Summary Formula sheet

TRIGONOMETRY
I. Measurement of an angle – radian Measure.
1. A radian is a measure of an angle subtended at the center of a circle by an arc of length equal to the radius
of the circle.
One radian is denoted by 1C and
1 Radian = 5716`22`; 1  0.01745 Radian
180
 Radian = 180 ; 1 radian =


2. Radian measure of some common angles
Degrees 15◦ 30◦ 45◦ 60◦ 75◦ 90◦ 105◦ 120◦ 135◦ 150◦ 165
◦ ◦ ◦
180 270 360
◦


Radian     5  7 2 3 5 11  3 2
12 6 4 3 12 2 12 3 4 6 12 2
II. Basic identities of trigonometric ratios
1. cos 2θ + sin 2θ = 1; cos 2θ = 1- sin 2θ; sin 2θ = 1- cos 2θ.
2. 1+ tan 2θ = sec 2θ; tan 2θ = sec2θ -1; sec 2θ - tan 2θ = 1.
3. 1+ cot 2θ = cosec 2θ; cot 2θ = cosec 2θ -1; cosec 2θ - cot 2θ = 1.
1  
4. a) secθ + tanθ   θ  n + 
secθ - tanθ  2
1
b) cosecθ + cotθ   θ  n 
cosecθ - cotθ
III. Domain and range of trigonometric ratios
Trigonometric ratios Domain Range
sinθ R 1  sin   1
cosθ R 1  cos   1
tanθ   
R  (2n  1) , n  z  R
 2 
cosecθ R   n , n  z  cos ec  1 or cosec  1
secθ   
R  (2n  1) , n  z  s ec  1 or sec  1
 2 
cotθ R   n , n  z  R


i.e., | sin  | 1, | cos | 1, | cosecθ | 1, | secθ | 1
IV. Sign of trigonometric ratios (Quadrant rule)
Quadrant I II III Iv
θ

Sin and cosec + + - -
Cos and sec + - - +
Tan and cot + - + -

V. Trigonometric ratios of some standard angles
θ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
0 15 30 45 60 75 90 120 135 150 180 360
◦
0
c
 c
c c
  5 c c 2 c 3 c 5 c c 2 c
12 6 4 3 12 2 3 4 6
Sin 3 1 1 1 3 3 1 3 1 1
θ 0
2 2
1
2 2 0 0
2 2 2 2 2 2

, cos θ 3 1 3 1 1 3 1 1 1  3
1 0 -1 1
2 2 2 3 2 2 2 2 2 2
tan 3 1 1 3 1 1
θ 0
3
1 3 -  3 -1
3
0 0
3 1 3 1

Note : sin(multiple of  ) = 0, tan (multiple of  )= 0 cos (odd multiple of ) = 0,
12
Cos (odd multiple of  ) =-1 cos(even multiple of  ) = 1.

i.e., sin n  = 0, tan n  = 0, cos (2n – 1) = 0, cos (2n – 1)  = -1 cos (2n  ) = 1.
12
VI. Trigonometric ratios of allied angles
Measure of angle Sin Cos Tan Cosec Sec cot
◦ -θ -sin θ cos θ -tan θ -cosec θ sec θ -cot θ
-θ
◦ ◦  cos θ sin θ cot θ sec θ cosec θ tan θ
90 - θ 
2
◦ ◦  cos θ
90 + θ  -sin θ -cot θ sec θ -cosec θ -tan θ
2
◦ ◦   sin θ -cos θ -tan θ cosec θ -sec θ -cot θ
180 - θ
◦ ◦   - sin θ -cos θ tan θ -cosec θ -sec θ cot θ
180 + θ
◦ ◦ 3 - cos θ -sin θ cot θ -sec θ -cosec θ tan θ
270 - θ 
2
◦ ◦ 3 -cos θ sin θ -cot θ -sec θ cosec θ -tan θ
270 + θ 
2
◦ ◦ 2   -sin θ cos θ -tan θ -cosec θ sec θ -cot θ
360 - θ
◦ ◦ 2   sin θ cos θ tan θ cosec θ sec θ cot θ
360 + θ

Note : (a) The trigonometric ratios of allied angles can be remembered by the following producer :
◦
The trigonometric ratios of angles 90   and 270    changes from sine to cosine, cosine to sine and tan
to cot, whereas the ratios of the angles 180   and 360    remains same. The positive or negative signs
can be determined by applying the quadrant rule.
(b) The trigonometric ratios of an angle of any magnitude can be expressed in terms of trigonometric ratio of
an acute angle (with proper sign). To this end the following producer may be useful .
Suppose we have to find value of t-ratio of the angle θ .
Step 1. Find the sign of the t-ratio of  ,by finding in which quadrant the angle  lies. This can be done by
applying the quadrant rule, i.e., ASTC Rule.
Step 2. Find the numerical value of the t-ratio of  using the following method:
(a) If  lies in the second quadrant, i.e., 90     180  , then t-ratio of ( 180    ) is the required numerical
value.
(b) If  lies in the third quadrant, i.e., 180     270  , then t- ratio of (   180 ) is the required numerical
value.
(c) If  lies in the fourth quadrant, i.e., 270     360  , then t- ratio of ( 360    ) is the required
numerical value.
(d) If  is greater than 360 , i.e.,   n (360  )   , then remove the multiple of 360 and find the t- ratio
of the remaining angle by applying the above method.
VII. Trigonometric ratio of compound angles.
(a) sin (A + B) = sinA cosB + cosA sinB
(b) sin(A – B) = sinA cosB – CosA sinB
(c) cos(A + B) = cosAcosB - sinAsinB
(d) cos(A – B) = cosA cosB + sinAsinB