Find the values of

• [math]\sin(-\pi)[/math] and [math]\cos(-\pi)[/math]
• [math]\sin(\frac34\pi)[/math] and [math]\cos(\frac34\pi)[/math]
• [math]\sin(-\frac{\pi}2)[/math] and [math]\cos(-\frac{\pi}2)[/math]
• [math]\sin(\frac{\pi}6)[/math] and [math]\cos(\frac{\pi}6)[/math]
• [math]\sin(\frac{\pi}3)[/math] and [math]\cos(\frac{\pi}3)[/math]
• [math]\sin(\frac{5\pi}4)[/math] and [math]\cos(\frac{5\pi}4)[/math].

Make a table like the one below showing the sign of [math]\cos t[/math] and [math]\sin t[/math] in each of the four quadrants. Put [math]+[/math] or [math]-[/math] in each entry of the table.

Find all values of [math]t[/math] such that

• [math]\sin t = 0[/math]
• [math]\cos t = 0[/math]
• [math]\sin t = 1[/math]
• [math]\cos t = 1[/math]
• [math]\sin t = -1[/math]
• [math]\cos t = -1[/math]
• [math]\sin t = 2[/math]
• [math]\cos t = 2[/math].

What is the domain and range of each of the functions of [math]\cos[/math] and [math]\sin[/math]?

If [math]k[/math] is an arbitrary integer, find

• [math]\cos k\pi[/math]
• [math]\sin k\pi[/math]
• [math]\cos \left( \frac{\pi}2 + k\pi \right)[/math]
• [math]\sin \left( \frac{\pi}2 + k\pi \right)[/math].

Remembering that [math]\frac{\pi}4 + \frac{\pi}6 = \frac{5\pi}{12}[/math] and [math]\frac{\pi}4 - \frac{\pi}6 = \frac{\pi}{12}[/math], find

• [math]\sin \frac{5\pi}{12}[/math]
• [math]\cos \frac{5\pi}{12}[/math]
• [math]\sin \frac{\pi}{12}[/math]
• [math]\cos \frac{\pi}{12}[/math].

If [math]f[/math] is a function with the property that [math]f(t+2\pi) = f(t)[/math], for every real number [math]t[/math], show from this that

• [math]f(t-2\pi) = f(t)[/math], for every real number [math]t[/math].
• [math]f(t+2\pi n) = f(t)[/math], for every real number [math]t[/math] and every integer [math]n[/math]. (Use induction.)

• lab{6.1.8a} Use \ref{thm 6.1.5} to write a formula for [math]\cos 2a[/math] in terms of [math]\cos a[/math] and [math]\sin a[/math].
• Similarly, use \ref{thm 6.1.7} to write a formula for [math]\sin 2a[/math].
• Write a formula for [math]\cos a[/math] and another for [math]\sin a[/math] in terms of [math]\cos \frac a2[/math] and [math]\sin \frac a2[/math].

Use the formula for [math]\cos 2a[/math] [Problem \ref{ex6.1.8a}] and identity [math]1 = \cos^2a + \sin^2a[/math] to derive a formula for

• [math]\cos^2a[/math] in terms of [math]\cos 2a[/math]
• [math]\sin^2a[/math] in terms of [math]\cos 2a[/math].

Let [math]f[/math] be a function which is periodic with period [math]2\pi[/math], i.e., [math]f(t+2\pi) = f(t)[/math], and suppose that the graph of [math]f[/math] for [math]0 \leq t \leq 2\pi[/math] is as shown in Figure. Draw the graph of [math]f[/math] for [math]-2\pi \leq t \leq 6\pi[/math].