Angles of Elevation and Depression

Angles of Elevation and Depression

Study the following figure carefully. angle of elevation angle of depression When we see an object above us, the angle between our line of sight and the horizontal is called the angle of elevation. When we see an object below us, the angle between our line of sight and the horizontal is called the angle of depression.

Consider two animals A and B as shown in the figure. angle of depression of A from B angle of elevation of B from A Are the angle of elevation of B from A and the angle of depression of A from B equal?

Consider two animals A and B as shown in the figure. angle of depression of A from B angle of elevation of B from A i.e. are BAD and ABC equal? CB and AD are two horizontal lines. They are parallel. Also, BAD and ABC form a pair of alternate angles. BAD = ABC

Refer to the figure. angle of depression of A from B angle of elevation of B from A Angle of elevation of B from A = angle of depression of A from B

In the figure, the height of the flower is 30 cm. The angle of elevation from the worm to the top of the flower is 50. A Consider right-angled triangle ABC. tan 50 30 BC BC 30 tan 50 C 50 B 30 cm BC 25.2 cm (cor. to 3 sig. fig.) The distance between the flower and the worm is 25.2 cm.

In the figure, AE and CF are two buildings. Find the Let s study one more example. (a) angle of elevation of A from C, (b) angle of depression of C from A. (Give your answers correct to 3 significant figures.) Solution (a) Consider ADC. tan ACD AD DC AE CF DC (60 49) m 77 m 1 7 60 m D E A 77 m C F B 49 m

In the figure, AE and CF are two buildings. Find the (a) angle of elevation of A from C, (b) angle of depression of C from A. (Give your answers correct to 3 significant figures.) Solution (a) tan ACD 7 1 (b) ACD 8.13 (cor. to 3 sig. fig.) The angle of elevation of A from C is 8.13. BAC ACD (alt. s, AB // DC) 60 m D 8.13 The angle of depression of C from A is 8.13. E A 77 m C F B 49 m

Follow-up question 3 The figure shows a monkey and a banana tree. Find the distance between the two bananas at points B and C. (Give you answer correct to 3 significant figures.) Solution Consider ADC. CD tan CAD AD CD tan 20 10 m CD 10 tan 20 m B 10 C A 20 D 10 m

Follow-up question 3 (cont d) The figure shows a monkey and a banana tree. Find the distance between the two bananas at points B and C. (Give you answer correct to 3 significant figures.) Solution Consider ADB. BD tan BAD AD BD tan (10 20 ) 10 m BD 10 tan 30 m B 10 C A 20 D 10 m

Follow-up question 3 (cont d) The figure shows a monkey and a banana tree. Find the distance between the two bananas at points B and C. (Give you answer correct to 3 significant figures.) Solution The distance between the two bananas BD CD B (10 tan 30 10 tan 20 ) m 10 C 2.13 m (cor. to 3 sig. fig.) A 20 D 10 m

Example 4 In the figure, Raymond s eye level is 1.5 m above the ground. A balloon C is fixed at 50 m vertically above the ground. If the horizontal distance BD between Raymond and the balloon is 28 m, find the angle of elevation of the balloon C from his eye at A correct to 3 significant figures.

Solution ED AB 1.5 m AE BD 28 m Consider right-angled triangle AEC. CE tan CAE AE CD ED AE (50 1.5) m 28 m 48.5 28 CAE 60.0 (cor. to 3sig. fig.) The angle of elevation of the balloon C from his eye at A is 60.0.

Example 5 In the figure, a tourist is travelling upwards from B to D via C in a sightseeing lift. The angle of depression of a point A on the ground from C is 45, while that of A from D is 70. If AB 20 m, find the vertical distance between C and D. (Give your answer correct to the nearest 0.1 m.)

Solution With the notations in the figure, CAB ACE (alt. s, AB // 45 Consider ABC. CB tan 45 AB CB AB tan 45 201m 20 m DAB ADF (alt. s, AB // 70 EC) FD)

Consider ABD. DB tan 70 AB DB AB tan 70 20 tan 70 m DC DB CB (20 tan 70 20) m 34.9 m (cor. to the nearest 0.1m) The vertical distance between C and D is 34.9 m.

Example 6 In the figure, PQ is a vertical flagpole. Nicole measures the angle of elevation of P from A to be 28. Then she walks 20 m towards Q and arrives at B, where she measures the angle of elevation of P from B to be 42. (a) Find the height of the flagpole PQ. (b) Find the horizontal distance QB. (Give your answers correct to 3 significant figures.)

Solution Let PQ h m. (a) In PQB, tan 42 QB In PQA, tan 28 QA PQ QB h tan 42 PQ QA h tan 28 m m

h QA QB BA h h 20 tan 28 tan 42 1 1 20 tan 28 tan 42 h 25.9701 26.0 (cor. to 3sig. fig.) The height of the flagpole PQ is 26.0 m. (b) h QB m (from (a)) tan 42 25.9701 m tan 42 28.8 m (cor. to 3 sig. fig.) The horizontal distance QB is 28.8 m.

(P.243) Yes 30 o, since 2 angles are alt. angles. They are equal. ID6 (P.243) 39 o ID7 (P.244) 170 m

ID8 (P.244) 3800 m ID9 (P.245) ID10 (P.247) 10.3 m PQ=QB-PB = 80(tan37 o -tan32 o ) = 10.3 m 86.6 m 51.6 m h = 50tan60 o h =50tan60 o -50tan35 o )

CP (P.247) 25 o 24.6 o

CP (P.248) 53 m 12.2 m

Example 7 The figure shows two buildings AB and CD. Kelvin measures the angle of elevation of C from A to be 40, and the angle of depression of D from A to be 65. If C is 400 m above the horizontal ground, find the horizontal distance between the two buildings, correct to the nearest 0.1 m.

Solution Let AE x m. In ACE, CE tan 40 AE CE x tan 40 m In AED, tan 65 DE DE AE x tan 65 m CE ED CD x tan 40 x tan 65 400 x(tan 40 tan 65) 400 x 134.1 (cor. to the nearest 0.1m) The horizontal distance between the two buildings is 134.1 m.