Solving in terms of a

Question

If tantheta is an acute and tantheta = a, express costheta, sintheta in terms of a

 

My solution was:

 

sintheta/costheta = tantheta

therefore, sintheta/costheta = a

 

hence, sintheta = acostheta or

sintheta/a = costheta

 

I included those two solutions since it asked for the answer in terms of a, is that correct? I am told that whenever a question says to solve in terms of something, it has to include that variable rather than make it the subject hence why I have two solutions.

Answer 1

If tan(θ) = a, what is cos(θ) and sin(θ)?

The question asks you to express cos(θ) and sin(θ) in terms of a, that is, it should be something like cos(θ)=g(a), an expression which contains only a. The same with sin(θ).

 

1+tan²(θ)=1/cos²(θ)

cos(θ)=±1/√(1+tan²(θ))=±1/√(1+a²).

Now you can find sin(θ).

 

Answer 2

Draw a right triangle with an acute angle θ, side opposite θ of length a and side adjacent θ of length 1.  

 

Then, tanθ = a/1 = a.  By the Pythagorean Theorem, the length of the hypotenuse is √(1+a²).

 

So, sinθ = a / √(1+a²)     cosθ = 1 / √(1+a²)