Chapter 6

Here is a proof that the cosine function has period \(2 \pi .\) We saw in the text that \(\cos (t+2 \pi)=\cos t\) for every \(t .\) We must show that there is no positive number smaller than \(2 \pi\) with this property. Do this as follows: (a) Find all numbers \(k\) such that \(0< k <2 \pi\) and \(\cos k=1\) [Hint: Draw a picture and use the definition of the cosine function. \(]\) (b) