# Need help with simplifying trigonometry expressions?

+1 vote

1. If x = cos theta simplify:

a) a^2 - x^2

b) 1/ square root (a^2 + x^2)

2. If x = a tan theta simplify

a) a^2 + x^2

b) 1/ square root (a^2 + x^2)

3. If x = 5 cos theta and y = 5 sin theta find an equation connecting x and y by eliminating theta.

+1 vote

1. If x = cosθ:

a) a2 - x2

Substitute x = cosθ in the given expression

= a2 – (a cosθ)2

= a2 – a2 cos2 θ

= a2 (1 – cos2 θ

Recall Pythagorean identity: sin2 θ + cos2 θ = 1 ---->  1 – cos2 θ = sin2θ

=  a2(sin2θ

I hope it helps!!!

selected Apr 26, 2013 by potatoes

1. If x = cosθ:

b)  1/( a2 +  x2)

Substitute x = cosθ in the given expression

= 1/(( a2(a cosθ)2)

= 1/(( a2a2cos2 θ)

= 1/√((a2 (1 + cos2 θ))

2. If x = a tanθ:

a) a2 + x2

Substitute x = atanθ in the given expression

= a2 + (a tanθ)2

= a2 + a2tan2θ

= a2 (1 + tan2 θ

Recall trignometric identity: sec2 θ - tan2 θ = 1 ---->  1 + tan2 θ = sec2θ

=  a2sec2θ

Simplified trignometric expression is a2sec2θ.

2. If x = a tanθ:

b)  1/( a2 +  x2)

Substitute x = atanθ in the given expression

= 1/(( a2(atanθ)2)

= 1/(( a2a2tan2 θ)

= 1/√((a2 (1 + tan2 θ))

Recall trignometric identity: sec2 θ - tan2 θ = 1 ---->  1 + tan2 θ = sec2θ

= 1/√((a2 (sec2 θ))

= 1/asecθ

+1 vote

3.

x = 5cosθ, y = 5sinθ

cosθ = x/5 and sinθ = y/5

Substitute in pythagorean identity: sin2 θ + cos2 θ = 1

(y/5)2 + (x/5)2 = 1

x2 +  y2 = 25

I hope it helps!!!