Using the measurements given in the figure

Using the measurements given in the figure alongside,

(a) Find the values of:

(i) sin ϕ

(ii) tan θ.

(b) Write an expression for AD in terms of θ.

Trigonometric Ratios Class 9 ICSE ML Aggarwal img 34

Answer :

BC = 12, BD = 13

In right angled ∆BCD

BD2 = BC2 + CD2

CD2 = BD2 – BC2

CD2 = (13)2 – (12)2

⇒ CD2 = 169 – 144 = 25

CD = √25 = 5

Trigonometric Ratios Class 9 ICSE ML Aggarwal img 35

CD = BE = 5 and EA = AE = 14 – 5 = 9

(a) 

(i) sin ϕ = perpendicular/hypotenuse

In right angled ∆BCD

sin ϕ = CD/BD = 5/13

(ii) tan θ = perpendicular/hypotenuse

In right angled ∆AED

tan θ = ED/AE = BC/AE = 12/9 = 4/3 (Since ED = BC)

(b) In right angled ∆AED

sin θ = perpendicular/hypotenuse

cos θ = base/perpendicular

sin θ = ED/AD or cos θ = AE/AD

AD = ED/sin θ or AD = AE/cos θ

AD = 12/sin θ or AD = 9/cos θ

Hence,

AD = 12/ sin θ or AD = 9/cos θ.

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