1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles.

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1 NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND OF REAL NUMBERS Name: Date: Mrs. Nguyen s Initial: LESSON 6.4 THE LAW OF SINES Review: Shortcuts to prove triangles congruent Definition of Oblique Triangles 1) SSS ) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles. Drawings: Drawings: Area of Oblique Triangles (SAS case) Law of Sines (in the case of ASA, SAA, SSA) The area A of a triangle with sides of lengths a and b and with included angle θ 1 is: A= absinθ In triangle ABC we have: sin A sin B sin C = = a b c Example: Example: Practice Problems: Find the missing sides and angles in each problem. Round to decimal places. 1. ABC, m A= 54, m B= 9, a= 10. AHS, m A = 5, m H = 111, a = B 43 9 A 58 C Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 1

2 Practice Problems: Find the area of each triangle. Round to decimal places. 4. ABC, if b = 10, c = 6, and m A = ABC, if a = 5, b = 10, and m C = The triangle has sides of length 10 cm, 3 cm, with included angle 10. Practice Problems: Applying what you know. 7. A water tower 30 m tall is located at the top of a hill. From a distance of 10 m down the hill, it is observed that the angle formed between the top and the base of the tower is 8. Find the angle of inclination of the hill. 8. A communications tower is located at the top of a steep hill. The angle of inclination of the hill is 58. A guy wire is to be attached to the top of the tower & to the ground, 100 m downhill from the base of the tower. The angle of elevation from the bottom of the guy wire to the top of the tower is 70. Find the length of the cable required for the guy wire. Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page

3 Ambiguous Case of Law of Sines If you re given SSA, then there can either be 0, 1, or triangles formed. This is the ambiguous case. Drawings: Practice Problems: Solve for all possible triangles that satisfy the given conditions. Round all answers to decimal places. 9. ERW, m R = 35, e = 5, r = DWC, m D = 1, d = 11, w = MLT, m M = 15, m = 10, l = A = 39, a = 10, b = 14 Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 3

4 LESSON 6.5 THE LAW OF COSINES Law of Cosines (in the case of SAS or SSS) In triangle ABC we have: = + cos a b c bc A = + cos b a c ac B = + cos c a b ab C Example: Practice Problems: Solve each triangle. 1. In ABC, b = 6, c = 8, and m A = 6. In ABC, a = 6, c = 8, and m B = In ABC, a = 3, b = 7, and c = 5 4. In BAT, b = 7, a = 9, and t = 1 Heron s Formula: (SSS case) The area A of triangle ABC is given by: A= s( s a)( s b)( s c) where Example: 1 s = ( a+ b+ c ) is the semi-perimeter of the triangle; that is, s is half the perimeter. Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 4

5 Practice Problems: Find the area of the triangle whose sides have the given lengths. 5. MAP, if m = 5, a = 8, and p = 1 6. MEW, if m = 5, e = 7, and w = CAT, c = 9, a = 45, and t = 18 Heading and Bearing is a direction of navigation indicated by an acute angle measured from due north or due south. Practice Problems: Solve each triangle. 8. A pilot sets out from an airport and heads in the direction N15 W, flying at 50 mph. After one hour, he makes a course correction and heads in the direction of N45 W. Half an hour after that, he must make an emergency landing. (A) Find the distance between the airport & his final landing point. (B) Find the bearing from the airport to his final landing point. Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 5

6 9. Airport B is 300 mi from airport A at a bearing of N50 E. A pilot wishes to fly from A to B mistakenly flies due east at 00 mph for 30 minutes, when he notices his error. (A) How far is the pilot from his destination at the time he notices the error? (B) What bearing should he head his plane in order to arrive at airport B? 10. Two ships leave a harbor at the same time. One ship travels on a bearing of S1 W at 14 mph. The other ship travels on a bearing of N75 E at 10mph. How far apart will the ships be after three hours? 11. You are on a fishing boat that leaves its pier and heads east. After traveling for 5 miles, there is a report warning of rough seas directly south. The captain turns the boat & follows a bearing of S40 W for 13.5 miles. (A) At this time, how far are you from the boat s pier? (B) What bearing could the boat have originally taken to arrive at this spot? Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 6

