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 Page 1


.
 
Trigonometrical Ratios & Identities
1. Basic Trigonometric Identities:
(A) sin²
?
? + cos²
?
?  = 1;  ?1 ? sin ? ? 1;  ?1 ? cos ? ? 1  ? ? ? ? R
(B) sec²
?
? ? tan²
?
?  = 1  ;  ?sec
 
? ? ? 1   ??  ? ? R – ? ?
?
?
?
?
?
?
? ?
?
? n ,
2
1 n 2
(C) cosec²
?
? ? cot²
?
? = 1  ;  ?cosec
 
? ? ? 1  ?  ? ? R – ? ? ? ? ? n , n
2. Circular Definition Of Trigonometric Functions:
sin ? =
OP
P M
cos ? = 
OP
OM
tan ? =
?
?
cos
s i n
, cos ? ? ? 0
cot ? =
?
?
s i n
cos
, sin ? ? ? 0
sec ? =
? cos
1
, cos ? ? ? 0cosec ? =
? s i n
1
, sin ? ? ? 0
3. Trigonometric Functions Of Allied Angles:
If ? is any angle, then ? ? ? ? 90 ± ?, 180 ± ?, 270 ± ?, 360 ± ? etc. are called ALLIED ANGLES.
(A) sin ( ? ?) = ? sin
 
? ; cos ( ? ?) = cos
 
?
(B)  sin (90° ? ?) = cos
 
? ;cos (90°
 
? ?) = sin
 
?
(C)  sin (90° + ?) = cos
 
? ;cos (90° + ?) = ? sin
 
?
(D)  sin (180° ? ?) = sin
 
? ;cos (180° ? ?) = ? cos
 
?
(e)  sin (180° + ?) = ? sin
 
? ;cos (180° + ?) = ? cos
 
?
Page 2


.
 
Trigonometrical Ratios & Identities
1. Basic Trigonometric Identities:
(A) sin²
?
? + cos²
?
?  = 1;  ?1 ? sin ? ? 1;  ?1 ? cos ? ? 1  ? ? ? ? R
(B) sec²
?
? ? tan²
?
?  = 1  ;  ?sec
 
? ? ? 1   ??  ? ? R – ? ?
?
?
?
?
?
?
? ?
?
? n ,
2
1 n 2
(C) cosec²
?
? ? cot²
?
? = 1  ;  ?cosec
 
? ? ? 1  ?  ? ? R – ? ? ? ? ? n , n
2. Circular Definition Of Trigonometric Functions:
sin ? =
OP
P M
cos ? = 
OP
OM
tan ? =
?
?
cos
s i n
, cos ? ? ? 0
cot ? =
?
?
s i n
cos
, sin ? ? ? 0
sec ? =
? cos
1
, cos ? ? ? 0cosec ? =
? s i n
1
, sin ? ? ? 0
3. Trigonometric Functions Of Allied Angles:
If ? is any angle, then ? ? ? ? 90 ± ?, 180 ± ?, 270 ± ?, 360 ± ? etc. are called ALLIED ANGLES.
(A) sin ( ? ?) = ? sin
 
? ; cos ( ? ?) = cos
 
?
(B)  sin (90° ? ?) = cos
 
? ;cos (90°
 
? ?) = sin
 
?
(C)  sin (90° + ?) = cos
 
? ;cos (90° + ?) = ? sin
 
?
(D)  sin (180° ? ?) = sin
 
? ;cos (180° ? ?) = ? cos
 
?
(e)  sin (180° + ?) = ? sin
 
? ;cos (180° + ?) = ? cos
 
?
(f)  sin (270° ? ?) = ? cos
 
? ;cos (270° ? ?) = ? sin
 
?
(g)  sin (270° + ?) = ? cos
 
? ;cos (270° + ?) = sin
 
?
(h)  tan (90° ? ? ?) = cot
 
? ;cot (90° ? ? ?) = tan
 
?
4. Graphs of Trigonometric functions:
(A)  y = sin x x ? R;  y ? [–1, 1]
(B)  y = cos x x ? R;  y ? [ – 1, 1]
(C)  y = tan x   x ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? R
(D)  y = cot x  x ? R – n ? , n ?? ?;  y ? R
(e)  y = cosec xx ? R  – n ? , n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
Page 3


.
 
