Question

In the diagram, PN¯¯¯¯¯ is the perpendicular bisector of AB¯¯¯¯¯ and is also the angle bisector of ∠CPD. If m∠CPD = x, which quantity is equal to sin ∠DPB?

sin x2

sinx2

cosx2

cos x2

Answer 1

Since ∠CPD = x and segment PN is the angle bisector of this angle, therefore segment PN equally divides ∠CPD into two angles. Which means that:

∠CPN = ∠NPD = x / 2

Further, segment PN is also the perpendicular bisector of AB which further means that the intersection formed by PN and AB creates a right angle (90°). Therefore:

∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

Therefore:

sin∠DPB = sin(90 – x/2) which is not in the choices

However we know that the relationship of sin and cos is:

sin(π/2 - θ) = cos θ

Where,

π/2 = 90

θ = x/2

Therefore:

sin(90 – x/2) = cos(x/2)

 

Answer:

cos(x/2)

Answer 2

The quantity which is equal to sin ∠DPB is:

• cos(x/2)

What is an Angle?

This refers to the figure which is formed by two rays with a common endpoint.

Hence, we know that

• ∠CPN = ∠NPD = x / 2

If we segment PN which is the bisector of AB, it would crerate angle 90 and this would give us:

• ∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

With this in mind, there is the relation between sin and cos, which would be:

• sin(π/2 - θ) = cos θ

We are aware that

• π/2 = 90
• θ = x/2

Hence,

sin(90 – x/2) = cos(x/2)

=cos(x/2)

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