TRIGONOMETRY HANDWRITTEN NOTES FOR NDA 2022 help4nda

TRIGONOMETRY :

HISTORY OF TRIGONOMETRY :

Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). These six trigonometric functions in relation to a right triangle are displayed in the figure.

For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.

Trigonometry developed from a need to compute angles and distances in such fields as astronomy, mapmaking, surveying, and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry. Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry.

CHAPTER NAME	TRIGONOMETRY
SUBJECT	MATHEMATICS
CLASS	11^th
NOTES LANGUAGE	ENGLISH
NO. OF PAGES	14

SYALLBUS FOR TRIGONOMETRY :

Angles and their measures in degrees and in radians.
Trigonometric ratios.
Trigonometric identities Sum and difference formulae.
Multiple and Sub-multiple angles.
Inverse trigonometric functions.
Applications-Height and distance, properties of triangles.

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NOTES PDF DOWNLOAD :

QUESTION FOR PRACTICE :

Q. No.1: In any triangle ABC, prove that a sin (B – C) + b sin (C – A) + c sin (A – B) = 0.

Solution:

In any triangle ABC,

a/sin A = b/sin B = c/sin C = k

a = k sin A, b = k sin B, c = k sin C

LHS

= a sin (B – C) + b sin (C – A) + c sin (A – B)

= k sin A [sin B cos C – cos B sin C] + k sin B [sin C cos A – cos C sin A] + k sin C [sin A cos B – cos A sin B]

= k sin A sin B cos C – k sin A cos B sin C + k sin B sin C cos A – k sin B cos C sin A + k sin C sin A cos B – k sin C cos A sin B

= 0

= RHS

Hence proved that a sin (B – C) + b sin (C – A) + c sin (A – B) = 0.

Q.No.2: Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm (use π = 22/7).

Solution:

Given,

Length of the arc = l = 37.4 cm

Central angle = θ = 60° = 60π/180 radian = π/3 radians

We know that,

r = l/θ

= (37.4) * (π / 3)

= (37.4) / [22 / 7 * 3]

= 35.7 cm

Hence, the radius of the circle is 35.7 cm.

Q. No.3: A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Solution:

Given,

Number of revolutions made by the wheel in 1 minute = 360

1 minute = 60 seconds

Number of revolutions in 1 second = 360/60 = 6

Angle made in 1 revolution = 360°

Angles made in 6 revolutions = 6 × 360°

Radian measure of the angle in 6 revolutions = 6 × 360 × π/180

= 6 × 2 × π

= 12π

Hence, the wheel turns 12π radians in one second.

Q. No. 4: Find the value of cos 570° sin 510° + sin (-330°) cos (-390°).

Solution:

LHS =cos (570)sin (510) + sin (- 330)cos (- 390)

= cos (570) sin (510) + [ – sin (330) ]cos (390) [ because sin( – x ) = – sin x and cos( – x ) = cos x ]

= cos (570)sin(510) – sin (330)

= cos (90 * 6 + 30) sin (90 * 5 + 60) – sin (90 * 3 + 60) cos (90 * 4 + 30)

= – cos (30) cos (60) – [ – cos (60) ] cos (30)

= – cos (30) cos (60) + cos (30) sin (60)

= 0

Q. No. 5: The value of cos 1° cos 2° cos 3° … cos 179° is
(a) 1/√2
(b) 0
(c) 1
(d) -1

Sol: (b)
Since cos 90° = 0, we have
cos 1° cos 2° cos 3° …cos 90°… cos 179° = 0

6.Derive the values of the angle

a. sin18∘sin18∘

b. tan15∘tan15∘

c. sin6712∘sin6712∘

7.Prove that

cos55∘+cos65∘+cos175∘=0cos55∘+cos65∘+cos175∘=0

8. If y=cos2x+sin4xy=cos2x+sin4x for all values of x, then prove that

34≤y≤134≤y≤1

9.Find the solution of the trigonometric equation 2tan2x+sec2x=22tan2x+sec2x=2 in the range [0,2π][0,2π]

11.Which of the following is not correct?
(A) sin A = –1/5
(B) cos A = 1
(C) sec A =1/2
(D) tan A = 20

10.Show that 2sin2y+4cos(x+y)sinxsiny+cos2(x+y)=cos2x

NDA MATHEMATICS NOTES :

#COMPLEX NO.

#SET THEORY

#RELATION

#PERMUTATION AND COMBINATION

#LOGARITHM