1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric

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1 1. The Si Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1. The Rectangular Coordinate Sstem 1.3 Definition I: Trigonometric Functions 1.4 Introduction to Identities 1.5 More on Identities 1

2 1.1 Angles, Degrees, and Special Triangles Angles in general Verte Initial side, terminal side Positive angle, negative angle Angle measurement: degrees Classifing angle b range of angle measurements Pthagorean theorem General triangles Special triangles triangle triangle

3 1.1 Angles, Degrees, and Special Triangles Angles in general Verte O Initial side OA, terminal side OB Positive angle: counter-clockwise; negative angle: clockwise B O verte θ A 3

4 1.1 Angles, Degrees, and Special Triangles Angles in general Right angle Straight angle Acute angle Obtuse angle Complementar angles Supplementar angles 4

5 1.1 Angles, Degrees, and Special Triangles Pthagorean Theorem. B a + b = c a C b c A Proof of Pthagorean Theorem. a b c b c a a c b c a b 5

6 1.1 Angles, Degrees, and Special Triangles General triangles Equilateral triangle Isosceles triangle Scalene triangle Acutetriangle Obtuse triangle Right triangle 6

7 1.1 Angles, Degrees, and Special Triangles Special triangles Triangle Triangle

8 1.1 Angles, Degrees, and Special Triangles Problems (1) Indicate whether the given angle is acute or obtuse; give the complement and supplement [, 6, 8] a) 50 b) 160 c) () Refer to figure below [10, 14] C αβ A D B Find B if β = 45 Find B if α + β = 80, and A = 80 8

9 1.1 Angles, Degrees, and Special Triangles Problems (3) Through how man degrees does the hour hand of a clock move in 4 hours? [1] (4) Find the remaining sides of a triangle if the shortest side is 3 [44] (5) Fill the remaining sides of a triangle if the longest side is 1 [58]

10 1. The Rectangular Coordinate Sstem Graphing line Graphing parabolas The distance formula Graphing circles Angles in standard position Using technolog 10

11 1. The Rectangular Coordinate Sstem (, +) -ais An point on the -ais has the form (0, b) (+, +) Quadrant II 3 Quadrant I 1 Origin An point on the -ais has the form (a, 0) ais 1 Quadrant III Quadrant IV 3 (, ) (+, ) 11

12 1. The Rectangular Coordinate Sstem (1) Graph line = [8] () Graph parabola = ( 1) + [E] (3) Find the distance between ( 3, 8) and ( 1, 6) [8] (4) Find the distance between the origin and (, ) [e.g.4] (5) Equation of a circle. ( 5, ) Verif the point is on the unit circle [40] Graph circle + = 36 [4] 3 3 1

13 1. The Rectangular Coordinate Sstem (1,0) Figure

14 1. The Rectangular Coordinate Sstem (6) Angle in standard position. An angle in standard position (initial side is on the positive -ais, and verte is the origin) What does it mean θ QII (read as angle θ belongs to QII) Name an angle between 0 and 360 that is co-terminal with the angle 300 [64] Draw angle 55 in standard position. Find a positive angle and a negative angle that are co-terminal with 55 [66] 14

15 1. The Rectangular Coordinate Sstem (7) Angle in standard position. [70] Draw angle 45 in standard position. Name a point on the terminal side of the angle Find the distance from the origin to that point Name another angle that is co-terminal with 45 (8) Find all angles that are co-terminal with 150 [78] (9) Using technolog. Graph a circle. 15

16 1.3 Definition I: Trigonometric Functions Define si trigonometric functions for angles in standard position (, ) r o θ Angle in standard position: initial side is the positive -ais and verte is the origin. 16

17 1.3 Definition I: Trigonometric Functions Function Abbreviation Definition The sine of θ sinθ = The cosine of θ cosθ = The tangent of θ tanθ = The cotangent of θ cotθ = The secant of θ secθ = The cosecant of θ cscθ = Where + = r, or r = +. That is, r is the distance from the origin to (, ). r r r r 17

18 1.3 Definition I: Trigonometric Functions For θ in QI QII QIII QIV r r sinθ = and cscθ = + + r r cosθ = and secθ = + + tanθ = and cotθ =

19 1.3 Definition I: Trigonometric Functions (1) Find si trigonometric functions of θ if (1, 5) is on the terminal side of θ. [4] () Draw the angle 135 in standard position. Find a point on the terminal side, then find the sine, cosine, and tangent of 135. [6] (3) Indicate the two quadrants the angle θ could terminate in if sinθ = 3/ 10 [4] (4) Find the remaining trig functions of θ If cosθ = 4/5 and θ QIV [5] cscθ = 13/5 and cosθ < 0 [60] 19

20 1.4 Introduction to Identities Reciprocal Identities Equivalent Form csc θ = sec θ = cot θ = 1 sinθ 1 cosθ 1 tanθ sin θ = cos θ = tan θ = 1 cscθ 1 secθ 1 cotθ TABLE 1 (MEMORIZE) 0

21 1.4 Introduction to Identities Ratio Identities tan θ = sinθ cosθ Because sinθ = cosθ / / r r = = tanθ cot θ = cosθ sinθ Because cosθ = sinθ / / r r = = cotθ TABLE (MEMORIZE) 1

22 1.4 Introduction to Identities Pthagorean Identities Equivalent Form sin θ + cos θ = 1 cosθ = ± sinθ = ± 1 sin θ 1 cos θ 1 + tan θ = 1 + cot θ = sec csc θ θ TABLE 3 (MEMORIZE) Meaning of sin. sin θ = sinθ θ ( )

23 1.4 Introduction to Identities 3 (1) If cos θ =, find secθ. [10] () If cotθ =, find tanθ. [14] 3 (3) Find the cotθ, if sin = and cos θ =. [18] θ (4) Find the cosθ if sinθ = and θ terminate in QII. [30] (5) Find secθ if tanθ = 7/4 and cosθ < 0. [4] (6) Find the rest trigonometric functions of θ, if sinθ = 1/13 with θ QI [44] cscθ = and cosθ is negative [48] 3

24 1.5 More on Identities Etension of 1.4 Use identities in 1.4 to prove more trig identities. 4

25 1.5 More on Identities (1) Write cscθ in terms of sinθ onl [7] () Write each epression in terms of sinθ and cosθ, then simplif cscθ cotθ (a) [16] (b) csc(θ) cot(θ)cos(θ) [6] (3) Add or subtract, simplif if possible, and write our answers in sinθ and/or cosθ onl [3, 34] (a) Add. (b) Subtract. 1 cos θ + 1 cosθ sinθ cosθ 5

26 1.5 More on Identities Prove (4) sinθ cotθ = cosθ [60] θ cos (5) = cos θ [64] secθ 6