Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: s θ = r This defintion of an angle in radian measure is independent of the radius of the circle. A larger circle with a longer radius, R, will also sustain a longer arc, S, S s s S and the ratio, will be the same as . In other words, θ = = R r r R S s θ θ r R For a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is πr θ = = π r Since a (semi-circle) straight angle has measure 180◦, π radian is equivalent to 180◦. π Given an angle measurement in degree, multiply that number by to find 180◦ the radian measure. 180◦ Given an angle measurement in radian, multiply that number by to find π the degree measure. Example: What is 55◦ in radian? π 55 11 Ans: 55◦ · = π = π ≈ 0:31π ≈ 0:96 180◦ 180 36 Example: What is 45◦ in radian? π 45 1 π Ans: 45◦ · = π = π = 180◦ 180 4 4 Unless a decimal approximation is desired, we should always leave the number 11 11π in exact format. That is, π or is the most desirable way of writing the 36 36 above angle measure in radian. You should note that radian measure is a (real) number, and is more canon- ical than degree measure when used in working with mathematical functions of real numbers. If an angle measurement is written without the ◦ symblo in the upper right, it is a radian measure. If the ◦ is present, it is a degree measure. Distingush the difference between 1◦ and 1 (radian). Example: What is 1◦ in radian? π π Ans: 1◦ · = ≈ 0:0056 (rad) 180◦ 180 Example: What is 1 (rad) in degree? 180◦ Ans: 1 · ≈ 57:3◦ π 2π Example: What is radian expressed in degree measure? 3 2π 180◦ 360◦ Ans: · = = 120◦ 3 π 3 π Example: what is radian in degree? 6 π 180◦ 180◦ Ans: · = = 30◦ 6 π 6 With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive side of the x−axis. If we obtained the other side (Called the Terminal Side) of the angle via a counter-clockwise rotation, we have a positive angle. If the terminal side of the angle is obtained via a clockwise rotation, we have a negative angle. Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained. π E.g. The terminal side of is in the first quadrant. 4 π E.g. The terminal side of − is in the fourth quadrant. 6 2π E.g. The terminal side of is in the second quadrant. 3 4π E.g. The terminal side of is in the third quadrant. 3 16π E.g. The terminal side of is in the first quadrant. 7 E.g. The terminal side of π is the negative x−axis. π E.g. The terminal side of − is the negative y−axis. 2 Two angles are co-terminal angles if they have the same terminal side. π 5π E.g. The two angles and − are co-terminal. 3 3 Notice that if θ is any angle, then the angles θ + 2π; θ + 4π; θ + 6π; θ + 8π; ··· are all co-terminal angles Given a Circle with radius r centered at the origin (The equation of this circle is x2 + y2 = r2), we define the terminal point of an angle θ to be the point of intersection of the circle with the terminal side of θ. π E.g. The terminal point P of θ = is in the first quadrant. 6 If the radius of the circle is r = 1, then the coordinate of this terminal point P p ! 3 1 is ; 2 2 If the radius of the circle is r = 3, then the coordinate of this terminal point P p ! 