Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lecture 1: Trigonometric Functions: Definitions 1.1 The sine, cosine, and tangent functions We say triangles ∆ABC and ∆DEF are similar if 6 A = 6 D, 6 B = 6 E, and 6 C = 6 F . An important fact about similar triangles is that the ratios of corresponding sides are equal. That is, AB DE = , AC DF AB DE = , BC EF and AC DF = . BC EF Note that two right triangles are similar if they have a pair of equal acute angles (the remaining two angles are then equal because one is a right angle and the sum of the angles of triangle must equal 180◦ ). It follows that the ratios of the sides of a right triangle are determined completely by either one of its acute angles. This is the basis for the following definition. Definition Suppose ∆ABC is a right triangle with right angle at A. Let θ be the angle at C. Then the sine of θ is AB sin(θ) = , BC the cosine of θ is AC cos(θ) = , BC and tangent of θ is AB . tan(θ) = AC Note that tan(θ) = sin(θ) . cos(θ) Example Suppose 4ABC is a right triangle with right angle at A and acute angles ◦ which are both √ 45 . If the legs AB and AC are each of length 1, then the hypotenuse BC is of length 2. Hence 1 sin(45◦ ) = √ , 2 1 cos(45◦ ) = √ , 2 and tan(45◦ ) = 1. 1-1 Lecture 1: Trigonometric Functions: Definitions 1-2 Example Suppose ∆ABC is an equilateral triangle with each side of length 1. Let D be the base of the perpendicular from B to the side AC. Then ∆DBC is a right triangle with acute angle 6 B = 30◦ and acute angle 6 C = 60◦ . Now BC has length 1, DC has length 12 , and BD has length r 1 1− = 4 r √ 3 3 = . 4 2 Hence √ 3 , 2 1 cos(60◦ ) = , 2 sin(60◦ ) = and tan(60◦ ) = We also see that √ 3. 1 , 2 √ 3 ◦ cos(30 ) = , 2 sin(30◦ ) = and 1 tan(30◦ ) = √ . 3 We complete the definition of sine, cosine, and tangent for all angles between 0◦ and 90◦ with the following: sin(0◦ ) = 0, cos(0◦ ) = 1, tan(0◦ ) = 0, sin(90◦ ) = 1, and cos(90◦ ) = 0. Note that tangent is not defined for 90◦ . 1.2 Extending the definitions We now extend the definitions of sine, cosine, and tangent to functions on all of (−∞, ∞). To do so, we first note that we may restrict our discussion to right triangles with hypotenuses of length 1. In that case, if we place one acute angle at the origin and the right angle on the x-axis, the other vertex of the triangle will lie on the unit circle C with Lecture 1: Trigonometric Functions: Definitions 1-3 equation x2 + y 2 = 1. In fact, choosing any point on C determines such a triangle. If (a, b) is a point on C, the acute angle θ determined by (a, b) is the angle between the line from (a, b) to the origin and the x-axis. Instead of measuring this with degrees, we will now measure the angle by the distance from (1, 0) to (a, b) measured along the circle in the counterclockwise direction. We can then allow for θ < 0, meaning that (a, b) is a distance |θ| from (1, 0) in the clockwise direction, and we may allow for |θ| ≥ 2π, meaning that we move around the circle one or more times before reaching (a, b). With these conventions, if we move a distance θ from (1, 0) to a point (a, b) on the unit circle C, we define sin(θ) = b, cos(θ) = a, and tan(θ) = b . a When measuring angles in this way, the units of measurement are called radians. Note that, since the circle has a circumference of 2π, we have 90◦ = 2π π radians = radians. 4 2 Hence 1◦ = and Example π radians 180 180 ◦ 1 radian = . π For our standard angles we have π radians, 6 π 45◦ = radians, 4 30◦ = and π radians. 3 Hence we may now restate our previous values of sine, cosine, and tangent: π √3 π π 1 π 1 sin = sin = 1. sin(0) = 0 sin = sin =√ 3 2 2 6 2 4 2 π √3 π π 1 π 1 cos(0) = 1 cos = cos =√ cos = cos = 0. 6 2 4 3 2 2 2 π π π √ 1 tan(0) = 0 tan =√ tan =1 tan = 3. 6 4 3 3 60◦ = Lecture 1: Trigonometric Functions: Definitions 1-4 Example Using the basic values from the previous example and the geometry of the circle, we may now find values for sine, cosine, and tangent for many different angles. For example, we have sin(π) = 0, cos(π) = −1, and tan(π) = 0. Example We have sin and cos Note that tangent is not defined at Example 3π 2 3π 2 = −1, = 0. 3π 2 . We have sin(2π) = 0, cos(2π) = 1, and tan(2π) = 0. Example We have sin(3π) = 0, cos(3π) = −1, and tan(3π) = 0. Example We have 5π 1 sin = , 6 2 √ 5π 3 cos =− , 6 2 5π 1 tan = −√ . 6 3 Lecture 1: Trigonometric Functions: Definitions Example We have 5π 1 sin − =− , 6 2 √ 5π 3 cos − =− , 6 2 5π 1 tan − =√ . 6 3 1.3 The other trigonometric functions We may now define the remaining three trigonometric functions: sec(θ) = 1 , cos(θ) csc(θ) = 1 , sin(θ) and cot(θ) = Example and 1 cos(θ) = . tan(θ) sin(θ) We have sec 2π 3 = csc 2π 3 = 1 1 = = −2, 1 2π − cos 2 3 1 1 2 = √ =√ , 2π 3 3 sin 3 2 2π 1 cos − 1 2π 3 = √2 = − √ . cot = 2π 3 3 3 sin 3 2 1-5