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Trigonometric ratios
(0,0)
P (x,y)
Study section
Table of contents
M1-2.a : Understand trigonometric ratios for a standard unit circle
M1-2.b : Know signs of trigonometric ratios
M1-2.c : Understand range of trigonometric ratios
M1-2.d : Know ratios of standard angles
M1-2.e : Learn the Fundamental identities
M1-2.f : Understand relation between ratios of Ɵ and -Ɵ
M1-2.a : Understand trigonometric ratios for a standard unit circle
Ratios are defined as co-ordinates of a point on a standard unit circle
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A (1,0)
O (0,0)
B (0,1)
C (-1,0)
D (0,-1)
Ɵ
P (x,y)
Sine Ɵ = sin Ɵ = y
Cosine Ɵ = cos Ɵ = x
Tangent Ɵ = tan Ɵ = sin Ɵcos Ɵ
= 𝑦𝑥
Cosecant Ɵ = cosec Ɵ = 1sin Ɵ
= 1𝑦
Secant Ɵ = sec Ɵ = 1cos Ɵ =
1𝑥
Cotangent Ɵ = cot Ɵ = cos Ɵsin Ɵ
= 𝑥𝑦
P (x,y) = P (cos Ɵ,sin Ɵ)
M1-2.b : Know signs of trigonometric ratios
o Different signs in different quadrants
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O
X axis
Y axis
1st quadrant (+,+)
2nd quadrant (-,+)
3rd quadrant (-,-)
4th quadrant (+,-)
Quadrant/Ratio 1st 2nd 3rd 4th
Sin x + + - -Cos x + - - +Tan x + - + -
Cosec x + + - -Sec x + - - +Cot x + - + -
M1-2.b : Know signs of trigonometric ratios
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(+,+) (-,+)
(-,-) (+,-)
M1-2.c : Understand range of trigonometric ratios
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(1,0)(0,0)
(0,1)
(-1,0)
(0,-1)
We observe that
– 1 ≤ sin x ≤ 1 and – 1 ≤ cos x ≤ 1
Since cosec x = (1/sin x)cosec x <= -1 or >= 1
Also, since sec x = (1/cos x)sec x <=-1 or >=1
tan x and cot x can take any real value
M1-2.d : Know ratios of standard angles
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A ngle/ Ratio 0 π/ 6 π/ 4 π/ 3 π/ 2 π 3π/ 2 2π
S in x 0 1/2
1/ξ2 ξ3/2
1 0 -1 0
C os x 1 ξ3/2
1/ξ2 1/2
0 -1 0 1
T an x 0 1/ξ3
1 ξ3 Not defined
0 Not defined
0
M1-2.e : Learn the Fundamental identities
From distance formula,
(x-0)2 + (y-0)2 = 1x2+ y2 = 1
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(0,0)
P (x,y)
Dividing by cos2 Ɵtan2 Ɵ + 1 = sec2 Ɵ
Dividing by sin2 Ɵ1+ cot2 Ɵ = cosec2 Ɵ
Thus, sin2 Ɵ + cos2 Ɵ = 1
M1-2.f : Understand relation between ratios of Ɵ and -Ɵ
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O (0,0)
P (x,y)
Q (x,-y)
A (1,0)
Ɵ
-Ɵ
For point P,sin Ɵ = y and cos Ɵ = x
For point Qsin (-Ɵ) = -y and cos (-Ɵ)
= x
Comparing the two, y = sin Ɵ = - sin (-Ɵ)
And x = cos Ɵ = cos (-Ɵ)
i.e. sin (-Ɵ) = - sin Ɵ
i.e. cos (-Ɵ) = cos Ɵ
End of study section
Quiz section
Calculate the length of the side AC, given that sin θ = 0.6
12 cm 16 cm
20 cm 8 cm
Question 1
A
B C12 cm
Ɵ
Next
12 cm 16 cm
20 cm 8 cm
Question 1
That is correct!
Calculate the length of the side AC, given that sin θ = 0.6A
B C12 cm
Ɵ
Explanation Next Q
12 cm 16 cm
20 cm 8 cm
Question 1
Next Q
Calculate the length of the side AC, given that sin θ = 0.6A
B C12 cm
Ɵ
12 cm 16 cm
20 cm 8 cm
Explanation
Question 1
Next Q
That is wrong, please try again…
Calculate the length of the side AC, given that sin θ = 0.6A
B C12 cm
Ɵ
12 cm 16 cm
20 cm 8 cm
Explanation
Question 1
Next Q
That is wrong, please try again…
Calculate the length of the side AC, given that sin θ = 0.6A
B C12 cm
Ɵ
Sin Ɵ = opposite/hypotenuse
Sin Ɵ = 12/AC
0.6 = 12/AC
AC =20 cm
Explanation to Question 1
Next
End of quiz section
Mind map section
Trigonometric ratios
Next
Ratios of standard angles
Next
End of Mind map section
Flash card section
Length of arc = s = r ƟLength of arc = s =________
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s
O
r Ɵ
Flash card 1
Length of arc = s = r ƟLength of arc = s =________
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s
O
r Ɵ
Flash card 1
Length of arc = s = r Ɵ
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s
O
r Ɵ
Flash card 1
Area of a sector = ½ r2ƟArea of a sector = _______
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O rƟ
Sector
Flash card 2
Area of a sector = ½ r2ƟArea of a sector = _______
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O rƟ
Sector
Flash card 2
Area of a sector = ½ r2Ɵ
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O rƟ
Sector
Flash card 2
1ᶜ = (180/ Π) o1ᶜ = ________ o
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Flash card 3
1ᶜ = (180/ Π) o1ᶜ = ________ o
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Flash card 3
1ᶜ = (180/ Π) o
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Flash card 3
1o = (Π /180)ᶜ1o = _______ᶜ
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Flash card 4
1o = (Π /180)ᶜ1o = _______ᶜ
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Flash card 4
1o = (Π /180)ᶜ
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Flash card 4
End of Flash card section