Unit 2 Intro to Angles and Trigonometry

Transcription

1 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of Units (6) Angles in Standard Position Quadrantal Angles; Coterminal Angles (8) Arcs and Sectors of Circles (10) Trigonometry of Right Triangles (14) Triangles Triangles (15) Trigonometry of Angles (17) Signed values of Trigonometric Ratios (18) Reference Angles (20) Reference Angles and Trigonometric Ratios Know the meanings and uses of these terms: Degree Radian Angle in standard position Quadrantal angle Coterminal angles Sector of a circle Reference angle Review the meanings and uses of these terms: Angle Vertex of an angle Ray Intecepted arc Central angle of a circle

2 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 2 Definition of an Angle (Geometric Definition) Definition 1: The composition of two rays with a common endpoint. Definition 2: The result of coincident rays where one ray has been rotated about its endpoint. Definition: The vertex of an angle is the endpoint shared by the rays of the angle. AOB at right R 1 and R 2 are the rays O is the vertex If R 2 is the ray that has been rotated out of coincidence with R 1, we say that R 1 is the initial side and R 2 is the terminal side. The measurement of an angle is quantified by the amount of rotation from the initial side to the terminal side. A counterclockwise rotation results in a positive measurement while a clockwise rotation results in a negative measurement.

3 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 3 Angle Measurements & Notation There are two primary units used for measuring angles: degrees and radians. (There is also a historically interesting but functionally irrelevant third unit called grads.) One degree is defined to be 1/360 th of a complete rotation about a vertex. Thus an angle measuring 360 would involve the terminal side rotating completely back into coincidence with the initial side. One radian is defined to be a rotation in which the intercepted arc of the unit circle is length 1. Thus an angle measuring 2 would involve the terminal side rotating completely back into coincidence with the initial side; by extension, this means one radian is exactly 1/2 of a complete rotation about a vertex. The most common symbol used to mark an angle and identify its measurement is the Greek letter (theta).

4 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 4 Basic Conversions of Degrees and Radians 180 = radians 1 = 180 radians radians 90 = radians 270 = radians 1 radian = = radians When working with degrees, always either use the symbol or write the word degree. 30 = radians 60 = radians When working with radians you may use the word radian, the abbreviation rad, or nothing at all.

5 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 5 More Converting of Measurements Convert from radians to degrees: To convert from degrees to radians, multiply by. 180 To convert from radians to degrees, multiply by 180. Ex. 1: 5 radians 6 Convert from degrees to radians: Ex. 1: 40 Ex. 2: 13 8 radians Ex. 2: 225 Ex. 3: 5 radians

6 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 6 Angles in Standard Position Quadrantal Angles and Coterminal Angles Definition: An angle is said to be in standard position if its vertex is at the origin of the coordinate plane and its initial side is on the positive x-axis. Definition: An angle is described as quadrantal if its terminal side is on an axis., -90, 0, 90, 180, 270, 360, 0,, 2 2,,, 3, 2 2, Definition: Angles are said to be coterminal if they have a common terminal side. Example: 70, 430, -290 are coterminal measurements

7 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 7 For an angle measurement of c degrees, c + 360n, where n is any integer, will be a coterminal angle measurement. For an angle measurement c radians, c + 2n, where n is any integer, will be a coterminal angle measurement. Find a coterminal angle measurement in [0, 360). Find a coterminal angle measurement in [0, 2). Ex. 1: = 1000 Ex. 1: = 23 3 radians Ex. 2: = 1975 Ex. 2: = 37 4 radians

8 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 8 Arcs and Sectors of Circles Definition: An arc is a portion of a circle between two endpoints. Definition: An intercepted arc is a portion of a circle whose endpoints are points on the rays of an angle. Definition: A central angle of a circle is an angle whose vertex is at the center of the circle. The length of an arc of a circle, represented by s, can be calculated using the radius r of the circle and the measurement in radians of the center angle which subtends the arc: s = r Definition: A sector of a circle is a region in the interior of a circle bounded by a central angle and the arc it subtends. The area of a sector of a circle, represented by A s, can be calculated using the radius r of the circle and the measurement in radians of the center angle which subtends the arc: A s = 1 2 r2

9 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 9 Calculate the length of the arc labeled s below and the area of the sector bounded by and s. Calculate the radius of the circle labeled r below and the area of the sector bounded by and s. Ex. 1: s Ex. 2: 16 ft 8 m r