Trigonometrical Ratios & Identities
1. Basic Trigonometric Identities:
(A) sin²
?
? + cos²
?
?  = 1;  ?1 ? sin ? ? 1;  ?1 ? cos ? ? 1  ? ? ? ? R
(B) sec²
?
? ? tan²
?
?  = 1  ;  ?sec
 
? ? ? 1   ??  ? ? R – ? ?
?
?
?
?
?
?
? ?
?
? n ,
2
1 n 2
(C) cosec²
?
? ? cot²
?
? = 1  ;  ?cosec
 
? ? ? 1  ?  ? ? R – ? ? ? ? ? n , n
2. Circular Definition Of Trigonometric Functions:
sin ? =
OP
P M
cos ? = 
OP
OM
tan ? =
?
?
cos
s i n
, cos ? ? ? 0
cot ? =
?
?
s i n
cos
, sin ? ? ? 0
sec ? =
? cos
1
, cos ? ? ? 0cosec ? =
? s i n
1
, sin ? ? ? 0
3. Trigonometric Functions Of Allied Angles:
If ? is any angle, then ? ? ? ? 90 ± ?, 180 ± ?, 270 ± ?, 360 ± ? etc. are called ALLIED ANGLES.
(A) sin ( ? ?) = ? sin
 
? ; cos ( ? ?) = cos
 
?
(B)  sin (90° ? ?) = cos
 
? ;cos (90°
 
? ?) = sin
 
?
(C)  sin (90° + ?) = cos
 
? ;cos (90° + ?) = ? sin
 
?
(D)  sin (180° ? ?) = sin
 
? ;cos (180° ? ?) = ? cos
 
?
(e)  sin (180° + ?) = ? sin
 
? ;cos (180° + ?) = ? cos
 
?
(f)  sin (270° ? ?) = ? cos
 
? ;cos (270° ? ?) = ? sin
 
?
(g)  sin (270° + ?) = ? cos
 
? ;cos (270° + ?) = sin
 
?
(h)  tan (90° ? ? ?) = cot
 
? ;cot (90° ? ? ?) = tan
 
?
4. Graphs of Trigonometric functions:
(A)  y = sin x x ? R;  y ? [–1, 1]
(B)  y = cos x x ? R;  y ? [ – 1, 1]
(C)  y = tan x   x ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? R
(D)  y = cot x  x ? R – n ? , n ?? ?;  y ? R
(e)  y = cosec xx ? R  – n ? , n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
(f)  y = sec x x ? ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
5. Trigonometric Functions of Sum or Difference of Two Angles:
(A) sin (A ± B) = sinA cosB ± cosA sinB
(B) cos (A ± B) = cosA cosB ? sinA sinB
(C) sin²A ? sin²B = cos²B ? cos²A = sin (A+B). sin (A ? B)
(D) cos²A ? sin²B = cos²B ? sin²A = cos (A+B). cos (A
 
? B)
(e) tan
 
(A ± B) = 
B tan A tan 1
B tan A tan
?
?
(f) cot (A ± B) =
A cot B cot
1 B cot A cot
?
?
(g) tan (A + B + C)
=
A tan C tan C tan B tan B tan A tan 1
C tan B tan A tan C tan B tan A tan
? ? ?
? ? ?
.
6. Factorisation of the Sum or Difference of Two Sines or Cosines:
(A) sinC + sinD = 2 sin
2
D C ?
 cos
2
D C ?
(B) sinC ? sinD = 2 cos
2
D C ?
 sin
2
D C ?
(C) cosC + cosD = 2 cos
2
D C ?
 cos
2
D C ?
(D) cosC
 
?
 
cosD = ?
 