3 3 3 is ; 2 2 π E.g. The terminal point of θ = is on the positive side of the y−axis. If the 2 radius of the circle is r = 1, then the coordinate of the terminal point of θ is (0; 1). Note: Any co-terminal angle has the same terminal point. If θ is an angle with terminal point P , then any angle of the form θ + 2π; θ + 4π; θ + 6π; θ + 8π; ··· all have P as the terminal point. To find the coordinate of the terminal point, we need to know the radius of the circle and the measurement of the angle. Definition of the Trigonometric Functions Using circle: Given an angle θ in standard position, on a circle with radius r, and the terminal point of θ is P , and (x; y) is the coordinate of P , we define the six trigonometric functions of θ by: x cos θ = This is the cosine function. r y sin θ = This is the sine function. r y tan θ = This is the tangent function. x r sec θ = This is the secant function. x r csc θ = This is the cosecant function. y x cot θ = This is the cotangent function. y Sometimes, for convenience, we assume a circle of radius r = 1, called a unit circle, when defining or evaluating the values of the trigonometric functions. Note that we only need the measurement of the angle (we do not need to know the radius of the circle) to find the values of the six tri functions (because the r cancles out when we take the ratio of x or y with each other or with r. Given an angle θ in standard position, the reference angle of θ is the acute angle that the terminal side of θ makes with the x−axis. To find the coordinate of the terminal point, it is most often easier to consider the length of the sides of the right triangle formed by the terminal side of θ and the x−axis with the reference angle of θ being one of the interior angles. In the above picture, θ is the angle in standard position and β is the reference π angle. Note that the reference angle must be angle between 0 to (0 to 90◦). 2 Example: The reference angle of a 120◦ angle is a 60◦ angle. Example: The reference angle of a −235◦ angle is a 55◦ angle. Example: The reference angle of a −30◦ angle is a 30◦ angle. π Example: For θ = , find the values of the six tri functions of θ. 3 π Notice that = 60◦. Assuming a circle of radius 1, the terminal point P of θ 3 p ! 1 3 has coordinate ; , so the values of the trig functions would be: 2 2 p p x 1 1 y 3 3 cos θ = = 2 = sin θ = = 2 = r 1 2 r 1 2 p p y 3 p x 1 1 3 tan θ = = 2 = 3 cot θ = = p2 = p = x 1 y 3 3 3 2 2 p r 1 r 1 2 2 3 sec θ = = = 2 csc θ = = p = p = x 1 y 3 3 3 2 2 Example: 30◦ 5π π Assuming a unit circle, if θ = = 150◦, its reference angle is = 30◦, and the 6 6 p ! 3 1 terminal point P of θ has coordinate − ; . Therefore, 2 2 5π 1 sin = 6 2 p 5π 3 cos = − 6 2 p 5π 1 1 3 tan = 2p = −p = − 6 3 3 3 − 2 Example: y p 2 2 π x 4 p 2 π r = 1 2 θ = − 4 p p ! 2 2 P = ; − 2 2 π π If we use a unit circle, when θ = − (45◦), then its reference angle is , and 4 4 p p ! 2 2 the terminal point P of θ has coordinate ; − . Therefore, 2 2 p π 2 sin − = − 4 2 p π 2 cos − = 4 2 p π − 2 tan − = p 2 = −1 4 2 2 Example: 7π y θ = 6 p 3 2 π x 1 6 2 r = 1 p ! 3 1 P = − ; − 2 2 7π π If we use a unit circle, when θ = (210◦), then its reference angle is (30◦), 6 6 p ! 3 1 and the terminal point P of θ has coordinate − ; − . Therefore, 2 2 7π 1 sin = − 6 2 p 7π 3 cos = − 6 2 7π −1 1 tan = p2 = p 6 3 3 − 2 Example: y p ! 1 3 2π P = − ; θ = 2 2 3 p 3 r = 1 2 π 3 1 x 2 2π π If we use a unit circle, when θ = (120◦), then its reference angle is (60◦), 3 3 p ! 1 3 and the terminal point P of θ has coordinate − ; . Therefore, 2 2 p 2π 3 sin = 3 2 2π 1 cos = − 3 2 p 3 p 2π 2 tan = 1 = − 3 3 −2 Example: y 23π θ = 6 p 3 2 π x 6 1 r = 1 2 p ! 3 1 P = ; − 2 2 23π π If we use a unit circle, when θ = (690◦), then its reference angle is , and 6 6 p ! 3 1 the terminal point P of θ has coordinate ; − . Therefore, 2 2 23π 1 sin = − 6 2 p 23π 3 cos = 6 2 23π −1 1 tan = p 2 = −p 6 3 3 2 23π 11π Notice that θ = = 2π + is greater than 2π, therefore when we draw 6 6 the terminal point of θ in standard position, we completed one cycle (2π), then 11π rotated another = 330◦ to reach the terminal point P .