Start with substituting 12tanθ for x and simplifying. Then use Trigonometric Identities to transform the function.
f(θ)=1/2sinθ
Practice makes perfect
We are given a function and want to write it in terms of a single trigonometric function of θ. Let's start by substituting 12tanθ for x. Here θ is greater than - π2 and less than π2.
We will now use the trigonometric identities to transform the function. First, consider the denominator. Recall one of the Pythagorean Identities.
tan^2θ+1=sec^2θ
Let's use this to transform our function.
f(θ)=12tanθ/sqrt(1+tan^2θ)=12tanθ/sqrt(sec^2θ)
We know that sqrt(x^2)=|x|. To determine if secθ is positive or negative we need to consider the possible values of θ. Since - π2<θ< π2, θ lies in Quadrant I or Quadrant IV. Recall the behavior of secant in each quadrant.
In Quadrant I and IV the value of secant is positive, so sqrt(sec^2θ)=secθ.
f(θ)=12tanθ/sqrt(sec^2θ)=12tanθ/secθ
Now recall the definition of secant and one of the Quotient Identities considering tangent.
secθ = & 1/cosθ
tanθ = & sinθ/cosθ
Finally, we will use both of these identities to transform our function.