2 sin
2
D C ?
 
sin
2
D C ?
7. Transformation of Products into Sum or Difference of Sines & Cosines:
(A) 2 sinA cosB = sin(A+B) + sin(A ?B)
(B) 2 cosA sinB = sin(A+B) ? sin(A ?B)
(C) 2 cosA cosB = cos(A+B) + cos(A ?B)
(D) 2 sinA sinB = cos(A ?B) ? cos(A+B)
8. Multiple and Sub-multiple Angles :
(A) sin 2A = 2 sinA cosA ;  sin
 
? = 2 sin
?
2
 cos
?
2
(B) cos 2A = cos²A ? sin²A = 2cos²A
 
?
 
1 = 1 ? 2 sin²A;
Page 4


.
 
Trigonometrical Ratios & Identities
1. Basic Trigonometric Identities:
(A) sin²
?
? + cos²
?
?  = 1;  ?1 ? sin ? ? 1;  ?1 ? cos ? ? 1  ? ? ? ? R
(B) sec²
?
? ? tan²
?
?  = 1  ;  ?sec
 
? ? ? 1   ??  ? ? R – ? ?
?
?
?
?
?
?
? ?
?
? n ,
2
1 n 2
(C) cosec²
?
? ? cot²
?
? = 1  ;  ?cosec
 
? ? ? 1  ?  ? ? R – ? ? ? ? ? n , n
2. Circular Definition Of Trigonometric Functions:
sin ? =
OP
P M
cos ? = 
OP
OM
tan ? =
?
?
cos
s i n
, cos ? ? ? 0
cot ? =
?
?
s i n
cos
, sin ? ? ? 0
sec ? =
? cos
1
, cos ? ? ? 0cosec ? =
? s i n
1
, sin ? ? ? 0
3. Trigonometric Functions Of Allied Angles:
If ? is any angle, then ? ? ? ? 90 ± ?, 180 ± ?, 270 ± ?, 360 ± ? etc. are called ALLIED ANGLES.
(A) sin ( ? ?) = ? sin
 
? ; cos ( ? ?) = cos
 
?
(B)  sin (90° ? ?) = cos
 
? ;cos (90°
 
? ?) = sin
 
?
(C)  sin (90° + ?) = cos
 
? ;cos (90° + ?) = ? sin
 
?
(D)  sin (180° ? ?) = sin
 
? ;cos (180° ? ?) = ? cos
 
?
(e)  sin (180° + ?) = ? sin
 
? ;cos (180° + ?) = ? cos
 
?
(f)  sin (270° ? ?) = ? cos
 
? ;cos (270° ? ?) = ? sin
 
?
(g)  sin (270° + ?) = ? cos
 
? ;cos (270° + ?) = sin
 
?
(h)  tan (90° ? ? ?) = cot
 
? ;cot (90° ? ? ?) = tan
 
?
4. Graphs of Trigonometric functions:
(A)  y = sin x x ? R;  y ? [–1, 1]
(B)  y = cos x x ? R;  y ? [ – 1, 1]
(C)  y = tan x   x ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? R
(D)  y = cot x  x ? R – n ? , n ?? ?;  y ? R
(e)  y = cosec xx ? R  – n ? , n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
(f)  y = sec x x ? ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
5. Trigonometric Functions of Sum or Difference of Two Angles:
(A) sin (A ± B) = sinA cosB ± cosA sinB
(B) cos (A ± B) = cosA cosB ? sinA sinB
(C) sin²A ? sin²B = cos²B ? cos²A = sin (A+B). sin (A ? B)
(D) cos²A ? sin²B = cos²B ? sin²A = cos (A+B). cos (A
 
? B)
(e) tan
 
(A ± B) = 
B tan A tan 1
B tan A tan
?
?
(f) cot (A ± B) =
A cot B cot
1 B cot A cot
?
?
(g) tan (A + B + C)
=
A tan C tan C tan B tan B tan A tan 1
C tan B tan A tan C tan B tan A tan
? ? ?
? ? ?
.
6. Factorisation of the Sum or Difference of Two Sines or Cosines:
(A) sinC + sinD = 2 sin
2
D C ?
 cos
2
D C ?
(B) sinC ? sinD = 2 cos
2
D C ?
 sin
2
D C ?
(C) cosC + cosD = 2 cos
2
D C ?
 cos
2
D C ?
(D) cosC
 
?
 
cosD = ?
 
