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

Angle of elevation of B from A = angle of depression of A from B 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 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 = Ð BC AB ACB 30 cm C  = BC 30 cm tan 50  50 B BC = tan 50 30 cm  fig.) sig. 3 to (cor. 25.2 cm = BC \ The distance between the flower and the worm is 25.2 cm. \

Let’s study one more example. 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.) Let’s study one more example. A Solution B (a) Consider △ADC. C D 60 m tan = Ð DC AD ACD 49 m - = DC CF AE E 77 m F 77 m 49) m (60 - = 7 1 =

In the figure, AE and CF are two buildings. Find the (a) 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.) A Solution B (a) ∵ tan ÐACD = 7 1 C D 60 m fig.) sig. 3 to (cor. 8.13 = Ð ACD  \ 49 m The angle of elevation of A from C is 8.13. \ E 77 m F (alt. ∠s, AB // DC) Ð BAC =Ð ACD (b) 8.13 =  The angle of depression of C from A is 8.13. \

Follow-up question 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 10° 20° 10 m A B C D Consider △ADC. tan = Ð AD CD CAD = 10 m CD tan 20° CD = 10 tan 20° m

Follow-up question (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 10° 20° 10 m A B C D Consider △ADB. tan = Ð AD BD BAD = 10 m BD tan (10°+20°) BD = 10 tan 30° m

Follow-up question (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 10° 20° 10 m A B C D ∴ The distance between the two bananas = BD - CD = (10 tan 30°-10 tan 20°) m = 2.13 m (cor. to 3 sig. fig.)