Two trigonometric inequalities Consider the angle θ in standard position in a unit circle, where 0 ≤ θ < π/2 or −π/2<θ<0 (use both figures).
a. Show that |AC| = |sin θ|, for −π/2<θ < π/2.(Hint: Consider the cases 0 ≤ θ < π/2 and −π/2 < θ < 0 separately.)
b. Show that |sin θ|<|θ|, for −π/2< θ < π/2.(Hint: The length of arc AB is θ, if 0 ≤ 9<π/2, and −θ, if −π/2<θ < 0.)
c. Conclude that −|θ| ≤ sin θ ≤ |θ|, for −π/2<θ<π/2.
d. Show that 0 ≤ 1 − cos θ ≤ |θ|, for −π/2 < θ<π/2.
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