Need help with simplifying trigonometry expressions?

Question

1. If x = a 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.

Answer 1

1. If x = a cosθ:

a) a² - x²

Substitute x = a 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!!!

Answer 2

1. If x = a cosθ:

b)  1/√( a² +  x²)

Substitute x = a cosθ in the given expression

= 1/√(( a² +  (a cosθ)²)      

= 1/√(( a² +  a²cos² θ)

= 1/√((a² (1 + cos² θ))

 

Answer 3

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.

Answer 4

2. If x = a tanθ:

b)  1/√( a² +  x²)

Substitute x = atanθ in the given expression

= 1/√(( a² +  (atanθ)²)      

= 1/√(( a² +  a²tan² θ)

= 1/√((a² (1 + tan² θ))

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

= 1/√((a² (sec² θ))

= 1/asecθ

Answer 5

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!!!