HARD TRIG QUESTIONS PDF

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Harder 2U and easier 3U trig questions
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Trigonometry 2unit including harder 3D Trig 1)

In the diagram below P

h 12

11

A

Q

1000 m

B

2)

The angle of elevation of a tower PQ of height h metres at a point A due east of it is 12. From another point B, the bearing of the tower is 051T and the angle of elevation is 11. The points A and B are 1000 metres apart and on the same level as the base Q of the tower. a. Show that AQB = 141. b. Consider the triangle APQ and show that AQ = h tan 78. c. Find a similar expression for BQ. d. Use the cosine rule in the triangle AQB to calculate h to the nearest metre.¤ In the diagram below P

100 m

A

3)

C 60

20

600 m

B

Two yachts A and B subtend an angle of 60 at the base C of a cliff. From yacht A the angle of elevation of the point P, 100 metres vertically above C, is 20. Yacht B is 600 metres from C. a. Calculate the length AC. b. Calculate the distance between the two yachts.¤ A tourist wants to estimate how tall a Pharaohs obelisk in some tourist site. T

hm

P

22 36 14

O

Q With the aid of a simple protractor, cardboard and a pencil, she estimated the angle of elevation of the top of the obelisk as 22° from an initial position P. She then walked 56 metres to another position Q, where she found the angle of elevation of the top of the obelisk to be 14°. She also found the size of POQ to be 33°. What did you think her estimation of the height of the obelisk was?

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4)

In the diagram below

45 O

30

100

5)

A surveyor stands at a point A, which is due south of a tower OT of height h m. The angle of elevation of the top of the tower from A is 45 . The surveyor then walks 100 m due east to point B, from where she measures the angle of elevation of the top of the tower to be 30 . a. Express the length of OB in terms of h. b. Show that ℎ  50√2. c. Calculate the bearing of B from the base of the tower to the nearest minute.¤ David is in a life raft and Anna is in a cabin cruiser searching for him. They are in contact by mobile telephone. David tells Anna that he can see Mt Hope. From David’s position the mountain has a bearing of 109, and the angle of elevation to the top of the mountain is 16. Anna can also see Mt Hope. From her position it has a bearing of 139, and the top of the mountain has an angle of elevation of 23. The top of Mt Hope is 1500 m above sea level. H A 23 1500 m 16 D

6)

B

Find the distance and bearing of the life raft from Anna’s position.¤ From a point A due south of a tower, the angle of elevation of the top of the tower T, is 23°. From another point B, on a bearing of 120° from the tower, the angle of elevation of T is 32°. The distance AB is 200 metres. T

North

O

A a. b.

120°

B

Copy or trace the diagram into your writing booklet, adding the given information to your diagram. Hence find the height of the tower.

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7)

The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below.  y 3  – 2

–2



0r



O

x

1 P(x,y) –  2



8)

As P, rotates clockwise on the circumference of the circle to complete one revolution, the angle  which OP makes with the initial position (positive direction of x-axis) is observed. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the y-coordinate of P. a. Plot the curve showing all necessary information b. Name a function that best represent the plotted curve and give its rule c. What is the domain and range of the function? The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below. y

D

C 

 2

E  F

B P(x,y)

1 

O

A r

0

2  I

x

3  2

G H As P, rotates anticlockwise on the circumference of the circle to complete one revolution, some locations of P were marked as points A, B, C, D, E, F, G, H and I. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the y-coordinate of P. a. Plot the curve showing all necessary information b. Name a function that best represent the plotted curve and write its rule c. Locate and label approximate positions of the points A, B, C, D, E, F, G, H and I on your curve.

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9)

The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below. y C 

D

B

 2

A E F



r

3  2

x

2  I

1 P(x,y)

G

10)

0



O

H

As P, rotates clockwise on the circumference of the circle to complete one revolution, some locations of P were marked as points A, B, C, D, E, F, G, H and I. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the y-coordinate of P. a. Redraw the unit circle and replace all boundary angle sizes with negative measures to comply with the orientation of P b. Plot the curve showing all necessary information c. Name a function that best represent the plotted curve and write its rule d. Locate and label approximate positions of the points A, B, C, D, E, F, G, H and I on your curve. The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below.