2 sin
2
D C ?
 
sin
2
D C ?
7. Transformation of Products into Sum or Difference of Sines & Cosines:
(A) 2 sinA cosB = sin(A+B) + sin(A ?B)
(B) 2 cosA sinB = sin(A+B) ? sin(A ?B)
(C) 2 cosA cosB = cos(A+B) + cos(A ?B)
(D) 2 sinA sinB = cos(A ?B) ? cos(A+B)
8. Multiple and Sub-multiple Angles :
(A) sin 2A = 2 sinA cosA ;  sin
 
? = 2 sin
?
2
 cos
?
2
(B) cos 2A = cos²A ? sin²A = 2cos²A
 
?
 
1 = 1 ? 2 sin²A;
2 cos²
2
?
 = 1 + cos
 
?, 2 sin²
2
?
 = 1 ? cos
 
?.
(C) tan 2A =
A tan 1
A tan 2
2
?
; tan
 
? =
2
2
2
t an 1
t an 2
?
?
?
(D) sin 2A =
A tan 1
A tan 2
2
?
, cos 2A =
A tan 1
A tan 1
2
2
?
?
(e) sin 3A = 3 sinA
 
? 4 sin
3
A
(f) cos 3A = 4 cos
3
A ? 3 cosA
(g) tan 3A = 
A tan 3 1
A tan A tan 3
2
3
?
?
9. Important Trigonometric Ratios:
(A) sin n
 
? = 0 ; cos n
 
? = ( ?1)
n
;  tan n
 
? = 0,  where n ? ?
(B) sin 15° or  sin
12
?
  = 
2 2
1 3 ?
 = cos 75° or  cos 
12
5 ?
;
 cos 15° or  cos
12
?
  = 
2 2
1 3 ?
 = sin 75°  or  sin 
12
5 ?
;
tan
 
15° =
1 3
1 3
?
?
 = 3 2 ? = cot 75° ; tan
 
75° =
1 3
1 3
?
?
 = 3 2 ? = cot 15°
(C) sin
10
?
 or sin 18° =
4
1 5 ?
  &  cos 36° or cos
5
?
 = 
4
1 5 ?
10. Conditional Identities:
If  A + B + C = ?  then :
(i) sin2A + sin2B + sin2C = 4 sinA sinB sinC
(ii) sinA + sinB + sinC = 4 cos
2
A
 cos
2
B
 cos
2
C
(iii) cos 2
 
A + cos 2
 
B + cos 2
 
C = ? 1 ? 4 cos A cos B cos C
(iv) cos A + cos B + cos C = 1 + 4 sin
2
A
 sin
2
B
 sin
2
C
(v) tanA + tanB + tanC = tanA tanB tanC
(vi) tan
2
A
 tan
2
B
 + tan
2
B
 tan
2
C
 + tan
2
C
 tan
2
A
 = 1
(vii) cot
2
A
 + cot
2
B
 + cot
2
C
 = cot
2
A
. cot
2
B
. cot
2
C
(viii) cot A cot B + cot B cot C + cot C cot A = 1
(ix) A + B + C =
?
2
  then  tan A tan B + tan B tan C + tan C tan A = 1
Page 5


.
 
