opp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles

Transcription

1 Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to θ the acent side, and the side osite the right angle is the hopotenuse. We define the si trigonometric functions of θ as the ratio of the lengths of two of the sides of the triangle: (the sine function) sin θ = hp (the cosine function) cos θ = hp (the tangent function) tan θ = (the secant function) sec θ = hp (the cosecant function) csc θ = hp (the cotangent function) cot θ = Using this definition, the si trigonometric functions are well-defined for all angles θ where 0 < θ < 90 In defining the si trigonometric functions using a right triangle, we wanted to have consistenc. That is, if we used a different right triangle with the same angle θ, we wanted to know that we will get the same value for each of the functions. This will be the case as illustrated b the following:

2 C C hp θ A B B Notice in the picture above that triangle ABC is similar to triangle AB C (wh?), therefore, their corresponding sides are in proportion, in particular, using the definition of the trig functions, sin θ = hp = CB AC = C B AC cos θ = hp = AB AC = AB AC This fact tells us that, in determining the value of a trig function on an angle θ, we ma use an right triangle. The value of the si trig functions is determined b the angle θ, not b the triangle we use.

3 Special Triangles: a a a In a 90 triangle, the two sides have the same length and the hpotenuse is times the length of the side. In other words, if an one of the side has length a, then the other side also has length a and the hpotenuse has length a a a a 0 In a 0 90 triangle, if the side osite the 0 angle has length a, then the side osite the angle has length a, and the hpotenuse has length a. In other words, the hpotenuse is twice the length of the shorter side. Also remember from geometr that in an triangle, the longest side is the side osite the largest angle, and the shortest side is the side osite the smallest angle. Using this fact and the above special triangle, we can find the eact value of the lengths of some triangles:

4 Eample: Find the value of the variables: 0 Ans: The shortest side in this triangle is the one osite the 0 angle, its length is. This is a 0 90 triangle, the hpotenuse is twice the shortest side, so = () = 6. The side osite the angle is times the length of the shortest side, so = =. Eample: Find the value of the variables: 7 0 Ans: In this triangle, is the shortest side (it is osite the smallest angle), the side osite the angle is, so we can set up the equation: = 7, solving for gives: = 7 The hpotenuse,, is twice, so = 7 = 14

5 Eample: Find the value of the variables: 0 11 Ans: The length of the hpotenuse is twice of the shortest side,, so = 11, is times the shorter side, so = 11 = 11 Eample: Find the value of the variables: 1 Ans: This is a 90 triangle, the two sides are equal in length, so = 1. The hpotenuse is times the length of the side, so = 1 = 1

6 Eample: Find the value of the variables: 6 Ans: In this 90 triangle, the hpotenuse is 6. The hpotenuse is times the length of the side, so we can set up the equation: = 6, solving for gives: = 6 The two sides have the same length, so = = 6 Using the special triangles, we can find the value of the trig functions on some special angles. Eample: Find sin(0 ), cos(0 ), tan(0 ), cot(0 ), sec(0 ), and csc(0 ) Ans: 1 0 For convenience, we can assign the side osite the 0 angle with a length of 1, then the acent side to the 0 angle will have a length of, and the hpotenuse will have a length of, and we have: sin(0 ) = hp = 1 cot(0 ) = = csc(0 ) = hp = 1 = cos(0 ) = hp = 1 = sec(0 ) = hp = tan(0 ) = = 1

7 Eample: Find sin( ), cos( ), tan( ), cot( ), sec( ), and csc( ) Ans: 1 1 For convenience, we can assign the side osite the angle with a length of 1, then the acent side to this angle will also have a length of 1, and the hpotenuse will have a length of, and we have: sin( ) = hp = 1 cot( ) = = 1 1 = 1 csc( ) = hp = 1 = cos( ) = hp = 1 sec(45 ) = hp = 1 = tan( ) = = 1 1 = 1

8 Eample: Find sin( ), cos( ), tan( ), cot( ), sec( ), and csc( ) Ans: 0 1 For convenience, we still assign the side osite the 0 angle (the side acent to the angle) with a length of 1, then the osite side to the angle will have a length of, and the hpotenuse will have a length of, and we have: sin( ) = hp = cot( ) = = 1 csc( ) = hp = cos( ) = hp = 1 sec( ) = hp = 1 = tan( ) = = 1 = The right triangle approach to defining trigonometric functions is eas to understand, and can be used convenientl to solve goemetric problems involving triangles. The drawback with this definition is the fact that the trig functions are defined onl for angle θ between 0 < θ < 90. We wanted the trig functions to be defined for all (or most) of the real numbers. In order to do that, we need a different approach to defining the trig functions.