θ = radians; where s = arc length, r = radius

Transcription

1 TRIGONOMETRY: 2.1 Degrees & Radians Definitins: 1 degree - 1 radian θ r s FORMULA: s θ = radians; where s = arc length, r = radius r IMPLICATION OF FORMULA: If s = r then θ = 1 radian EXAMPLE 1: What is the radian measure f a central angle subtended by an arc f 32 cm in a circle f radius 8 cm.? THINK!! What is the radian measure f an angle f 180? Of 90? Of 60? Of 45? Of 30? * Cnverting Radians Degrees π Deg Rad: multiply by Rad Deg: multiply by π EXAMPLE 2a: Find the degree measure f 1.5 radians in exact frm and in decimal frm t 4 places. EXAMPLE 2b: Find the radian measure f 120 in exact frm and in decimal frm t 4 places.

2 Angles in Standard Psitin: vertex is at rigin, initial side is psitined alng the x- axis ( 0 ) LABEL: α is psitive (cunterclckwise) β is negative (clckwise) 0 EXAMPLE 3: Sketch the fllwing angles in standard psitin: π (A) (B) Cterminal angles have the same initial & terminal sides. Therefre, the measures f cterminal angles differ by integer multiples f r. EXAMPLE 4: Which f the fllwing pairs f angles are cterminal? Create a labeled sketch f each angle n the same axis. (A) α = 90, β = 90 (B) α = 750, β = 30 π 25π 3π 7π (C) α =, β = (D) α =, β =

3 FORMULA t find arc length: s = rθ where s = arc length, r = radius, θ = central angle IN RADIANS EXAMPLE 5: In a circle f radius 6 ft, find the arc length subtended by a central angle f: (A) θ = 1.7 radians (B) θ = 40 *CIRCLE: AREA OF A SECTOR* part whle : A πr = θ 2 2π Nw slve this equatin fr A: EXAMPLE 6: In a circle f radius 7 in, find the area f the sectr with central angle: (A) radians (B) 110

4 2.3 Trignmetric Functins: Unit Circle Apprach Algebra 2 Review: The graph f a 2 + b 2 = 1 is a circle with center at (0,0) and radius = 1 This is the definitin f a UNIT CIRCLE P(a, b) WITHIN A UNIT CIRCLE: a = cs x, and b = sin x r P(a, b) = P(cs x, sin x) EXPLAIN WHY: (1,0) *What is the dmain & range fr sine (y = sin x) & csine (y = cs x)? DOMAIN: RANGE: Definitins: is an angle in DEGREES. is an angle in RADIANS. represents a randm pint n the terminal side f angle θ r angle x. is the distance frm the rigin t pint P. Reference triangle: Reference angle: Define the 6 trig functins: Fill in + r values fr the functins in relatin t the 4 quadrants sinθ = cscθ = = sin sin csθ = secθ = = cs cs tanθ = ctθ = = tan tan sin sin tanθ = ctθ = = cs cs tan tan

5 EXAMPLE 1: Find the exact values f each f the 6 trig functins fr the angle x with terminal side cntaining P(- 6, 8). Be sure t sketch the triangle n the crdinate plane. EXAMPLE 2: Find the exact values f each f the 6 trig functins fr the angle θ with terminal side cntaining P(- 4, - 3). Be sure t sketch the triangle n the crdinate plane. EXAMPLE 3: Find the exact value f each f the ther 5 trig functins fr the angle x (withut finding x) given that the terminal side f x lies in quadrant I and sin x = 5 13 EXAMPLE 4: Find the exact value f each f the ther 5 trig functins fr the angle θ (withut finding θ ) given that the terminal side f θ lies in quadrant II and tanθ = 3 4 EXAMPLE 5: Use a calculatr and evaluate t 4 decimal places (this is where the calculatr mde matters) (A) cs = (B) sec( 2.805) = (C) tan ( ') = (D) sin12 = (E) csc ' 43" = (F) ct 9 =

6 TRIGONOMETRY: 2.5 Exact Values and Prperties f Trignmetric Functins Quadrantal angle: Example 1: Evaluate each functin at the given quadrantal angle. Sketch each angle. (A) sin 3π/2 (B) sec (- π) (C) tan 90 (D) ct (- 270 ) Develping the Unit Circle: Evaluating Trig Functins f Multiples f π/4 Draw the reference triangle with a reference angle f 45 in each quadrant. Recrd the sine and csine values at the given angle. Quadrant II (135 ) Quadrant I (45 ) Quadrant III (225 ) Quadrant IV (315 ) Example 2: Evaluate each functin at the given angle. Give exact answers nly. (A) cs (5π/4) (B) tan (3π/4) (C) csc (45 ) (D) sec (- π/4) Develping the Unit Circle: Evaluating Trig Functins f Multiples f π/6 Draw tw reference triangles with reference angles f 30 and 60 in each quadrant. Recrd the sine and csine values at the given angle. Quadrant II (120 & 150 ) Quadrant I (30 & 60 ) Quadrant III (210 & 240 ) Quadrant IV (300 & 330 ) Example 3: Evaluate each functin at the given angle. Give exact answers nly. (A) ct (5π/6) (B) csc (330 ) (C) sin (2π/3) (D) tan (4π/3)

7 Finding Special Angles Using the given rati fr each trig functin, determine the least psitive θ in degree and radian measure. Suppse sin θ = 3. Draw a reference triangle in the first quadrant with side ppsite reference angle 3 and 2 hyptenuse 2. Observe that this is a special triangle: Example 4: Find the least psitive angle fr sec θ = 2. Peridic Functins: Nte: Bth the sine and csine functin have a perid f 2π. Tangent and ctangent functins have a perid f. Let Q(a, b) be the pint n the unit circle that lies n the terminal side f an angle having Radian measure x. Then, since there are 2π radians in the ne cmplete rtatin, the pint Q(a, b) lies n the terminal side f x + 2π. b = sin x = sin (x+2π) a = cs x = cs (x + 2π) Example 5: If sin x = , what is the value f each f the fllwing? (A) sin (x + 2π) (B) sin (x 2π) (C) sin (x + 14π) (D) sin (x 26π) Fundamental Identities: Reciprcal Identities Tangent Identities Odd/Even Identities Pythagrean Identities csc x =! sec x =! ct x =! tan x = ct x = sin( x) = sin x sin! x + cs! x = 1 cs( x) = cs x tan( x) = tan x Example 6: Simplify each expressin t ne trignmetric functin using the fundamental identities. (A)!!!"#!!! (B) tan( x) cs ( x) (C) (!)!"!!