1.
New Era Progressive School
Session 2015-16
PowerPoint Presentation
Mathematics
Submitted By :
Priyanka Sahu
Submitted To :
Mrs. Mona Thakur

2.
Trigonometry…

3.
Contents …
 Introduction To Trigonometry
 Trigonometric Ratios
•Trigonometric Ratios of Some Specific Angles
• Trigonometric Ratios of Complementary Angles
Trigonometric Identities

4.
Introduction To Trigonometry
• Trigonometry is a branch of mathematics
which deals with the measurement of the
sides and the angles of a triangle and the
problems allied with angles.
• The word ‘Trigonometry’ is derived from the
Greek words TRIGONON (means triangle) and
METRON (means measure).

5.
Trigonometric Ratios
• Let us consider a right triangle ABC :
• Here angle A is an acute angle. Note the position of the side
BC with respect to angle A. We will call it the perpendicular to
angle A. AC is the hypotenuse of the right angled ∆ and the side
AB is the part of angle A. So, we will call it as the base to angle
A.
• The trigonometric ratios of angle A in right angle ABC are
defined as follows :
• Sine of angle A = Perpendicular = BC
Hypotenuse AC
• Cosine of angle A = Base = AB
Hypotenuse AC
• Tangent of angle A = Perpendicular = BC
Base AB

6.
• Cosecant of angle A = Hypotenuse = AC
Perpendicular BC
• Secant of angle A = Hypotenuse = AC
Base AB
• Cotangent of angle A = Base ____ =AB
Perpendicular AC
• The ratios defined above are abbreviated as sin A, cos A, tan A
,cosec A, sec A, and cot A respectively. Note that the ratios cosec A ,
sec A and cot A are respectively, the reciprocals of the ratios sin A,
cos A, and tan A.
• Also, observe that tan A= BC/AC = sin A and
AB/AC cos A
cot A = AB/AC = cos A
BC/AC sin A
So, the trigonometric ratios of an acute angle in a right –angled
triangle express the relationship between the angle and the length of
its sides.

7.
Example
Q. In a triangle ABC, right angled at a , if AB=12cm, AC=5cm and
BC=13cm ,find all the six trigonometric ratios of angle B .
Sol. With reference to angle B ,we have
Base=AB=12,Perpendicular=AC=5,Hypotenuse=BC=5cm
Using the definitions of trigonometric ratios, we have
sin B = AC/BC = 5/13
cos B = AB/BC = 12/13
tan B = AC/AB =5/12
cosec B = BC/AC = 13/5
sec B = BC/AB = 13/12
cot B = AB/AC =12/5

8.
Trigonometric Ratios of Some Specific Angles

9.
Example…
Q. If x=30°,verify that cos3x=4cos³x-3cosx
Sol. We have,
x = 30° ⇒ 3x = 90°
Cos3x = cos90°=0
And, 4cos³x - 3cosx = 4cos³30°- 3cos30°
⇒ 4cos³x – 3cosx = 4 (√3/2)³ - 3(√3/2)
=4 X 3√3/8 - 3√3/2
=3√3/2 - 3√3/2
=0

10.
Trigonometric ratios of Complementary Angles
• Complementary angle : Two angles are said to be
complementary, if their sum is 90°.
• It follows from the above definition that θ and (90° - θ ) are
complementary angles for an acute angle θ.
• So, if θ is an acute angle, then
sin (90° – θ) = cos θ, cos (90° – θ) = sin θ,
tan (90° – θ) = cot θ, cot (90° – θ) = tan θ,
sec (90° – θ) = cosec θ, cosec (90° – θ) = sec θ,

11.
Example
Q. Prove that : tan10°tan15°tan75°tan80° = 1
We have,
LHS =tan10°tan15°tan75°tan80°
= tan(90° - 80°) tan (90° - 75°) tan75°tan80°
= cot80° cot75° tan75° tan80°
= (cot80° tan80°) (cot75° tan75°)
= 1 x 1 =1
Sol.

12.
Trigonometric Identities
• An equation involving trigonometric ratios of an
angle θ (say) is said to be a trigonometric identity if it
is satisfied for all values of θ for which the given
trigonometric ratios are defined.
• We can obtain some more identities by using some
fundamental trigonometric identities.

13.
1) cos2 θ + sin2 θ = 1
2) cos2 θ =1 - sin2 θ
3) sin2 θ =1 - cos2 θ
4) 1 + tan2 θ = sec2 θ
5) sec2 θ - tan2 θ = 1
6) tan2 θ = sec2 θ – 1
7) cot2 θ + 1 = cosec2 θ
8) cosec2 θ - cot2 θ = 1
9) cot2 θ = cosec2 θ – 1
Some Fundamental Trigonometric Identities

14.
Example
Q.
Sol.
Q.
Sol.

15.
Thankyou…