Trigonometric ratios

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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)


o Different signs in different quadrants


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.c : Understand range of trigonometric ratios


(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


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


(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


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


B C12 cm

Ɵ

12 cm 16 cm

20 cm 8 cm

Explanation

Question 1

Next Q

That is wrong, please try again…


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

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End of Mind map section

Flash card section

Length of arc = s = r ƟLength of arc = s =________

See back Next

s

O

r Ɵ

Flash card 1

Length of arc = s = r Ɵ

See back Next

s

O

r Ɵ

Flash card 1

Area of a sector = ½ r2ƟArea of a sector = _______

See back NextPrev

O rƟ

Sector

Flash card 2

Area of a sector = ½ r2Ɵ

See back NextPrev

O rƟ

Sector

Flash card 2

1ᶜ = (180/ Π) o1ᶜ = ________ o

See back NextPrev

Flash card 3

1ᶜ = (180/ Π) o

See back NextPrev

Flash card 3

1o = (Π /180)ᶜ1o = _______ᶜ

See back NextPrev

Flash card 4

1o = (Π /180)ᶜ

See back NextPrev

Flash card 4

End of Flash card section

Trigonometric ratios

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