Download Trigonometry

What is Trigonometry?
Trigonometry is a branch
of mathematics that studies triangles and the
relationships between their sides and the
angles between these sides.
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.
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.
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
Line of Sight
Horizontal
SHIVANI SLAUJA
Angle 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.
Angel of Elevation
SHIVANI SLAUJA
Angle of Depression
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
Angel of Depression
Trigonometric Ratios
TYPES
Student’S work
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 ´ 0.5317) + 2 ← tan 28˚ = 0.5317
= 17.951
The height of the tree is approximately 17.95 m.
Tarunikka singh
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 = 173.20
The Height of the tower from ground is 173.20 meter.
Amisha aggarwal
Two Angle of Elevation
Soumya narula
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
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.