We can evaluate the six trigonometric functions for t = -7π/4.
First, we start by transforming it into a positive angle, adding a full circle (2π):
[tex]t=-\frac{7\pi}{4}+2\pi=-\frac{7}{4}\pi+\frac{8}{4}\pi=\frac{\pi}{4}[/tex]
Now, we can evaluate the basic 3 trigonometric functions as:
[tex]\begin{gathered} \sin (t)=\frac{\sqrt[]{2}}{2} \\ \cos (t)=\frac{\sqrt[]{2}}{2} \\ \tan (t)=\frac{\sin(t)}{\cos(t)}=\frac{\frac{\sqrt[]{2}}{2}}{\frac{\sqrt[]{2}}{2}}=1 \end{gathered}[/tex]
and the other 3 can be evaluated as:
[tex]\begin{gathered} \csc (t)=\frac{1}{\sin(t)}=\frac{2}{\sqrt[]{2}}\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}}=\frac{2\sqrt[]{2}}{2}=\sqrt[]{2} \\ \sec (t)=\frac{1}{\cos(t)}=\sqrt[]{2} \\ \cot (t)=\frac{1}{\tan (t)}=1 \end{gathered}[/tex]
Answer: if t = -7π/4, we have
sin(t) =√2/2
cos(t) = √2/2
tan(t) = 1
csc(t) =√2
sec(t) =√2
cot(t) = 1
