blank A: a^2 + b^2 = c^2
blank B: Definition of unit circle
blank C: sin θ = y/1 = y
Explanation:
In order to prove the identity given, we first start with Pythagoras's theorem
[tex]a^2+b^2=c^2[/tex]which is blank a.
Next, we apply the theorem to the circle to get
[tex]x^2+y^2=r^2[/tex]then we make the substitutions.
Since it is a unit circle r = 1 (blank B) and using trigonometry gives
[tex]\cos \theta=\frac{x}{r}=\frac{x}{1}=x[/tex][tex]\boxed{x=\cos \theta}[/tex]and
[tex]\sin \theta=\frac{y}{r}=\frac{y}{1}=y[/tex][tex]\boxed{y=\sin \theta}[/tex]which is blank C.
With the value of x, y and r in hand, we now have
[tex]x^2+y^2=1[/tex][tex]\rightarrow\sin ^2\theta+\cos ^2\theta=1[/tex]Hence, the identity is proved.