2

y P(x,y)

1



 O

0r 2

x

3 2

As P, rotates anticlockwise on the circumference of the circle to complete one revolution, the angle  which OP makes with the initial position (positive direction of x-axis) is observed. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the x-coordinate of P. a. Plot the curve showing all necessary information b. Name a function that best represent the plotted curve and give its rule c. What is the domain and range of the function?

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11)

The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below. y C 

D

B

 2

A E F



r

3  2

x

2  I

1 P(x,y)

G

12)

0



O

H

As P, rotates clockwise on the circumference of the circle to complete one revolution, some locations of P were marked as points A, B, C, D, E, F, G, H and I. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the x-coordinate of P. a. Redraw the unit circle and replace all boundary angle sizes with negative measures to comply with the orientation of P b. Plot the curve showing all necessary information c. Name a function that best represent the plotted curve and write its rule d. Locate and label approximate positions of the points A, B, C, D, E, F, G, H and I on your curve. The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below.



2

y P(x,y)

1



 O

0r 2

x

3 2 As P, rotates anticlockwise on the circumference of the circle to complete one revolution, the angle  which OP makes with the initial position (positive direction of x-axis) is observed. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the ratio of the y : x coordinates of P. a. Plot the curve showing all necessary information b. Name a function that best represent the plotted curve and give its rule c. What is the domain and range of the function?

PAGE 5 OF 23

13)

The point P(x, y) is on the circumference of a unit circle with centre O, as shown in the diagram below. y C 

D

E

B

 2

P(x,y)

1

A r





O

F

0 2 

I

x

3  2

G H As P, rotates anticlockwise on the circumference of the circle to complete one revolution, some locations of P were marked as points A, B, C, D, E, F, G, H and I. A curve is plotted on a set of axes with horizontal axis representing  and the vertical axis representing the ratio of the y : x coordinates of P. a. Plot the curve showing all necessary information b. Name a function that best represent the plotted curve and write its rule c. Locate and label approximate positions of the points A, B, C, D, E, F, G, H and I on your curve. Comment on the points which you cannot locate. 14)

06r

90 cm

A

15)

B

A pendulum is 90 cm long and swings through an angle of 0·6 radians. The extreme positions of the pendulum are indicated by the points A and B in the diagram. a. Find the length of the arc AB. b. Find the straight-line distance between the extreme positions of the pendulum. c. Find the area of the sector swept out by the pendulum.¤ The diagram shows a circle with centre O and radius 2 centimetres. The points A and B lie on the circumference of the circle and  AOB = θ. A O 

B

a. b.

There are two possible values of θ for which the area of  AOB is √3 square centimetres. One  value is  . Find the other value. 

Suppose that 𝜃   1. Find the area of the sector AOB.

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2.

16)

Find the exact length of the perimeter of the minor segment bounded by the chord AB and the arc AB.¤ The diagram shows a circle with centre O and radius 5 cm. The length of the arc PQ is 9 cm. Lines drawn perpendicular to OP and OQ at P and Q respectively meet at T.

O 5 cm P

Q

9 cm

T

17)

a. Find POQ in radians. b. Prove that OPT is congruent to OQT. c. Find the length of PT. d. Find the area of the shaded region.¤ In the figure below B

A

12

r

33 m

C

18)

AB and AC are radii of length r metres, of a circle with centre A. The arc BC of the circle subtends an angle of measure 12 radians at A. The arc length of the sector is 33 m. a. Find the exact length of the radius of the circle A b. Find the area of the segment of the chord BC In the figure below B

A

26 m

43

C

19) 20)

AB and AC are radii of length r metres, of a circle with centre A. The arc BC of the circle subtends an angle of measure 43 at A. The arc length of the sector is 26 m. a. Find the length of the radius of the circle A, correct to 2 decimal places b. Find the area of the segment of the chord BC Find all possible values of θ if tan θ = √3 and 0 ≤ θ ≤ 360°.  Find all possible values of θ if cosec θ =  and 0 ≤ θ ≤ 360°. √

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21) 22) 23) 24) 25) 26) 27) 28) 29)

Find all possible values of θ if sec θ = 2 and 0 ≤ θ ≤ 360°. Find all possible values of θ if cot θ = 1 and 0 ≤ θ ≤ 360°. Find all possible values of θ if cosec2θ = 2 and 0 ≤ θ ≤ 360°. Find all possible values of θ if 2 cos 2θ  1 = 0 and 0 ≤ θ ≤ 360°. Find all possible values of θ if 3 tan2 θ  1 = 0 and 0 ≤ θ ≤ 360°. Find all possible values of θ (to the nearest minute) if 6 cos 2 θ + sin θ = 5 and 0 ≤ θ ≤ 360°. Find all possible values of θ if cot θ = 0 and 0 ≤ θ ≤ 360°. On the same set of axes sketch the functions y = sin  and its reciprocal, where 0    2π. State the domain and range of both functions. The diagram below presents parts of the curve of a trigonometric function and parts of the curve of its reciprocal in the domain [0, 360]. Identify the function and its reciprocal, compete the graph of each and label the curves with the correct function names.

