Question

on the following unit circle, θ is in radians. without a calculator, evaluate the following expressions to the nearest hundredth. expression value cos(π + θ) cos(2π - θ) (unit circle diagram with point (0.83, -0.55))

Explanation:

Step1: Recall the unit circle coordinates

On the unit circle, the coordinates of a point corresponding to an angle \(\theta\) are \((\cos\theta, \sin\theta)\). From the given point \((0.83, -0.55)\), we know that \(\cos\theta = 0.83\) and \(\sin\theta=- 0.55\).

Step2: Evaluate \(\cos(\pi+\theta)\)

Use the cosine addition formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\). For \(A = \pi\) and \(B=\theta\), we have \(\cos(\pi+\theta)=\cos\pi\cos\theta-\sin\pi\sin\theta\). Since \(\cos\pi=- 1\) and \(\sin\pi = 0\), this simplifies to \(\cos(\pi+\theta)=-1\times\cos\theta-0\times\sin\theta=-\cos\theta\). Substituting \(\cos\theta = 0.83\), we get \(\cos(\pi+\theta)=- 0.83\).

Step3: Evaluate \(\cos(2\pi-\theta)\)

Use the cosine subtraction formula \(\cos(A - B)=\cos A\cos B+\sin A\sin B\). For \(A = 2\pi\) and \(B = \theta\), we have \(\cos(2\pi-\theta)=\cos2\pi\cos\theta+\sin2\pi\sin\theta\). Since \(\cos2\pi = 1\) and \(\sin2\pi=0\), this simplifies to \(\cos(2\pi - \theta)=1\times\cos\theta+0\times\sin\theta=\cos\theta\). Substituting \(\cos\theta = 0.83\), we get \(\cos(2\pi-\theta)=0.83\).

Answer:

For \(\cos(\pi+\theta)\), the value is \(-0.83\).
For \(\cos(2\pi - \theta)\), the value is \(0.83\).