10 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 10 Trigonometry of Right Triangles Trigonometry of right triangles is based on relationships between an acute angle and the ratio formed by two sides of the triangle. sin opposite leg hypotenuse cos adjacent leg hypotenuse With respect to the angle chosen in the triangle at left, the trigonometric ratios of are defined: tan sec opposite leg adjacent leg hypotenuse adjacent leg cot csc adjacent leg opposite leg hypotenuse opposite leg The sine of is the ratio of the opposite leg to the hypotenuse. The cosine of is the ratio of the adjacent leg to the hypotenuse. The tangent of is the ratio of the opposite leg to the adjacent leg. The cotangent of is the ratio of the adjacent leg to the opposite leg. The secant of is the ratio of the hypotenuse to the adjacent leg. The cosecant of is the ratio of the hypotenuse to the opposite leg. It is important to remember that the ratios are based on the relative position of the legs of the right triangle with respect to the angle chosen. If changes from one of the acute angles to the other acute angle, the roles of opposite and adjacent are switched.

11 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 11 Define the trigonometric ratios of below. Ex: Find the third side of length using the Pythagorean Theorem, then define the trigonometric ratios. Ex.: sin cos sin cos tan co t tan co t sec csc sec csc

12 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 12 Sketch a triangle which satisfies the given ratio & then define the remaining trigonometric ratios. Ex.: cos 2 5 Observe that a physical model of a triangle demonstrates the consistency between the ratios and the angles: 1 unit sin tan co t sec csc

13 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 13 Express x and y as ratios in terms of. Express x and y as ratios in terms of. Ex. 1: 15 x Ex. 2: 12 y x y

14 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE Triangle Triangle sin 45 sin 30 sin 60 cos 45 cos 30 cos 60 tan 45 tan 30 tan 60 cot 45 cot 30 cot 60 sec 45 sec 30 sec 60 csc 45 csc 30 csc 60

15 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 15 Trigonometry of Angles By placing one of the acute angles of a triangle at the origin in standard position, it is possible to relate the trigonometry of right triangles to angles in general. Definition: Let (x, y) be a point on the terminal side of an angle in standard position. Let r be the distance from the origin to (x, y). Then: (x, y) sin tan sec y r y x r x cos cot csc x r x y r y We can eventually extend the definition of the trigonometric ratios by noting that it is possible to form a consistent definition even when the angle is outside the interval (0, 90).

16 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 16 Find the trigonometric ratios of. Find the trigonometric ratios of. Ex. 1: Ex. 2: sin co s sin (-4, -10) co s tan cot tan cot se c csc se c csc

17 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 17 Signed values of trigonometric ratios From the definition of each trigonometric ratio, it is possible to know in advance whether the ratio is going to be positive or negative. Without determining the exact value, determine whether the trigonometric ratio is positive or negative. Ex. 1 sin 200 Quadrant I sin, cos, tan, cot, sec, csc positive x > 0, y > 0 (0, 90) or 0, 2 Ex. 2 sec 300 Quadrant II sin, csc positive x < 0, y > 0 cos, tan, cot, sec negative (90, 180) or, 2 Quadrant III tan, cot positive x < 0, y < 0 sin, cos, sec, csc negative 3 (180, 270) or, 2 Quadrant IV cos, sec positive x > 0, y < 0 sin, tan, cot, csc negative (270, 360) or 3 2,2 Ex. 3 tan 80 Ex. 4 cos 1180

18 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 18 Reference Angles Definition: Let be an angle in standard position. Then the reference angle associated with is the acute angle formed by the terminal side of and the x-axis. To find a reference angle for some angle : If is in (0, 360) or (0, 2), and the terminal side is in QI, then. the terminal side is in QII, then the terminal side is in QIII, then the terminal side is in QIV, then If is not in (0, 360) or (0, 2), find a coterminal angle c that is and then apply the rules above substituting c for.

19 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 19 Find the reference angle for each given. Find the reference angle for each given. Ex. 1: 290 Ex. 3: 2390 Ex. 2: 570 Ex. 4: 27 5

20 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 20 Reference angles and trigonometric ratios Find the sine, cosine, and tangent of. Any trigonometric ratio involving will have the same absolute value as the same trigonometric ratio involving. Ex.: Since is acute, any trigonometric ratio involving will have a positive value. Thus any trigonometric ratio of can be defined in terms of a trigonometric ratio of with an appropriate accommodation of its sign value based on the quadrant where the terminal side of is found. sin = cos = tan =

21 HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 21 Rewrite each trigonometric ratio using a reference angle and then evaluate as possible. Rewrite each trigonometric ratio using a reference angle and then evaluate as possible. Ex. 1: cos 290 Ex. 3: sin 2390 Ex. 2: tan 570 Ex. 4: 27 cos 5