3 2 1 0

90

180

270

360

°

–1 –2 –3

30) 31)

On the same set of axes sketch the functions y = cos  and its reciprocal, where 0    360. State the domain and range of both functions. On the same set of axes sketch the functions y = cos  and its reciprocal, where 0    2π. State the domain and range of both functions.

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32)

The diagram below presents parts of the curve of a trigonometric function and parts of the curve of its reciprocal in the domain [0, 360]. Identify the function and its reciprocal, compete the graph of each and label the curves with the correct function names.

3 2 1 0

90

180

270

360  °

–1 –2 –3

33)

The diagram below presents parts of the curve of a trigonometric function and parts of the curve of its reciprocal in the domain [0, 360]. Identify the function and its reciprocal, compete the graph of each and label the curves with the correct function names.

3 2 1 0

90

180

270

360

°

–1 –2 –3

34)

On the same set of axes sketch the functions y = tan  and its reciprocal, where 0    360. State the domain and range of both functions.

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35)

The diagram below presents parts of the curve of a trigonometric function and parts of the curve of its reciprocal in the domain [0, 360]. Identify the function and its reciprocal, compete the graph of each and label the curves with the correct function names.

3 2 1 0

90

180

270

360

 °

–1 –2 –3

36) 37) 38) 39)

40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52)

Use Pythagoras theorem to prove the Pythagorean identity sin 2  + cos 2  = 1 for any acute angle  . Use the unit circle definition of the trigonometric ratios to prove the Pythagorean identity sin 2  + cos2  = 1, illustrate your proof geometrically.   Prove that  𝑡𝑎𝑛  for an acute angle   

Use the fact that sin2  + cos2  = 1, for any angle , to prove the Pythagorean identities a. tan2  + 1 = sec2  b. 1 + cot2  = cosec 2  Prove that (sec θ + tan θ) (sec θ – tan θ) = 1 Prove that sin 2 θ  cos2 θ = 1  2 cos 2 θ Prove that (sec2 θ  1) cot2 θ = 1 Prove that sec2 θ + cosec2 θ = sec2 θ cosec 2 θ Prove that √1  𝑡𝑎𝑛 𝜃  𝑠𝑒𝑐  Prove that √1  𝑡𝑎𝑛 𝜃 √1  𝑠𝑖𝑛  𝜃  1 Prove that 4  3sec2 θ + 3tan 2 θ = 1 Prove that √𝑠𝑒𝑐 𝜃 𝑐𝑠𝑐 𝜃 𝑐𝑜𝑡 𝜃  1  𝑐𝑜𝑡      Prove that  1  𝑡𝑎𝑛   Prove that

  

 sin 𝑥 cos 𝑥

    

Prove that 

    

  1 



Find the exact value of cos sin  

Find the exact value of cosec 

 

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53) 54) 55) 56) 57) 58) 59)



Find the exact value of sec cos  Find the exact value of cosec  45° cot 60°  Find the exact value of cosec cot  Find the exact value of cos 45° sin 30°   Find the exact value of cos sin   Find the exact value of

 °  °

D

C

2 cm x°

60° A

B

2 cm P 4 cm

In the figure (not to scale), ABCD is a parallelogram in which AB = 6 cm, AD = 2 cm, and DAB = 60°. The point P on AB is such that AP = 2 cm, and DPC = x°. a. Write down the length of DP. b. Use the cosine rule for each of the triangles PBC, PCD to show that 𝑐𝑜𝑠 𝑥°   .¤ √

60)

B

40·5 m A 20·0 m D

100 70 C

47·3 m

In the above figure (not drawn to scale), ABCD is a quadrilateral in which AB = 40·5 m, AD = 20·0 m, DC = 47·3 m, A = 100° and C = 70°. a. Use the cosine rule for triangle ABD to find the length BD. b. Use the sine rule to find DBC, correct to the nearest degree.¤ 61) N

