1. Quarter 4 - Module 4 Word Problems Involving Right Triangles

2. Introduction
Trigonometry is the branch of mathematics that deals with triangles particularly right triangles.
They are behind how sound and light move and are also involved in our perceptions of beauty and other facets on how our mind works.
So trigonometry turns out to be the fundamental to pretty much everything!

3. Most Essential Learning Competency:
Uses trigonometric ratios to solve real-life problems involving right triangles (M9GE-IVe-1)

4. Subtasks:
1. recall the trigonometric ratios and angles of elevation and depression;
2. uses trigonometric ratios to solve real-life problems involving right triangles

5. A trigonometric function is a ratio of certain parts of a triangle
A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent.
Let us look at this triangle…
Given the assigned letters to the sides and angles, we can determine the following trigonometric functions. a c b ө A B C
Sinθ=
Cos θ=
Tan θ=
Side Opposite
Side Adjacent
Hypotenuse = a b c

6. some trigonometric ratios
Values of
some trigonometric ratios
Example 1
Example 2
Example 3

7. USES OF TRIGONOMETRY One of the most ancient subjects studied
by scholars all over the world, astronomers have
used trigonometry to calculate the distance
from the earth to the planets and stars.
Its also used in geography and in
navigation.
The knowledge of trigonometry is used
to construct maps, determine the
position of an island in
relation to longitudes and latitudes.
Trigonometry is used in almost every
sphere of life around you.
Angle of depression
Angle of elevation

8. Basic Fundamentals Line of sight Angle of elevation Horizontal
Angle of Elevation: In the picture below, an observer is standing at the top of a building is looking straight ahead (horizontal line). The observer must raise his eyes to see the airplane (slanting line). This is known as the angle of elevation.
Line of sight
Angle of elevation
Horizontal

9. Horizontal Angle of depression
Angle of Depression: The angle below horizontal that an observer must look to see an object that is lower than the observer.
Line of sight
Angle of depression
Horizontal
Object

10. Example 1 :-
The angle of elevation of the top of a pole measures 45° from a point on the ground 18 ft. away from its base. Find the height of the flagpole.
Solution
Let’s first visualize the situation
Let ‘x’ be the height of the flagpole.
From triangle ABC, tan 45 ° =x/18
x = 18 × tan 45° = 18 × 1=18ft
So, the flagpole
is 18 ft. high.
45 °

11. Example 2 :- A tower stands on the ground. The angle of elevation
from a point on the ground which is 30 metres away from the foot
Of the tower is 30⁰. Find the height of the tower. (Take √3 = )
30⁰
30 m h A B C
Solution .
Let AB be the tower h metre high.
Let C be a point on the ground which is 30 m away from point B, the foot of the tower.
So BC = 30 m
Then ACB = 30⁰
Now we have to find AB i.e. height ‘h’ of the tower .

12. In right angled triangle ABC
Now we shall find the trigonometric ratio combining AB and BC .
In right angled triangle ABC AB B
tan 30⁰
30⁰
30 m h A B C 1 √3 h 30 30 √3 h
30 x √3
√3 x √3
30 x √3 3
10√3 m
10 x m
17.32 m
Hence, height of the tower = m

13. Example 3:-
An airplane is flying at a height of 2 miles above the level ground.
The angle of depression from the plane to the foot of a tree is 30°.
Find the distance that the air plane must fly to be directly above the tree.
Step 1: Let ‘x’ be the distance the airplane must fly to be directly above the tree. Step 2: The level ground and the horizontal are parallel, so the alternate interior angles are equal in measure.
Step 3: In triangle ABC, tan 30=AB/x.
Step 4: x = 2 / tan 30
Step 5: x = (2*31/2)  
Step 6: x = 3.464
So, the airplane must fly about 3.464
miles to be directly above the tree.
30 ° D

14. THE END