1. Some
Applications of
Trigonometry

2. What is Trigonometry?
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. 

3. The word trigonometry is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides’ ) and ‘metron’ (meaning measure).
Trigonometric ratios of an angle are some ratios of the sides of a right triangle with respect to its acute angles.
Trigonometric identities are some trigonometric ratios for some specific angles and some identities involving these ratios.

4. Origin of Trigonometry
The word trigonometry comes from the Greek words trigonon(“triangle”) and metron (“to measure”). Until about the 16th century, trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle (or any shape that can be dissected into triangles) when the values of other parts were given. For example, if the lengths of two sides of a triangle and the measure of the enclosed angle are known, the third side and the two remaining angles can be calculated. Such calculations distinguish trigonometry from geometry, which mainly investigates qualitative relations. Of course, this distinction is not always absolute: the Pythagorean theorem, for example, is a statement about the lengths of the three sides in a right triangle and is thus quantitative in nature. Still, in its original form, trigonometry was by and large an offspring of geometry; it was not until the 16th century that the two became separate branches of mathematics.

5. Trigonometry in the modern sense began with the Greeks. Hipparchus(c
Trigonometry in the modern sense began with the Greeks. Hipparchus(c. 190–120 bc) was the first to construct a table of values for a trigonometric function. He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord. To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends it—or, equivalently, the length of a chord as a function of the corresponding arc width. This became the chief task of trigonometry for the next several centuries. As an astronomer, Hipparchus was mainly interested in spherical triangles, such as the imaginary triangle formed by three stars on the celestial sphere, but he was also familiar with the basic formulas of plane trigonometry. In Hipparchus’s time these formulas were expressed in purely geometric terms as relations between the various chords and the angles (or arcs) that subtend them; the modern symbols for the trigonometric functions were not introduced until the 17th century.

6. Example
Suppose the students of a school are visiting Eiffel tower . Now, if a student is looking at the top of the tower, a right triangle can be imagined to be made as shown in figure.
Can the student find out the height of the tower, without actually measuring it?
Yes the student can find the height of the tower with the help of trigonometry

7. Line of Sight Line of Sight Horizontal
We observe generally that children usually look up to see an aero plane when it passes overhead. This line joining their eye to the plane, while looking up is called Line of sight
Line of Sight
Horizontal
SHIVANI SLAUJA

8. Angle of Elevation Line of Sight Horizontal Angel of Elevation
The angle which the line of sight makes with a horizontal line drawn away from their eyes is called the angle of Elevation of aero plane from them.
Line of Sight
Angel of Elevation
Horizontal
SHIVANI SLAUJA

9. Angle of Depression Horizontal Angel of Depression Line of Sight
The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel).
Horizontal
Line of Sight
Angel of Depression

10. Trigonometric Ratios

11. ANGLE OF ELEVATION AND ANGLE OF TWO ANGLE OF DEPRESSION
TYPES
ANGLE OF ELEVATION
ANGLE OF DEPRESSION
ANGLE OF ELEVATION AND ANGLE OF
DEPRESSION
TWO ANGLE OF DEPRESSION
TWO ANGLE OF ELEVATION

12. Student’s work

13. Angle of Elevation
A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree.
Solution:
Let the height of the tree be h. Sketch a diagram to represent the situation.
tan 28˚ = 
h – 2 = 30 tan 28˚
h = (30 ´ ) + 2 ← tan 28˚ = =
The height of the tree is approximately 17.95 m.
Tarunikka singh

14. Angle of Depression
The angle of depression of a vehicle on the ground from the top of tower is 60. If the vehicle is at distance of 100 meters away from the building, find the height of the tower.
Solution:
PQ is the height of the tower and RQ is the distance between the tower and the vehicle whereas PS is the line of sight.
Angle of depression, ∠∠ SPR 60 degree.
tanθ= Opposite side\Adjacent side Tan 60° = PQ\RQ 
h = 100 * tan 60°
h =
The Height of the tower from ground is meter.
Amisha aggarwal

15. Two Angle of Elevation
Soumya narula

16. Two Angle of Depression
Question: From the top of a tower
100 m high,a man observes two cars on
the opposite sides of the tower with
angles of depression 30 (degree) and
45 (degree)respectively. Find the
distance between the cars.
Aditi jauhari

17. Ques- A straight highway leads to the foot of a tower
Ques-  A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
Answer Let AB be the tower. D is the initial and C is the final position of the car respectively. Angles of depression are measured from A. BC is the distance from the foot of the tower to the car. A/q, In  right ΔABC, tan 60° = AB/BC ⇒ √3 = AB/BC ⇒ BC = AB/√3 m also, In  right ΔABD, tan 30° = AB/BD ⇒ 1/√3 = AB/(BC + CD) ⇒ AB√3 = BC + CD ⇒ AB√3 = AB/√3 + CD ⇒ CD = AB√3 - AB/√3 ⇒ CD = AB(√3 - 1/√3) ⇒ CD = 2AB/√3 Here, distance of BC is half of CD. Thus, the time taken is also half. Time taken by car to travel distance CD = 6 sec. Time taken by car to travel BC = 6/2 = 3 sec.

18. TEACHER-SARIKA SATIJA SHIVANI SALUJA
GROUP MEMBERS-
TEACHER-SARIKA SATIJA
SHIVANI SALUJA
ADITI JAUHARI SOUMYA NARULA AMISHA AGGARWAL TARUNIKKA SINGH

19. The End