use the unit circle to find the exact value of each of the following.\n33. ( sin left( \frac { 21 pi } { - 2…

Explanation:

Step1: Simplify the angle for (\sin\left(\frac{21\pi}{- 2}\right))

Use the property (\sin(x + 2k\pi)=\sin x). (\frac{21\pi}{-2}=-10\pi-\frac{\pi}{2}). Since (\sin(x + 2k\pi)=\sin x) ((k = - 5) here), (\sin\left(\frac{21\pi}{-2}\right)=\sin\left(-10\pi-\frac{\pi}{2}\right)). And (\sin(x + 2k\pi)=\sin x), so (\sin\left(\frac{21\pi}{-2}\right)=\sin\left(-\frac{\pi}{2}\right)). Also, (\sin(-x)=-\sin x), so (\sin\left(-\frac{\pi}{2}\right)=-\sin\frac{\pi}{2})

Step2: Calculate the value of (\sin\left(\frac{21\pi}{-2}\right))

We know that on the unit - circle, for the angle (\theta=\frac{\pi}{2}), the (y) - coordinate (since (\sin\theta=y) on the unit - circle (x^{2}+y^{2}=1)) is (1). So (\sin\left(\frac{21\pi}{-2}\right)=-1)

Step3: Simplify the angle for (\tan(1050^{\circ}))

Use the property (\tan(x + 360^{\circ}k)=\tan x). (1050^{\circ}=3\times360^{\circ}-30^{\circ}). So (\tan(1050^{\circ})=\tan(3\times360^{\circ}-30^{\circ})). Since (\tan(x + 360^{\circ}k)=\tan x) ((k = 3) here), (\tan(1050^{\circ})=\tan(-30^{\circ})). And (\tan(-x)=-\tan x), so (\tan(1050^{\circ})=-\tan30^{\circ})

Step4: Calculate the value of (\tan(1050^{\circ}))

On the unit - circle, for the angle (\theta = 30^{\circ}), (\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}). So (\tan(1050^{\circ})=-\frac{\sqrt{3}}{3})

Step5: Simplify the angle for (\sec(-900^{\circ}))

Use the property (\sec(x + 360^{\circ}k)=\sec x) and (\sec(-x)=\sec x). (-900^{\circ}=-3\times360^{\circ}+180^{\circ}). (\sec(-900^{\circ})=\sec(-3\times360^{\circ}+180^{\circ})). Since (\sec(x + 360^{\circ}k)=\sec x) ((k=-3)) and (\sec(-x)=\sec x), (\sec(-900^{\circ})=\sec(180^{\circ})). And (\sec\theta=\frac{1}{\cos\theta})

Step6: Calculate the value of (\sec(-900^{\circ}))

On the unit - circle, for the angle (\theta = 180^{\circ}), (\cos\theta=-1). So (\sec(-900^{\circ})=\frac{1}{\cos(180^{\circ})}=-1)

Answer:

1. (\sin\left(\frac{21\pi}{-2}\right)=-1)
2. (\tan(1050^{\circ})=-\frac{\sqrt{3}}{3})
3. (\sec(-900^{\circ})=-1)