Trigonometrical Ratios & Identities
1. Basic Trigonometric Identities:
(A) sin²
?
? + cos²
?
?  = 1;  ?1 ? sin ? ? 1;  ?1 ? cos ? ? 1  ? ? ? ? R
(B) sec²
?
? ? tan²
?
?  = 1  ;  ?sec
 
? ? ? 1   ??  ? ? R – ? ?
?
?
?
?
?
?
? ?
?
? n ,
2
1 n 2
(C) cosec²
?
? ? cot²
?
? = 1  ;  ?cosec
 
? ? ? 1  ?  ? ? R – ? ? ? ? ? n , n
2. Circular Definition Of Trigonometric Functions:
sin ? =
OP
P M
cos ? = 
OP
OM
tan ? =
?
?
cos
s i n
, cos ? ? ? 0
cot ? =
?
?
s i n
cos
, sin ? ? ? 0
sec ? =
? cos
1
, cos ? ? ? 0cosec ? =
? s i n
1
, sin ? ? ? 0
3. Trigonometric Functions Of Allied Angles:
If ? is any angle, then ? ? ? ? 90 ± ?, 180 ± ?, 270 ± ?, 360 ± ? etc. are called ALLIED ANGLES.
(A) sin ( ? ?) = ? sin
 
? ; cos ( ? ?) = cos
 
?
(B)  sin (90° ? ?) = cos
 
? ;cos (90°
 
? ?) = sin
 
?
(C)  sin (90° + ?) = cos
 
? ;cos (90° + ?) = ? sin
 
?
(D)  sin (180° ? ?) = sin
 
? ;cos (180° ? ?) = ? cos
 
?
(e)  sin (180° + ?) = ? sin
 
? ;cos (180° + ?) = ? cos
 
?
(f)  sin (270° ? ?) = ? cos
 
? ;cos (270° ? ?) = ? sin
 
?
(g)  sin (270° + ?) = ? cos
 
? ;cos (270° + ?) = sin
 
?
(h)  tan (90° ? ? ?) = cot
 
? ;cot (90° ? ? ?) = tan
 
?
4. Graphs of Trigonometric functions:
(A)  y = sin x x ? R;  y ? [–1, 1]
(B)  y = cos x x ? R;  y ? [ – 1, 1]
(C)  y = tan x   x ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? R
(D)  y = cot x  x ? R – n ? , n ?? ?;  y ? R
(e)  y = cosec xx ? R  – n ? , n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
(f)  y = sec x x ? ? R – (2n + 1) ? ?/2, n ?? ? ;  y ? ( ? ? ?, ? 1] ? [1, ?)
5. Trigonometric Functions of Sum or Difference of Two Angles:
(A) sin (A ± B) = sinA cosB ± cosA sinB
(B) cos (A ± B) = cosA cosB ? sinA sinB
(C) sin²A ? sin²B = cos²B ? cos²A = sin (A+B). sin (A ? B)
(D) cos²A ? sin²B = cos²B ? sin²A = cos (A+B). cos (A
 
? B)
(e) tan
 
(A ± B) = 
B tan A tan 1
B tan A tan
?
?
(f) cot (A ± B) =
A cot B cot
1 B cot A cot
?
?
(g) tan (A + B + C)
=
A tan C tan C tan B tan B tan A tan 1
C tan B tan A tan C tan B tan A tan
? ? ?
? ? ?
.
6. Factorisation of the Sum or Difference of Two Sines or Cosines:
(A) sinC + sinD = 2 sin
2
D C ?
 cos
2
D C ?
(B) sinC ? sinD = 2 cos
2
D C ?
 sin
2
D C ?
(C) cosC + cosD = 2 cos
2
D C ?
 cos
2
D C ?
(D) cosC
 
?
 
cosD = ?
 
2 sin
2
D C ?
 
sin
2
D C ?
7. Transformation of Products into Sum or Difference of Sines & Cosines:
(A) 2 sinA cosB = sin(A+B) + sin(A ?B)
(B) 2 cosA sinB = sin(A+B) ? sin(A ?B)
(C) 2 cosA cosB = cos(A+B) + cos(A ?B)
(D) 2 sinA sinB = cos(A ?B) ? cos(A+B)
8. Multiple and Sub-multiple Angles :
(A) sin 2A = 2 sinA cosA ;  sin
 
? = 2 sin
?
2
 cos
?
2
(B) cos 2A = cos²A ? sin²A = 2cos²A
 
?
 
