Answer:
a)
[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]
b)
[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]
c)
[tex]\tan \dfrac{7\pi}{4}=-1[/tex]
Step-by-step explanation:
We are asked to find the value of:
a)
[tex]\sin \dfrac{7\pi}{4}[/tex]
We know that:
[tex]\dfrac{7\pi}{4}=2\pi-\dfrac{\pi}{4}[/tex]
Hence, we have:
[tex]\sin (\dfrac{7\pi}{4})=\sin (2\pi-\dfrac{\pi}{4})\\\\\\\sin (\dfrac{7\pi}{4})=-\sin (\dfrac{\pi}{4})[/tex]
Since,
[tex]\sin (2\pi-\theta)=-\sin \theta[/tex]
Hence, we have:
[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]
b)
[tex]\cos \dfrac{7\pi}{4}[/tex]
[tex]\cos (\dfrac{7\pi}{4})=\cos (2\pi-\dfrac{\pi}{4})\\\\\\\cos (\dfrac{7\pi}{4})=\cos (\dfrac{\pi}{4})[/tex]
Since,
[tex]\cos (2\pi-\theta)=\cos \theta[/tex]
Hence, we have:
[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]
c)
[tex]\tan \dfrac{7\pi}{4}[/tex]
[tex]\tan (\dfrac{7\pi}{4})=\tan (2\pi-\dfrac{\pi}{4})\\\\\\\tan (\dfrac{7\pi}{4})=\tan (\dfrac{\pi}{4})[/tex]
Since,
[tex]\tan (2\pi-\theta)=-\tan \theta[/tex]
Hence, we have:
[tex]\tan \dfrac{7\pi}{4}=-1[/tex]
