IB Mathematics HL—Year 1Unit 4: Trigonometry (Core Topic 3)

Homework for Unit 4

Lesson 14Review I:radian measure,definitionsof trig ratios,and areas oftriangles

11D.2: 1–5; 11E.1: 1–5; 11E.2: 1–4, 6; 11F: 1–3, 5, 7, 8.

(A) Using the diagram to theright show that

sin θθ

< 1 and that

sin θθ cos θ

> 1.unit circle

θ

1

Lesson 15Review II:sectors, angularand linear ve-locity

11G: 1–6, 9, 13.

(A) The picture to the right represents ancircle with radius r having an angle(in radian measure) subtending anarc of length s. The region subtendedhas area A. Fill in the missing infor-mation:

sθ

r

1. r = 2, θ = π/4, s = , A =

2. r = 3, θ = 2, s = , A =

3. r = , θ = π/6, s = 2π/3, A

4. r = , s = , θ = π/6, A = π.

(B) Refer again to the above picture and fill in the missing informa-tion:

r θ s A

4 3ππ/6 12π

(C)1 Three planets revolve about a star in coplanar circular orbitswith the star at the center. All planets revolve in the samedirection, each at a constant speed, and the periods of theirorbits are 60, 84, and 140 years. The positions of the starand all three planets are currently collinear. They will nextbe collinear after how many years?

1A.I.M.E. 2007, #4.

Homework for Unit 4, cont’d

Lesson 15cont’dReview II:sectors, angularand linear ve-locity

(C) The larger circle in thepicture to the rightis revolving at a rateof 3 revolutions persecond. 3 cm

10 cm

1. Find the angular velocity of the larger circle.

2. Find the angular velocity of the smaller circle.

3. How many revolutions per second is the smaller circlemaking?

4. Do any of the above calculations depend on the distancebetween the two circles?

(D) Suppose that an automobile is traveling at a speed of 60 km/hr.Assuming that each of the four tires has a radius of .5 m,

1. how many revolutions per minute is each tire making?

2. What is the angular velocity of each tire?

(E) Suppose I am spinning a rock at the end of a 2 meter rope ata rate of 1 revolution per second. If I suddenly let go of therope, how fast will the rock travel?

(F) MPOS (Trigonometry): #12, #13, #22, #24

Lesson 16Review III:Laws of sinesand cosines

12A: 1, 3, 5; 12B.1: 1, 2; 12B.2: 1–4; 12C: 1, 4, 7, 8, 14;

(A) Find the area of the parallelogram below.

24

2015

(B) The picture to the right depictsthree mutually tangent circleswith radii 1, 2 and 3. Sketchthe triangle formed by joiningthe centers of the circles andthen completely solve this tri-angle.

Homework for Unit 4, cont’d

Lesson 16cont’dReview III:Laws of sinesand cosines

(C) The picture to the right depictsthree circles of radius r placedso as to be mutually tangent.

(i) Compute the area of the triangle formed by the centers ofthe three circles.

(ii) Compute the area of the interior region not contained inany of the three circles.

(iii) Compute the area of the triangle formed by the threepoints of tangency.

(iv) Compute the area of the largest circle that can be placedin the region described in (ii).

(D) From the top of a 10 m building the elevation to the top of aflagpole is 11◦. From the base of the same building the eleva-tion to the top of the same flagpole is 40◦. Compute the heightof the flagpole and the distance of the flagpole to the building.

11°

40° flagpolebuilding

10 m

Homework for Unit 4, cont’d

Lesson 16cont’dReview III:Laws of sinesand cosines

(E) Consider the configuration below.

D

EA

B

C

O

In the above, the lines through A,B and A,C are tangent tothe circle at points B and C, respectively. Also, the line seg-ment DE is tangent to the circle. Given that m∠DOE = 75◦,compute sinA.

(F) In the figure to the right, theslope of the line segment oflength r is m. The perpendic-ular line segment has length c.Show that

c =r(1 +m2)

m.

c

r

y

x

(G) In the quadrilateral depictedat the right, the lengths of thediagonals are a and b, andmeet at an angle θ. Show thatthe area of this quadrilateral is12ab sin θ.

θ

ba

(H) Prove the Angle Bisector Theo-rem. Namely, in the diagramwe have the angles θ = φ pre-cisely when a

b = cd . (Hint: use

the Law of Sines.)

φθa

c d

b

(I) MPOS (Trigonometry): #1, #6, #8, #11, #16, #17, #19, #23, #30,#31

Homework for Unit 4, cont’d

Lesson 17:Periodic func-tions; the sinefunction indetail

13A: 1 (a), (c), 2, 3 (in each of these problems, determine the period);13B.2: 1–6 (What are the periods for the two functions of #5?): 13B.2:1–5.