7 LESSON 7.1 THE UNIT CIRCLE The Unit Circle The unit circle is the circle of radius 1 centered at the origin. The equation of the unit circle is: x + y = 1 Note: Every point on the unit circle can be linked to the values of cos θ and sinθ. If point P whose coordinates are (x, y) lies on the unit circle for a given angle θ, then we know that x= cos θ and y = sinθ Practice Problems: Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant P, 5 in QIV. P, 5 in Q II Practice Problems: Find (a) the reference angle for each value of t, and (b) find the terminal point P(x, y) on the unit circle determined by the given value of t. 3. t = 4. 3 t = t = 6. 6 t = t = 8. 3 t = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 7

8 9. t = 10. t = 3 THE UNIT CIRCLE y x Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 8

9 Review: Definition of Reference Angle Let θ be an angle in standard position. Its reference angle is the acute angle θ ' formed by the terminal side of θ and the x-axis. Quadrant II ( ) ( θ) θ' = θ rad θ' = 180 deg ree Quadrant III ( ) ( θ ) θ' = θ rad θ' = 180 deg ree Quadrant IV ( ) ( θ) θ' = θ rad θ' = 360 deg ree Practice Problems: Find the reference angle for each of the given angles. 11. t = t = t = t = t = t = 5.8 Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 9

10 LESSON 7. TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS Definitions of Trigonometric Functions Cofunctions Let t be a real number and let x y be the point on the unit circle corresponding to t. sin t = y 1 csc t =, y y 0 cost = x 1 sec t =, x x 0 y x tan t =, x 0, cot t =, x y y 0 ( θ ) sin 90 = cosθ sin θ = cos θ Remember: SOH CAH TOA opp hyp sinθ = cscθ = hyp opp adj cosθ = secθ = hyp hyp adj opp adj tanθ = cotθ = adj opp ( θ ) cos 90 = sinθ cos θ = sin θ ( θ ) tan 90 = cotθ tan θ = cot θ ( θ ) cot 90 = tanθ cot θ = tan θ ( θ ) sec 90 = cscθ sec θ = csc θ ( θ ) csc 90 = secθ csc θ = sec θ Fundamental Trigonometric Identities Reciprocal Identities 1 sinθ = cscθ 1 cosθ = secθ 1 cscθ = sinθ 1 secθ = cosθ Quotient 1 tanθ = cotθ sinθ tanθ = cosθ 1 cotθ = tanθ cosθ cotθ = sinθ Pythagorean sin θ + cos θ = 1 1+ tan θ = sec θ 1+ cot θ = csc θ Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 10

11 Practice Problems: Evaluate the six trig functions at each real number without using a calculator. Plot the ordered pair. 1. t = 5 6 sin = csc = cos = sec = tan = cot =. t = 3 sin = csc = cos = sec = tan = cot = 3. t = 3 sin = csc = cos = sec = tan = cot = Domain of the Trigonometric Functions Definition of Periodic Function sin, cos: All real numbers tan, sec: All real numbers other than + n for any integer n. cot, csc: All real numbers other than n for any integer n. A function f is periodic if there exists a positive real number such that ( ) ( ) f t+ c = f t for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. Practice Problems: Evaluate the trigonometric function using its period as an aid. 4. cos5 = 5. 9 sin 4 = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 11

12 6. sin( 3 ) = 7. 8 cos = 3 Even and Odd Trigonometric Functions The cosine and secant functions are even. cos( t) = cost sec( t) = sect The sine, cosecant, tangent, and cotangent functions are odd. sin( t) = sint csc( t) = csct tan( t) = tant cot( t) = cott Remember: Even f ( t) = f ( t) Odd f ( t) = f ( t) Practice Problems: Use the value of the trig function to evaluate the indicated functions. 8. sin( t) = 3 8 sint = csct = 9. cos 4 5 cos t = ( t) = cos( t ) + = Practice Problems: Use a calculator to evaluate. Round to 4 decimal places. 11. csc1.3 = 1. cos 10. sin = (.5) = 13. cot1 = 4 Practice Problems: Use a calculator to evaluate. Round to 4 decimal places. 14. cos80 = 15. cot 66.5 = 16. sec 7 = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 1