1 = 1 ? 2 sin²A;
2 cos²
2
?
 = 1 + cos
 
?, 2 sin²
2
?
 = 1 ? cos
 
?.
(C) tan 2A =
A tan 1
A tan 2
2
?
; tan
 
? =
2
2
2
t an 1
t an 2
?
?
?
(D) sin 2A =
A tan 1
A tan 2
2
?
, cos 2A =
A tan 1
A tan 1
2
2
?
?
(e) sin 3A = 3 sinA
 
? 4 sin
3
A
(f) cos 3A = 4 cos
3
A ? 3 cosA
(g) tan 3A = 
A tan 3 1
A tan A tan 3
2
3
?
?
9. Important Trigonometric Ratios:
(A) sin n
 
? = 0 ; cos n
 
? = ( ?1)
n
;  tan n
 
? = 0,  where n ? ?
(B) sin 15° or  sin
12
?
  = 
2 2
1 3 ?
 = cos 75° or  cos 
12
5 ?
;
 cos 15° or  cos
12
?
  = 
2 2
1 3 ?
 = sin 75°  or  sin 
12
5 ?
;
tan
 
15° =
1 3
1 3
?
?
 = 3 2 ? = cot 75° ; tan
 
75° =
1 3
1 3
?
?
 = 3 2 ? = cot 15°
(C) sin
10
?
 or sin 18° =
4
1 5 ?
  &  cos 36° or cos
5
?
 = 
4
1 5 ?
10. Conditional Identities:
If  A + B + C = ?  then :
(i) sin2A + sin2B + sin2C = 4 sinA sinB sinC
(ii) sinA + sinB + sinC = 4 cos
2
A
 cos
2
B
 cos
2
C
(iii) cos 2
 
A + cos 2
 
B + cos 2
 
C = ? 1 ? 4 cos A cos B cos C
(iv) cos A + cos B + cos C = 1 + 4 sin
2
A
 sin
2
B
 sin
2
C
(v) tanA + tanB + tanC = tanA tanB tanC
(vi) tan
2
A
 tan
2
B
 + tan
2
B
 tan
2
C
 + tan
2
C
 tan
2
A
 = 1
(vii) cot
2
A
 + cot
2
B
 + cot
2
C
 = cot
2
A
. cot
2
B
. cot
2
C
(viii) cot A cot B + cot B cot C + cot C cot A = 1
(ix) A + B + C =
?
2
  then  tan A tan B + tan B tan C + tan C tan A = 1
 11. Range of Trigonometric Expression:
E = a sin
 
? + b cos
 
?
E =
2 2
b a ? sin ( ? + ?), where tan ? =
a
b
  =
2 2
b a ? cos (? ?? ??), where tan ? =
b
a
Hence for any real value of ? ?, 
2 2 2 2
ba E ba ? ? ? ? ?
12. Sine and Cosine Series:
sin
 
