(x) = cos(x + 2π)). This is due to the fact that the functions are periodic, with a period of 2π. tan and cot are periodic, with a period of π (e.g. tan(x) = tan(x + π)). - sin and cos are the same shape, shifted 90 or π/2 radians (i.e. sin(x) = cos(x - π/2)). Similarly, csc and sec are the same shape shifted 90 or π radians (i.e. csc(x) = sec(x - π/2)). 6

9 - cos and sec are symmetric across the y-axis. This means they are even functions, where cos(-x) = cos(x) and sec(-x) = sec(x). The other trigonometric functions are odd functions, which means sin(-x) = -sin(x), tan(-x) = -tan(x), csc(-x) = -csc(x), and cot(-x) = -cot(x). 7

10 Glossary Acute angle: Asymptote: CAST rule: Circle: Complementary angles: Conjugate angles: Degree: Hypotenuse: Obtuse angle: Periodic function: Pythagorean triple: Pythagorean Theorem: Radian: angle measuring less than 90 degrees. line which the function approaches but never reaches. tells which trigonometric function is positive in a given quadrant. the set of points which are distance r (called the radius) from a point c (called the center). a pair of angles whose measures add up to 90 degrees. a pair of angles whose measures add up to 360 degrees. symbolized ; a unit of measurement equal to 1/360 of a circle. the side of a right triangle opposite the right angle. angle measuring more than 90 degrees and less than 180 degrees. function for which f(x) = f(x + p) for some period p. a set of three integers a, b and c such that a 2 + b 2 = c 2. Given a right triangle with side lengths of a and b and a hypotenuse of c, a 2 + b 2 = c 2. a unit of angular measure, used with circles. It is equivalent to 180/π degrees. 8

11 Radius: Ray: Reciprocal: Reflex angle: Right angle: Right triangle: Square root: Straight angle: Supplementary angles: the distance between the center of a circle and a point on the circle. portion of a line starting with a point and extending to infinity. the reciprocal of x is 1/x, and vice-versa. angle measuring more than 180 degrees and less than 360 degrees. a 90 or π/2 radian angle. a triangle with a right angle. the square root of x is a number which when multiplied by itself equals x. 180 degree angle. a pair of angles whose measures add up to 180 degrees. Unit circle: a circle with radius 1. 9

12 References 10