S

B L From a lighthouse L a ship S bears 053°T and is at a distance of 8 nautical miles. From L a boat bears 293°T and is at a distance of 6 nautical miles. a. Draw a diagram marking on it the information supplied. b. Find the distance of ship S from boat B. Give your answer as a surd. c. Find the bearing of the ship S from boat B. Give your answer to the nearest degree.¤ [[End Of Qns] ]

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[Solutions] «1) a)  APQ = 90 – 12 = 78°  𝑡𝑎𝑛 ∠𝐴𝑃𝑄  , 𝐴𝑄  ℎ 𝑡𝑎𝑛 78°  c) Similarly BQ = h tan 79   d) 𝐴𝑄   4 ∙ 71ℎ m and 𝐵𝑄    5 ∙ 14ℎ m  (Correct to 2 decimal places) Apply cosine rule to  AQB AB2 = AQ2 + BQ 2 – 2×AQ×BQ×cos  AQB 1000  󰇛4 ∙ 71ℎ󰇜  󰇛5 ∙ 14ℎ󰇜  2  󰇛4 ∙ 71ℎ󰇜󰇛5.14ℎ󰇜 cos 141 Solving for h  106 = 86∙23h2 

 ℎ  .  ℎ  108 m (correct to the nearest metre) » 

«2) a) 𝐴𝐶    274 ∙ 75 m (to 2 d.p.) 

b) Apply cosine rule to  ABC AB2 = 274∙752 + 6002 – 2×274∙75×600 cos 60  AB = 520 m (to the nearest metre) »  «3) POT is right angled at O  𝑃𝑂    2 ∙ 48 

BOC is right angled at O  𝑄𝑂 



 4 ∙ 01

Applying cosine rule to POQ  󰇛56󰇜  󰇛2 ∙ 48ℎ󰇜  󰇛4 ∙ 01ℎ 󰇜  2󰇛2 ∙ 48ℎ󰇜󰇛4 ∙ 01ℎ󰇜𝑐𝑜𝑠33 562 = 5∙55h2  h = 23∙77 m » «4) a) Δ𝑇𝑂𝐵 is right angled at O  ⇒ 𝑡𝑎𝑛 30   ⇒ 𝑂𝐵  √3ℎ b) AO = h since AOT is right-angled isosceles triangle AOB is right angled at A ⇒ 3ℎ   ℎ  100 ⇒ 2ℎ   100 ⇒ ℎ  50√2   c) 𝑡𝑎𝑛 𝐴𝑂𝐵  ⇒ ∠𝐴𝑂𝐵  𝑡𝑎𝑛 󰇡√󰇢  54° 44′ √

Bearing of B from the base of the tower  180  54° 44󰆒  125° 16′ » N

A

139

41

N 109

D

41

19

30 19

B «5) ABH is right-angled at B  𝐴𝐵 





 3534 m (to the nearest metre)

PAGE 12 OF 23



DBH is right-angled at B  𝐷𝐵   5231 m (to the nearest metre) Applying cosine rule to ABD  2 2 2  AD = 3534 + 5231 – 2 × 3534 × 5231 × cos 30° Distance AD= 2799 m Using sine rule on ABD         𝑠𝑖𝑛  𝐷𝐴𝐵  







  𝐷𝐴𝐵  𝑠𝑖𝑛    69° or 111° (to the nearest degree) If  DAB = 69°   ADB = 81° i.e.  DAB is less than  ADB Then the length of BD should be less than the length of AB which is not the case Hence  DAB = 111° and the bearing = 139 + 111 = 250  » T

h

North 120°

O 60° 23° A «6) a) b) 𝐴𝑂 





32° 200 m

B

 2 ∙ 36ℎ and 𝐵𝑂    1 ∙ 6ℎ 

Apply cosine rule on the ABO AB2 = AO2 + BO 2 – 2×AO×BO×cos  AOB 200  󰇛2 ∙ 36ℎ󰇜  󰇛1 ∙ 6ℎ󰇜  2  󰇛2 ∙ 36ℎ󰇜󰇛1.6ℎ󰇜 𝑐𝑜𝑠 60 Solving for h  2002 = 4∙4496h2, h = 95 m » y=sin 1



–2 

3  2

   2

–