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TRIGONOMETRIC RATIOS
SYNOPSIS AND FORMULAE
π
1. One right angle = radians = 90o.
2
π radians = 2 right angles = 180o
1o = 601, 11 = 6011
1o = 0.01745 radians.
1c = 57o1714511 (approx)
2. Relations:
i) sin θ cosec θ = 1
ii) cos θ sec θ = 1
iii) tan θ cot θ = 1
iv) sin2 θ + cos2 θ = 1
v) 1 + tan2 θ = sec2 θ
→ (sec θ + tan θ) (sec θ – tan θ) = 1.
1
→ sec θ + tan θ = =1
sec θ − tan θ
vi) 1 + cot2 θ = cosec2 θ
→ (cosec θ + cot θ) (cosec θ – cot θ) = 1
1
→ cosec θ + cot θ =
cosec θ − cot θ
vii) sec2 θ + cosec2 θ = sec2 θ . cosec2 θ
viii) tan2 θ – sin2 θ = tan2 θ . sin2 θ;
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cot2 θ – cos2 θ = cot2 θ . cos2 θ
ix) sin2 θ + cos4 θ = 1 – sin2 θ cos2 θ
= sin4 θ + cos2 θ
x) sin4 θ + cos4 θ = 1 – 2sin2 θ cos2 θ
xi) sin6 θ + cos6 θ = 1 – 3sin2 θ cos2 θ
xii) sin2 x + cosec2 x ≥ 2
xiii) cos2 x + sec2 x ≥ 2
xiv) tan2 x + cot2 x ≥ 2.
3. Values of trigonometric ratios of certain angles
angle
↓ 0o π/6 π/4 π/3 π/2
→
ratio
sin 0 1/2 1/ 2 3/2 1
cos 1 3/2 1/ 2 1/2 0
tan 0 1/ 3 1 3 undefined
cot undefined 3 1 1/ 3 0
cosec undefined 2 2 2/ 3 1
sec 1 2/ 3 2 2 undefined
4. Signs of Trigonometric Ratios: If θ lies in I, II, III, IV quadrants then the signs of
trigonometric ratios are as follows.
II I
90o < θ < 180o 0o < θ < 90o
Sin θ and cosec θ all the ratios
III IV
180o < θ < 270o 270o < θ < 360o
tan θ and cot θ cos θ and sec θ
Note: i) 0o, 90o, 180o, 270o. 360o, 450o, ….. etc. are called quadrant angles.
ii) With “ALL SILVER TEA CUPS” symbol we can remember the signs of trigonometric
ratios.
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5. Increasing and Decreasing Behavior of Trigonometrical Ratios:
In Q1: sin θ, tan θ, sec θ are increasing functions and cos θ, cot θ, cosec θ are decreasing
functions.
In Q2: sin θ, cos θ, cot θ are decreasing functions and tan θ , sec θ, cosec θ are increasing.
In Q3: sin θ, cot θ, sec θ are decreasing functions and tan θ, cos θ, cosec θ are increasing
functions.
In Q4: sin θ, cos θ, tan θ are increasing functions and cosec θ, sec θ, cot θ are decreasing
functions.
6. Coterminal Angles: If two angles differ by an integral multiples of 360o then two angles are
called coterminal angles.
Thus 30o, 390o, 750o, 330o etc., are coterminal angles.
Fn 90 ∓ θ 180 ∓ θ 270 ∓ θ 360 ∓ θ
sin θ cos θ ± sin θ − cos θ ∓ sin θ
cos θ ± sin θ − cos θ ∓ sin θ cos θ
tan θ ± cot θ ∓ tan θ ± cot θ ∓ tan θ
cosec θ sec θ ± cosec θ − sec θ ∓ cosec θ
sec θ ± cosec θ − sec θ ∓ cosec θ sec θ
cot θ ± tan θ ∓ cot θ ± tan θ ∓ cot θ
7. Sin (n. 360o + θ) = sin θ
Cos (n. 360o + θ) = cos θ
Tan (n. 360o + θ) = tan θ
Sin (n. 360o – θ) = sin (–θ) = –sin θ
Cos (n. 360o – θ) = cos (–θ) = cos θ
Tan (n. 360o - θ) = tan (–θ) = –tan θ
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8. Complementary Angles: Two Angles A, B are said to complementary ⇒ A + B = 90o
1) sin A = cos B and cos A = sin B.
2) sin2 A + sin2 B = 1, and cos2 A = sin2 B.
3) tan A . tan B = 1 and cot A cot B = 1.
9. Supplementary angles: Two angles A, B are said to be supplementary ⇒ A + B = 180o.
1) sin A – sin B = 0
2) cos A + cos B = 0
3) tan A + tan B = 0
Note: 1) If A – B = 180o then i) cos A + cos B = 0
ii) sin A + sin B = 0
iii) tan A – tan B = 0
2) If A + B = 360o then i) sin A + sin B = 0
ii) cos A – cos B = 0
iii) tan A + tan B = 0
10.
Functons Domain Range
Sin R [−1,1]
Cos R [−1,1]
π
Tan R −{(2n+1) , n ∈ z} R
2
Cot R −{nπnn ∈ z} R
π
Sec R −{(2n+1) , n ∈ z} (−∞ −1]∪[1,∞)
2
Cosec R −{nπnn ∈ z} (−∞ −1]∪[1,∞)
Note: 1) If a cos θ + b sin θ = c then
a sin θ - b cos θ = ± a 2 + b 2 − c 2
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2) If a cos θ - b sin θ = c then
a sin θ + b cos θ = ± a 2 + b2 − c2
11. sin θ + sin(π + θ) + sin (2π + θ) + ….. ….. + sin(nπ + θ) = 0, if n is odd
= sin θ, if n is even.
12. cos θ + cos(π + θ) + cos(2π + θ) + ….. + cos(nπ + θ) = 0, if n is odd
= cos θ, if n is even.
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