Miniproject:Modeling usingthe sine func-tion

13C: 1–3. You are to use the graphics software Autograph to displaythe data given in problems 1–3 against your theoretical sine functionmodel. I have indicated the output of the result of the example onpage 274, below:

10

20

30

t (month)

T (temperature, celsius)

Jan Mar

y=6.5sin ¤(t-10)/6 + 21.5

May July Sept Nov Jan Mar May July

(estimated sinusoidal regression)

In the graph below, I’ve indicated the plot of the true sinusoidalregression (as calculated from the TI-83); note that there’s not muchdiscrepancy between this and the sine model given above.

10

20

30

t (month)

T (temperature, celsius)

Jan Mar

y=6.32sin(.524x-5.21)+21.5

May July Sept Nov Jan Mar May July

(true sinusoidal regression)

Your Project: Construct the above two graphs for each of the threedata sets indicated above.

Homework for Unit 4, cont’d

Lesson 18:Equations andexpessionsinvolving sineand cosine

13D.3: 1–4; 13D.4: 2, 3; 13F: 3, 4; 13G.1: 1–6 (these are somewhatrepetitive; do as many as you need to get the hang of what’s goingon); 13G.2: 1–3.

(A) MPOS (Trigonometry): #7, #15, #28.

(B) (Sadler and Thorning, Examination questions, pg 128:) #2(b)(solve this exactly, not approximately), (Sadler and Thorning,Examination questions, pg 378:) 9(a) (one solution can befound exactly, the other is to be found approximately, correctto 3 decimal places.)

Lesson 19:The additionrule; multiple-angle formulas

13H: 1–5 (these are routine and repetitive), 6–13 (these are moreimportant; I’ll try to allow some class time to work on these).

(A) Using problem 13H #11, compute the period of the function

f(x) = sin(x+

π

3

)sin(x+

π

2

).

13I: 1–7 (again, time-consuming, but very important practice!).

Phase representations. We shall show in this short discussion howto represent the expression a cosx± b sinx in the form R cos(x∓ α),where α is a suitable phase angle. The idea is simple enough: startby writing

a cosx± b sinx =√a2 + b2

(a√

a2 + b2cosx± b√

a2 + b2sinx

)=√a2 + b2 cosα cosx± sinα sinx

=√a2 + b2 cos(x∓ α)

(B) Write 2 sinx + 5 cosx asR cos(x−α) (that is, findR and α).

(C) Write 2 sinx + 5 cosx asR sin(x+ α)

b

aα

(D) Look at the diagram at the end of this syllabus for a “visual”proof that cos(α− β) = cosα cosβ + sinα sinβ.

(E) Without using a calculator compute and simplify(cos 36◦)(cos 108◦). (Hint: multiply and divide by sin 36◦

and see what happens!)

(E) MPOS (Trigonometry): #5.

(F) (Sadler and Thorning, Examination questions, pg 128:) 4(b),#6, #8, #9, #11, #13 pg 377:) #2.

Homework for Unit 4, cont’d

Lesson 20:The tangentfunction

13K.1: 1–3; 13K.2: 1–3, 4–5 (these are important!)

(A) Consider two lines L1 and L2 having slopes m1 and m2 as in-dicated.

θ

m2m1 Prove that tan θ =

m2 −m1

1 +m1m2.

(B) Compute the quantity

(1 + tan 1◦)(1 + tan 2◦) · · · (1 + tan 45◦).

(Hint: use the addition formula for tangent:

tan(A±B) =tanA± tanB

1∓ tanA tanB.

(C) Show that tan−1

(12

)+ tan−1

(13

)=π

4.

(D) Solve for n in the equation

tan−1

(12

)+ tan−1

(14

)+ tan−1

(1n

)=π

4.

(C) MPOS (Trigonometry): #4.

Lesson 21:Quadratictrigonometricequations &c

13M: 1–2; 13N: 3, 4, 6, 8; 13O: 1–3.

Unit 4 Test

Visual proof that cos(α − β) = cos α cos β + sin α sin β; note thatthe area of the inscribed rhombus is cos(α− β).

Area of inscribed rhombus is cos(α−β)

cos(α−β)=cos(α)cos(β)+sin(α)sin(β)

sin(α)

cos(α)

cos(β)

β

β

α

α

1

1

sin(β)

α−β

cos(α−β)1

11

1

α

α

β

β