We wish to express $\cos 400^\circ$ in terms of $\cos\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 400^\circ.$

First, we find an angle that's coterminal with $400^\circ$ that lies in the range $[0, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle: \[ 400^\circ - 360^\circ = 40^\circ\quad{\color{green}\checkmark} \]

Notice that our coterminal angle is acute and lies in the first quadrant. Therefore, $\theta_R = 40^\circ,$ and we conclude that

\[ \cos 400^\circ = \cos 40^\circ. \]

We wish to express $\sin{770^\circ}$ in terms of $\sin{\theta_R},$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 770^\circ.$

First, we find an angle that's coterminal with $770^\circ$ that lies in the range $[0, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle:

\begin{align*} 770^\circ - 360^\circ &=410^\circ\\[5pt] 770^\circ - 2\cdot 360^\circ &= 50^\circ\quad{\color{green}{\checkmark}} \end{align*}

Notice that our coterminal angle is acute and lies in the first quadrant. Therefore, $\theta_R=50^\circ,$ and we conclude that

\[ \sin 770^\circ = \sin 50^\circ. \]

We wish to express $\sin 515^\circ$ in terms of $\sin\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 515^\circ.$

To express $\sin 515^\circ $ in terms of $\sin\theta_R,$ we proceed as follows:

• Step 1: Find the corresponding coterminal angle that lies in the range $[0^\circ, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle. \[ 515^\circ - 360^\circ = 155^\circ\quad{\color{green}\checkmark} \] Therefore, \[ \sin 515^\circ = \sin 155^\circ. \]
  

• Step 2: Compute the reference angle. Since $\theta = 155^\circ$ is in the $2$nd quadrant, the reference angle is \[ \theta_R = 180^\circ - 155^\circ= 25^\circ. \]

• Step 3: Determine the correct sign. The sine function is positive $({\color{blue}{+}})$ in the second quadrant. Therefore, \[ \sin 515^\circ = {\color{blue}{+}}\sin\theta_R = {\color{blue}{+}}\sin25^\circ. \]

Therefore, we conclude that

\[ \sin 515^\circ = \sin 25^\circ. \]

We wish to express $\sin 575^\circ$ in terms of $\sin\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 575^\circ.$

To express $\sin 575^\circ$ in terms of $\sin\theta_R,$ we proceed as follows:

• Step 1: Find an angle that's coterminal with $575^\circ$ that lies in the range $[0, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle: \[ 575^\circ - 360^\circ = 215^\circ\quad{\color{green}\checkmark} \] Therefore, \[ \sin 575^\circ = \sin 215^\circ. \]

• Step 2: Compute the reference angle. Since $\theta = 215^\circ$ is in the $3$rd quadrant, the reference angle is \[ \theta_R = 215^\circ - 180^\circ= 35^\circ. \]

• Step 3: Determine the correct sign. The sine function is negative $({\color{red}{-}})$ in the third quadrant. Therefore, \[ \sin (575^\circ) = {\color{red}{-}}\sin\theta_R = {\color{red}{-}}\sin{35^\circ}. \]

Therefore, we conclude that

\[ \sin 575^\circ = -\sin 35^\circ. \]

We wish to express $\cos 1\,050^\circ$ in terms of $\cos\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 1\,050^\circ.$

To express $\cos 1\,050^\circ $ in terms of $\cos\theta_R,$ we proceed as follows:

• Step 1: Find the corresponding coterminal angle that lies in the range $[0^\circ, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle. \begin{align*} 1\,050^\circ - 360^\circ &= 690^\circ\\[5pt] 1\,050^\circ - 2\cdot 360^\circ &= 330^\circ\quad{\color{green}\checkmark} \end{align*} Therefore, \[ \cos 1\,050^\circ = \cos 330^\circ. \]

• Step 2: Compute the reference angle. Since $\theta = 330^\circ$ is in the $4$th quadrant, the reference angle is \[ \theta_R = 360^\circ - 330^\circ= 30^\circ. \]

• Step 3: Determine the correct sign. The cosine function is positive $({\color{blue}{+}})$ in the fourth quadrant. Therefore, \[ \cos 1\,050^\circ = {\color{blue}{+}}\cos\theta_R = {\color{blue}{+}}\cos 30^\circ. \]

Therefore, we conclude that

\[ \cos 1\,050^\circ = \cos 30^\circ. \]

We wish to express $\cos 552^\circ$ in terms of $\cos \theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 552^\circ.$

To express $\cos 552^\circ $ in terms of $\cos\theta_R,$ we proceed as follows:

• Step 1: Find the corresponding coterminal angle that lies in the range $[0^\circ, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle. \[ 552^\circ - 360^\circ = 192^\circ\quad{\color{green}\checkmark} \] Therefore, \[ \cos 552^\circ = \cos 192^\circ. \]

• Step 2: Compute the reference angle. Since $\theta = 192^\circ$ is in the $3$rd quadrant, the reference angle is \[ \theta_R = 192^\circ - 180^\circ= 12^\circ \]

• Step 3: Determine the correct sign. The cosine function is negative $({\color{red}{-}})$ in the third quadrant. Therefore, \[ \cos 552^\circ = {\color{red}{-}}\cos\theta_R = {\color{red}{-}}\cos 12^\circ. \]

Therefore, we conclude that

\[ \cos 552^\circ = -\cos 12^\circ. \]

We wish to express $\tan 611^\circ$ in terms of $\tan\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 611^\circ.$

To express $\tan 611^\circ $ in terms of $\tan\theta_R,$ we proceed as follows:

• Step 1: Find the corresponding coterminal angle that lies in the range $[0^\circ, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle. \begin{align*} 611^\circ - 360^\circ &= 251^\circ\quad{\color{green}{\checkmark}} \end{align*} Therefore, \[ \tan 611^\circ = \tan 251^\circ. \]

• Step 2: Compute the reference angle. Since $\theta = 251^\circ$ is in the $3$rd quadrant, the reference angle is \[ \theta_R =251^\circ-180^\circ= 71^\circ. \]

• Step 3: Determine the correct sign. The tangent function is positive $({\color{blue}{+}})$ in the third quadrant. Therefore, \[ \tan 611^\circ = {\color{blue}{+}}\tan\theta_R = {\color{blue}{+}}\tan 71^\circ. \]

Therefore, we conclude that

\[ \tan 611^\circ= \tan 71^\circ. \]

We wish to express $\tan 860^\circ$ in terms of $\tan\theta_R,$ where $\theta_R$ is the reference angle corresponding to the angle $\theta = 860^\circ.$

To express $\tan 860^\circ $ in terms of $\tan\theta_R,$ we proceed as follows:

• Step 1: Find the corresponding coterminal angle that lies in the range $[0^\circ, 360^\circ).$ We do this by subtracting integer multiples of $360^\circ$ until we get our desired angle. \begin{align*} 860^\circ - 360^\circ &=500^\circ\\[5pt] 860^\circ - 2\cdot 360^\circ &= 140^\circ\quad{\color{green}{\checkmark}} \end{align*} Therefore, \[ \tan 860^\circ = \tan 140^\circ. \]

• Step 2: Compute the reference angle. Since $\theta = 140^\circ$ is in the $2$nd quadrant, the reference angle is \[ \theta_R =180^\circ-140^\circ= 40^\circ. \]

• Step 3: Determine the correct sign. The tangent function is negative $({\color{red}{-}})$ in the second quadrant. Therefore, \[ \tan 860^\circ = {\color{red}{-}}\tan\theta_R = {\color{red}{-}}\tan 40^\circ. \]

Therefore, we conclude that

\[ \tan 860^\circ= -\tan 40^\circ. \]