IB Mathematics HL Year 1 Unit 4: Trigonometry (Core Topic 3) Homework for Unit 4. (A) Using the diagram to the right show that

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1 IB Mathematics HL Year Unit 4: Trigonometry (Core Topic 3) Homework for Unit 4 Lesson 4 Review I: radian measure, definitions of trig ratios, and areas of triangles D.2: 5; E.: 5; E.2: 4, 6; F: 3, 5, 7, 8. (A) Using the diagram to the right show that sin < and that sin cos >. unit circle Lesson 5 Review II: sectors, angular and linear velocity G: 6, 9, 3. (A) The picture to the right represents an circle with radius r having an angle (in radian measure) subtending an arc of length s. The region subtended has area A. Fill in the missing information: r s. r = 2, = π/4, s =, A = 2. r = 3, = 2, s =, A = 3. r =, = π/6, s = 2π/3, A 4. r =, s =, = π/6, A = π. (B) Refer again to the above picture and fill in the missing information: r s A 4 3π π/6 2π (C) Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 40 years. The positions of the star and all three planets are currently collinear. They will next be collinear after how many years? A.I.M.E. 2007, #4.

2 Lesson 5 cont d Review II: sectors, angular and linear velocity (C) The larger circle in the picture to the right is revolving at a rate of 3 revolutions per second. 0 cm 3 cm. 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 circle making? 4. Do any of the above calculations depend on the distance between 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,. 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 at a rate of revolution per second. If I suddenly let go of the rope, how fast will the rock travel? (F) MPOS (Trigonometry): #2, #3, #22, #24 Lesson 6 Review III: Laws of sines and cosines 2A:, 3, 5; 2B.:, 2; 2B.2: 4; 2C:, 4, 7, 8, 4; (A) Find the area of the parallelogram below (B) The picture to the right depicts three mutually tangent circles with radii, 2 and 3. Sketch the triangle formed by joining the centers of the circles and then completely solve this triangle.

3 Lesson 6 cont d Review III: Laws of sines and cosines (C) The picture to the right depicts three circles of radius r placed so as to be mutually tangent. (i) Compute the area of the triangle formed by the centers of the three circles. (ii) Compute the area of the interior region not contained in any of the three circles. (iii) Compute the area of the triangle formed by the three points of tangency. (iv) Compute the area of the largest circle that can be placed in the region described in (ii). (D) From the top of a 0 m building the elevation to the top of a flagpole is. From the base of the same building the elevation to the top of the same flagpole is 40. Compute the height of the flagpole and the distance of the flagpole to the building. 0 m building 40 flagpole

4 (E) Consider the configuration below. B D O C E A In the above, the lines through A, B and A, C are tangent to the circle at points B and C, respectively. Also, the line segment DE is tangent to the circle. Given that m DOE = 75, compute sin A. Lesson 6 cont d Review III: Laws of sines and cosines (F) In the figure to the right, the slope of the line segment of length r is m. The perpendicular line segment has length c. Show that c = r( + m2 ). m y r c x (G) In the quadrilateral depicted at the right, the lengths of the diagonals are a and b, and meet at an angle. Show that the area of this quadrilateral is 2 ab sin. a b (H) Prove the Angle Bisector Theorem. Namely, in the diagram we have the angles = φ precisely when a b = c d. (Hint: use the Law of Sines.) a c φ d b (I) MPOS (Trigonometry): #, #6, #8, #, #6, #7, #9, #23, #30, #3

5 Lesson 7: Periodic functions; the sine function in detail 3A: (a), (c), 2, 3 (in each of these problems, determine the period); 3B.2: 6 (What are the periods for the two functions of #5?): 3B.2: 5. Miniproject: Modeling using the sine function 3C: 3. You are to use the graphics software Autograph to display the data given in problems 3 against your theoretical sine function model. I have indicated the output of the result of the example on page 274, below: 30 T (temperature, celsius) 20 0 y=6.5sin (t-0)/ (estimated sinusoidal regression) Jan Mar May July Sept Nov Jan Mar May July t (month) In the graph below, I ve indicated the plot of the true sinusoidal regression (as calculated from the TI-83); note that there s not much discrepancy between this and the sine model given above. 30 T (temperature, celsius) 20 0 y=6.32sin(.524x-5.2)+2.5 (true sinusoidal regression) Jan Mar May July Sept Nov Jan Mar May July t (month) Your Project: Construct the above two graphs for each of the three data sets indicated above.

6 Lesson 8: Equations and expessions involving sine and cosine 3D.3: 4; 3D.4: 2, 3; 3F: 3, 4; 3G.: 6 (these are somewhat repetitive; do as many as you need to get the hang of what s going on); 3G.2: 3. (A) MPOS (Trigonometry): #7, #5, #28. (B) (Sadler and Thorning, Examination questions, pg 28:) #2(b) (solve this exactly, not approximately), (Sadler and Thorning, Examination questions, pg 378:) 9(a) (one solution can be found exactly, the other is to be found approximately, correct to 3 decimal places.) 3H: 5 (these are routine and repetitive), 6 3 (these are more important; I ll try to allow some class time to work on these). Lesson 9: The addition rule; multipleangle formulas (A) Using problem 3H #, compute the period of the function ( f(x) = sin x + π ) ( sin x + π ) I: 7 (again, time-consuming, but very important practice!). Phase representations. We shall show in this short discussion how to represent the expression a cos x ± b sin x in the form R cos(x ), where is a suitable phase angle. The idea is simple enough: start by writing a cos x ± b sin x = ( ) a 2 + b 2 a a 2 + b cos x ± b 2 a 2 + b sin x 2 = a 2 + b 2 cos cos x ± sin sin x = a 2 + b 2 cos(x ) (B) Write 2 sin x + 5 cos x as R cos(x ) (that is, find R and ). (C) Write 2 sin x + 5 cos x as R sin(x + ) a b (D) Look at the diagram at the end of this syllabus for a visual proof that cos( β) = cos cos β + sin sin β. (E) Without using a calculator compute and simplify (cos 36 )(cos 08 ). (Hint: multiply and divide by sin 36 and see what happens!) (E) MPOS (Trigonometry): #5. (F) (Sadler and Thorning, Examination questions, pg 28:) 4(b), #6, #8, #9, #, #3 pg 377:) #2.

7 3K.: 3; 3K.2: 3, 4 5 (these are important!) (A) Consider two lines L and L 2 having slopes m and m 2 as indicated. m m 2 m Prove that tan =. m 2 + m m 2 (B) Compute the quantity Lesson 20: The tangent function ( + tan )( + tan 2 ) ( + tan 45 ). (Hint: use the addition formula for tangent: tan(a ± B) = tan A ± tan B tan A tan B. (C) Show that tan ( 2 ) ( ) + tan = π 3 4. (D) Solve for n in the equation tan ( 2 (C) MPOS (Trigonometry): #4. ) ( ) ( ) + tan + tan = π 4 n 4. Lesson 2: Quadratic trigonometric equations &c 3M: 2; 3N: 3, 4, 6, 8; 3O: 3. Unit 4 Test

8 Visual proof that cos( β) = cos cos β + sin sin β; note that the area of the inscribed rhombus is cos( β). Area of inscribed rhombus is cos( β) cos() β cos(β) β β β cos( β) β sin() sin(β) cos( β)=cos()cos(β)+sin()sin(β)