13 Practice Problem 17: Let θ be an acute angle such that. Find: cosθ = 0.6 sinθ = cscθ = cosθ = secθ = tanθ = cotθ = Practice Problem 18: Given cscθ = 13 and secθ = 13 3, find Practice Problem 19: Given, find tan β = 5 sinθ = tanθ = cosθ = sec( 90 θ ) = cot β = tan( 90 β ) = cos β = csc β = Practice Problems: Evaluate the value of θ in degrees ( 0 θ 90 ) a calculator. < < and radians 0 < θ < without using 0. cscθ = 1. tanθ = 1. cotθ = 1 3. sinθ = 3 Practice Problems: Use a calculator to evaluate the value of θ in degrees ( 0 θ 90 ) 0 < θ <. Round to the nearest degrees and 3 decimal places for radians. 4. secθ = sinθ = cotθ = sinθ = < < and radians Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 13

14 LESSON 7.3 TRIGONOMETRIC GRAPHS Graph of Sine Function y = sin x Graph of Cosine Function y = cos x Domain: Range: x-intercepts: Relative Minima: Relative Maxima: Domain: Range: x-intercepts: Relative Minima: Relative Maxima: x 0 3 x 0 3 y y Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 14

15 Transformations y = d + asin( bx c) y = d + acos( bx c) Vertical stretch: a > 1 a Scaling factor a Vertical shrink: a < 1 Reflection over the x-axis b = b period c Horizontal translation (left or right) d Vertical translation (up or down) Definition of Amplitude of Sine and Cosine Curves The amplitude of y asin x y = a x represents half the distance between the minimum and the maximum value of the function and is given by Amplitude = a. = and cos Practice Problems: Write an equation for each dashed curve. 1. y =. y = Practice Problem 3: Write equations for both curves y solid y dashed = = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 15

16 Period of Sine and Cosine Functions Let b be a positive real number. The period of. b y = asinbx and y acosbx = is Practice Problems: Write and equation for each dashed curve. 4. y = 5. y = Graphs of Sine and Cosine Functions The graph of y = asin( bx c) and y acos( bx c) b > ). Amplitude = a Period = b characteristics. (Assume 0 = have the following The left and right endpoints of a one-cycle interval can be determined by solving the equations: bx c = 0 bx c =. Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully. 6. y = sin( 4x+ ) Amplitude: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection: Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 16

17 x 7. y = 3cos + Amplitude: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection: Practice Problem 8: When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approximated by y = 0.001sin880 t, where t is time in seconds. a. What is the period of the function? b. The frequency f is given by f 1 p =. What is the frequency of the note? Practice Problems: Write an equation for each curve. 9. y = 10. y = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 17

18 Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully x y = sin 3 Amplitude: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection: y = 3+ 4cos ( x 5) Amplitude: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection: y = 4+ sin ( x+ 3) Amplitude: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection: Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 18

19 LESSON 7.4 MORE TRIGONOMETRIC GRAPHS Graphs of y = csc xand y = sec x y = csc x y = sec x In order to graph y = csc xand y = sec x, use the graphs of the other sine & cosine as base models. Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully. 1. y = csc x. y = sec 3x 3. y = csc4x 4. y = sec x+ 3 Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 19

20 5. y = csc x y = 1+ sec ( x+ 4) Graph of Tangent Function y = tan x x 0 y Domain: Range: x-intercepts: 3 Vertical asymptotes: Two standard consecutive vertical asymptotes: bx c = bx c = Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 0

21 Practice Problem 7: Write an equation for each dashed curve. y = Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully. 8. y = 3tan( x) Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: x 9. y = 3tan + Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 1

22 Graph of Tangent Function y = cot x x 0 y Domain: Range: 3 x-intercepts: Vertical asymptotes: Two standard consecutive vertical asymptotes: bx c = 0 bx c = Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully. 10. y = cot x Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page

23 11. y = cot4x Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: 1. y = cot 3x+ 6 Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: 13. y = 3+ 5cot ( x ) 4 Amplitude: Left vertical asymptote: Right vertical asymptote: Vertical shift: Reflection: Mrs. Nguyen Honors Algebra II Chapter 6 & 7 Notes Page 3