Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians = 180º Therefore, o 180 π 1 rad =, or π 1º = 180 rad Angle Measure Conversions π 1. To convert degrees to radians, multiply by. 180 2. To convert radians to degrees, multiply by 180. π : (a) 90º = / 2 rad (b) 60º = / 3 rad (c) 5 / 4 rad = 225º
An angle is in if it is drawn on the -plane with its vertex at the origin and its initial side on the positive (i.e., the right half of) -axis. Coterminal Angles Two angles are if their sides coincide. Coterminal angles differ from each other by an integer multiple of 360º (respectively, by a multiple of 2 radians). Conversely, every pair of angles that differ by an integer multiple of 360º or 2 radians are coterminal. : 30º, 390º, 750º, 330º are coterminal (they are all in the form of 30º ± 360 º). Similarly, / 2, 5 / 2, 3 / 2, and -11 / 2 radians are all coterminal (they are all in the form of / 2 ± 2 radians). Area of a Circular Sector
The area of a circular sector with a central angle is the fraction / (angle of one revolution) of the area of the entire circle. It is found by the formulas: In a circle of radius, the area of a sector with a central angle, is = = θ ( π 2π θ ( π 360 2 θ 2 ) = ( in radians), or 2 2 θπ 2 ) = ( in degrees). 360 Length of a Circular Arc Similarly, the length of the arc subtending a central angle is the fraction / (angle of one revolution) of the circumference of the circle. It is found by the formulas: In a circle of radius, the length of the arc subtending a central angle, is = θ (2π ) = θ 2π ( in radians), or = θ θπ (2π ) = ( in degrees). 360 180 : A circular sector has radius 10 and central angle / 3. Find its area and the subtending arc. A = [( / 3) / 2] 4 2 = 16 / 6 = 8 / 3 = ( / 3) 4 = 4 / 3
Circular Motion motion along a circular path Linear Speed vs. Angular Speed: Linear speed = distance traveled / elapsed time Angular speed = change in central angle / elapsed time Suppose a point moves along a circle of radius and the ray from the center of the circle to the point traverses (in radians) in time. Let = be the distance the point travels along the arc in time. Then the speed of the object is given by Angular speed θ ω = (radians per unit ) Linear speed = If a point moves along a circle of radius with angular speed linear speed is given by =, then its : [#79] A circular saw has a blade with a 6-in radius. Suppose the blade spins at 1000 rpm. (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the blade in ft/sec.
6.2 Trigonometry of Right Triangles [See Trig. function summary handout.] Special Triangles 45º - 45º - 90º triangle 30º - 60º - 90º triangle Special Angles, in, in degrees radians sin cos tan csc sec cot 0º 0 0 1 0-1 - 30º / 6 1 / 2 3 / 2 3 / 3 2 2 3 / 3 3 45º / 4 2 / 2 2 / 2 1 2 2 1 60º / 3 3 / 2 1 / 2 3 2 3 / 3 2 3 / 3 90º / 2 1 0-1 - 0
Solving a Right Triangle Consider the right triangle: We have 2 = 2 + 2 = sin = cos : Let = 4 and = 7, find and derive the values of the six trigonometric functions for. : [#48] From the top of a 200-ft lighthouse, the to a ship in the ocean is 23º. How far is the ship from the base of the lighthouse? : [#49] A 20-ft ladder leans against a building so that the angle between the ground and the ladder is 72º. How high does the ladder reach on the building?
6.3 Trigonometric Functions of Angles Let be a right triangle with an acute angle θ shown below Then = + 2 2 sin θ = cos θ = tan θ = sec θ =, 0 cot θ = csc θ =, 0
Signs of the Trigonometric Functions (vs. Quadrants) Quadrant I: Quadrant II: Quadrant III: Quadrant IV: All positive Sin and csc positive Tan and cot positive Cos and sec positive Reference Angle Let be an angle in standard position. The associated with is the acute angle formed by the terminal side of and the -axis. Therefore, the reference angle is always measured against the nearer half of the -axis. Evaluating Trigonometric Functions for Any Angle For any angle : 1. Find the reference angle associated with the angle. 2. Determine the sign of the trigonometric function of by noting the quadrant in which lies. 3. The value of the trigonometric function at the desired angle is equal to its value at the reference angle times the sign found in step 2. : (a) Evaluate cos(210º), tan(210º), and csc(210º). (b) Evaluate sin(315º), cot(315º), and sce(315º).
Trigonometric Functions on the Unit Circle: the Pythagorean Identities For any angle : cos 2 θ + sin 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ : (a) If cos = 1 / 3, and is in quadrant IV, find sin and cot. (b) If tan = 3 / 5, and is in quadrant III, find sin and cos. Area of a Triangle The area A of a triangle with sides of lengths angle is 1 A = sinθ 2 and, and with included