Question

on the following unit circle, θ is in radians. without a calculator, evaluate the following expressions to the nearest hundredth. expression value sin(π − θ) sin(π + θ)

Explanation:

Step1: Recall unit circle coordinates

For a point \((x,y)\) on the unit circle, \(x = \cos\theta\) and \(y=\sin\theta\). Given the point \((0.54, 0.84)\), so \(\cos\theta = 0.54\) and \(\sin\theta=0.84\).

Step2: Evaluate \(\sin(\pi - \theta)\)

Use the identity \(\sin(\pi - \alpha)=\sin\alpha\). So \(\sin(\pi - \theta)=\sin\theta = 0.84\).

Step3: Evaluate \(\sin(\pi + \theta)\)

Use the identity \(\sin(\pi+\alpha)=-\sin\alpha\). So \(\sin(\pi + \theta)=-\sin\theta=- 0.84\).

Answer:

\(\sin(\pi - \theta)=\boldsymbol{0.84}\)
\(\sin(\pi + \theta)=\boldsymbol{-0.84}\)