? + sin
 
( ??+ ? ?) + sin
 
( ? + 2 ? ) +...... + sin ? ? ? ? ? ? 1 n =
2
2
n
s i n
s i n
?
?
 sin
 
?
?
?
?
?
?
?
?
? ?
2
1 n
cos
 
? + cos
 
( ??+ ? ?) + cos
 
( ? + 2 ? ) +...... + cos ? ? ? ? ? ? 1 n =
2
2
n
s i n
s i n
?
?
 cos
 
?
?
?
?
?
?
?
?
? ?
2
1 n
 1. DEFINITION
The equations involving trigonometric function of unknown angles are known as Trigonometric equations
e.g.  cos ? = 0 , cos
2
? – 4cos ? =1 , sin
2
? + sin ? = 2 , cos
2
 ? – 4sin ? =1
A solution of a trigonometric equation is the value of the unknown angle that satisfies the equation.
e.g.,   
1 3 9 11
sin o r , , , .. . .. ....
4 4 4 4 4 2
? ? ? ? ?
? ? ? ? ? ? ?
 2. PERIODIC FUNCTION
A function f(x) is said to be periodic if there exists T > 0 such that f(x + T) = f(x) for all x in the domain of
definitions of f(x). If T is the smallest positive real numbers such that f(x + T) = f(x) , then it is called the period of
f(x)
Since sin (2n ? + x ) = sinx , cos (2n ? + x) = cos x     ;     tan (n ? + x) = tan x  for all  n ? Z
Therefore sinx , cosx and tanx are perodic function, the period of sinx and cos x is 2 ? and that of tanx is ? .
Function Period
sin (ax + b) , cos (ax +b), sec (ax + b) , cosec (ax +b ) 2 ?/a
tan (ax + b), cot (ax +b) ?/a
| sin (ax + b)| , | cos (ax +b) | , | sec (ax +b) | , | cosec (ax +b ) | ?/a
| tan (ax + b ) | , | cot (ax +b ) | ?/2a
Trigonometrical Equations

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FAQs on Important Formulas: Trigonometric Ratios, Functions and Equations - Physics for JEE Main & Advanced

1. What are the trigonometric ratios?

Ans. Trigonometric ratios are ratios of the sides of a right triangle with respect to its angles. The main trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). They are defined as follows: - Sine (sin) of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. - Cosine (cos) of an angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. - Tangent (tan) of an angle is equal to the ratio of the length of the side opposite the angle to the length of the adjacent side.

2. How are trigonometric functions related to trigonometric ratios?

Ans. Trigonometric functions are mathematical functions that are defined based on the ratios of sides in a right triangle. The main trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are defined as follows: - Sine (sin) of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. - Cosine (cos) of an angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. - Tangent (tan) of an angle is equal to the ratio of the length of the side opposite the angle to the length of the adjacent side. - Cosecant (csc) of an angle is equal to the reciprocal of the sine of the angle. - Secant (sec) of an angle is equal to the reciprocal of the cosine of the angle. - Cotangent (cot) of an angle is equal to the reciprocal of the tangent of the angle.

3. How can trigonometric ratios be used to solve equations?

Ans. Trigonometric ratios can be used to solve equations involving trigonometric functions. By applying the trigonometric ratios and identities, we can manipulate the equations to find the values of the angles or sides involved. This is particularly useful in solving problems related to triangles, angles, and distances. The equations can be solved using algebraic techniques such as substitution, simplification, and factoring.

4. What are the important trigonometric identities to remember?

Ans. Some important trigonometric identities to remember are: - Pythagorean Identities: sin^2θ + cos^2θ = 1 and 1 + tan^2θ = sec^2θ. - Reciprocal Identities: cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ. - Quotient Identities: tanθ = sinθ/cosθ and cotθ = cosθ/sinθ. - Co-Function Identities: sin(π/2 - θ) = cosθ, cos(π/2 - θ) = sinθ, and tan(π/2 - θ) = cotθ. - Even-Odd Identities: sin(-θ) = -sinθ, cos(-θ) = cosθ, and tan(-θ) = -tanθ. - Double Angle Identities: sin(2θ) = 2sinθcosθ, cos(2θ) = cos^2θ - sin^2θ, and tan(2θ) = 2tanθ/1 - tan^2θ.

5. How can trigonometric ratios be applied in real-life scenarios?

Ans. Trigonometric ratios have various applications in real-life scenarios. Some examples include: - In navigation and surveying, trigonometric ratios are used to determine distances, angles, and heights of objects. - In physics and engineering, trigonometric ratios are used to analyze and calculate forces, velocities, and trajectories. - In architecture and construction, trigonometric ratios are used to design and build structures with specific angles and dimensions. - In astronomy, trigonometric ratios are used to calculate the positions and movements of celestial objects. - In sound and wave analysis, trigonometric ratios are used to study the properties of sound waves and analyze their frequencies and amplitudes.

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