Question

Using the measurements given in the following figure:

1. Find the value of sin Φ and tan θ.
2. Write an expression for AD in terms of θ.

Using the measurements given in the following figure.

1. Find the value of (i) sin Φ, (ii) tan θ
2. Write an expression for AD in terms of θ.

Answer 1

Consider the figure:

A perpendicular is drawn from D to the side AB at point E which makes BCDE is a rectangle.

Now in right-angled triangle BCD using Pythagorean Theorem

⇒ BD² = BC² + CD²    ...( AB is hypotenuse in ΔABD)

⇒ CD² = 13² – 12²

⇒ CD = 25

∴ CD = 5

Since BCDE is rectangle so ED 12 cm, EB = 5 and AE = 14 – 5 = 9 

(i) sin Φ = `"CD"/"BD" = (5)/(13)`

tan θ = `"ED"/"AE" = (12)/(9) = (4)/(3)`

(ii) sec θ = `"AD"/"AE"`

sec θ = `"AD"/(9)`

AD = 9 sec θ

Or

cosec θ = `"AD"/"ED"`

cosec θ = `"AD"/(12)`

 AD = 12 cosec θ

Answer 2

Given: From the figure: AB = 14, BC = 12, CD = 5, BD = 13. Angle at B is φ (between BC and BD); angle at A is θ between AB and AD.

Step-wise calculation:

1. sin φ: In right triangle BCD (right angle at C), for angle φ at B the opposite side = CD = 5 and hypotenuse = BD = 13.

Therefore, sin φ = `5/13`.

2. tan θ: Use triangle A–E–D where E is the point on AB with the same y-coordinate as D. So, AE = vertical drop from A to D = 14 – 5 = 9, and ED = horizontal = 12.

In that right triangle, tan θ = `("Opposite to"  θ)/("Adjacent to"  θ)`

= `"ED"/"AE"`

= `12/9`

= `4/3`

3. AD in terms of θ: From the same right triangle AED, adjacent (AE) = 9 and opposite (ED) = 12, hypotenuse = AD.

`"AD" = "AE"/(cos θ)`

= `9/(cos θ)`

= 9 sec θ

Equivalently `"AD" = "ED"/(sin θ)`

= `12/(sin θ)`

sin φ = `5/13`.

tan θ = `4/3`.

AD = 9 sec θ (equivalently AD = 12 cosec θ).