Question

question 10
for the quadrantal angle 7π/2, give the coordinates of the point where the terminal side of the angle intersects unit circle. then give the cosine and tangent of the angle.
a) (0, -1), cos(7π/2)=0, tan(7π/2)=undefined
b) (-1, 0), cos(7π/2)=0, tan(7π/2)= - 1
c) (0, -1), cos(7π/2)=undefined, tan(7π/2)=0
d) (-1, 0), cos(7π/2)= - 1, tan(7π/2)=undefined
e) (0, 1), cos(7π/2)=undefined, tan(7π/2)=undefined
question 11
evaluate: sin(150°)
a) √3/2
b) 1/2
c) -√3/2

Explanation:

Step1: Analyze the angle $\frac{7\pi}{2}$

The angle $\frac{7\pi}{2}= 3\pi+\frac{\pi}{2}$. One - full rotation is $2\pi$, and $\frac{7\pi}{2}$ is equivalent to rotating $3$ full - half - circles and an additional $\frac{\pi}{2}$ in the clock - wise direction. The terminal side of the angle $\frac{7\pi}{2}$ lies on the negative $y$ - axis. The point of intersection of the terminal side of the angle $\frac{7\pi}{2}$ and the unit circle is $(0, - 1)$.

Step2: Recall the definitions of cosine and tangent

For a point $(x,y)$ on the unit circle corresponding to an angle $\theta$, $\cos\theta=x$ and $\tan\theta=\frac{y}{x}$. For $\theta = \frac{7\pi}{2}$, $x = 0$ and $y=-1$. So, $\cos(\frac{7\pi}{2})=0$ and $\tan(\frac{7\pi}{2})=\frac{-1}{0}$, which is undefined.

Step3: Evaluate $\sin(150^{\circ})$

We know that $150^{\circ}=180^{\circ}-30^{\circ}$. Using the identity $\sin(A - B)=\sin A\cos B-\cos A\sin B$ with $A = 180^{\circ}$ and $B = 30^{\circ}$, or simply using the unit - circle definition. The reference angle for $150^{\circ}$ is $30^{\circ}$, and in the second quadrant, $\sin\theta>0$. Since $\sin30^{\circ}=\frac{1}{2}$, then $\sin150^{\circ}=\frac{1}{2}$.

Answer:

a) $(0, - 1),\cos(\frac{7\pi}{2}) = 0,\tan(\frac{7\pi}{2})$ is undefined
b) $\frac{1}{2}$