The document focuses on circular functions and their applications in trigonometry, particularly angles of elevation and depression. It includes definitions, properties of right triangles, and various example problems involving calculating heights and distances using trigonometric ratios. Students will learn to solve real-world problems related to angles and distances using these concepts.

1. CIRCULAR
FUNCTIONS
PRE-CALCULUS

2. PRE-CALCULUS
At the end of the lesson, students will be able to:
❑illustrate the different circular functions;
❑use reference angle to find the exact values of a
circular function;
❑define angle of elevation and angle of depression; and
❑solve problems on angle of elevation and depression
by applying concept of trigonometric functions.
✓
✓

3. Consider a right triangle with 𝜃 as one of its
acute angles. The trigonometric ratios are
defined as follows:
sin 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
csc 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sec 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cot 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
SOH
CAH
TOA
CHO
SHA
CAO 𝜃
side adjacent to 𝜃
side
opposite
to 𝜃

4. ANGLE OF ELEVATION
It is defined as the angle formed
by a horizontal line from an
observer’s eye and the line of
sight to an object above the
horizontal line.
𝜃ANGLE OF ELEVATION
HORIZONTAL LINE OF SIGHT

5. PRE-CALCULUS
1. A kite is flying with string inclined at 70 degrees to the horizon. What is
the height of the kite above the ground when the string is 86m long?
2. A tree casts a shadow 23 m long. The angle of elevation of a sun is 46
degrees. Find the height of the tree.

6. ANGLE OF DEPRESSION
It is defined as the angle formed
by a horizontal line from an
observer’s eye and the line of
sight to an object below the
horizontal line.
𝜃ANGLE OF DEPRESSION
HORIZONTAL LINE OF SIGHT

7. PRE-CALCULUS
1. The angle of depression of a vehicle on the ground from the top of the
tower is 60 degrees. If the vehicle is at a distance of 100 meters away from
the tower, find the height of the tower.
2. From the top of a vertical cliff 40 m high, the angle of depression of an
object that is level with the base is 34 degrees. How far is the object from
the base of the cliff?

8. 1. A man standing on a cliff sees two cars approaching him on a road. The angles of
depression to the two cars from the man’s line of sight are 25 and 40 respectively.
If the height of the cliff is 87 ft, find the distance between the two cars.
2. A photographer who is 6 foot tall is standing away from the base of a 60-foot tall
building. The photographer looks up to the top of the building and measures the angle
of elevation to be 53.47 degrees. How far is the photographer from the building?
3. A hospital has a wheelchair ramp leading up to its entrance. The entrance is 5
feet above ground level, and the ramp is designed to make an angle of elevation of
15 degrees with the ground. How long is the ramp?

9. Continue Accomplishing
Module 3 on Aleks
Review on Angle of Elevation
and Depression

10. PRE-CALCULUS
1. A kite is flying with string inclined at 70 degrees to the horizon. What is
the height of the kite above the ground when the string is 86m long?
70°
86 𝑚eters
?
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin 70° =
𝑦
86
[ ]
86 86
86 sin 70° = 𝑦
𝑦 = 𝟖𝟎. 𝟖𝟏 𝒎𝒆𝒕𝒆𝒓𝒔 𝒉𝒊𝒈𝒉

11. PRE-CALCULUS
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 46° =
𝑦
23
[ ]
23 23
23 tan 46° = 𝑦
𝑦 = 𝟐𝟑. 𝟖𝟐 𝒎𝒆𝒕𝒆𝒓𝒔 𝒉𝒊𝒈𝒉
2. A tree casts a shadow 23 m long. The angle of elevation of a sun is 46
degrees. Find the height of the tree.
46°
23 𝑚eters
?
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

12. PRE-CALCULUS
1. The angle of depression of a vehicle on the ground from the top of the tower
is 53 degrees. If the vehicle is at a distance of 120 meters away from the tower,
find the height of the tower.
53°
120 𝑚𝑒𝑡𝑒𝑟𝑠
?
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 53° =
𝑦
120
[ ]
120 120
120 tan 53° = 𝑦
𝑦 = 𝟏𝟓𝟗. 𝟐𝟒 𝒎𝒆𝒕𝒆𝒓𝒔 𝒉𝒊𝒈𝒉
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

13. PRE-CALCULUS
2. From the top of a vertical cliff 40 m high, the angle of depression of an object
that is level with the base is 34 degrees. How far is the object from the base of
the cliff?
34°
40 𝑚
?
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 34° =
40
𝑥
[ ]
x x
x tan 34° = 40
𝑥 = 𝟓𝟗. 𝟑𝟎 𝒎𝒆𝒕𝒆𝒓𝒔
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 34° tan 34°

14. 1. A man standing on a cliff sees two cars
approaching him on a road. The angles of
depression to the two cars from the man’s line of
sight are 25 and 40 respectively. If the height of
the cliff is 87 ft, find the distance between the two
cars.
25°
40°
87 𝑓𝑡
?
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 25° =
87
𝑥
[ ]
x x
x tan 25° = 87
𝑥1 = 186.57 𝒎𝒆𝒕𝒆𝒓𝒔
tan 25° tan 25°
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 40° =
87
𝑥
[ ]
x x
x tan 40° = 87
𝑥2 = 𝟏𝟎𝟑. 𝟔𝟖 𝒎𝒆𝒕𝒆𝒓𝒔
tan 40° tan 40°
186.57 m −𝟏𝟎𝟑. 𝟔𝟖 𝒎 = 𝟖𝟐. 𝟖𝟗 𝒎

15. 2. A photographer who is 6 foot tall is standing
away from the base of a 60-foot tall building. The
photographer looks up to the top of the building
and measures the angle of elevation to be 53.47
degrees. How far is the photographer from the
building?
53.47°
60 𝑓𝑡
tan θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 53.47° =
54
𝑥
[ ]
x x
x tan 53.47° = 54
𝑥 = 40 𝒎𝒆𝒕𝒆𝒓𝒔
tan 53.47° tan 53.47°
?
60 − 6 = 54 ft

16. 3. A hospital has a wheelchair ramp leading up to
its entrance. The entrance is 5 feet above ground
level, and the ramp is designed to make an angle
of elevation of 15 degrees with the ground. How
long is the ramp?
sin θ =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin 15° =
5
𝑟
[ ]
r r
r sin 15° = 5
𝑟 = 19.32 𝒇𝒕
sin 15° sin 15°
𝟏𝟓° ]5 ft
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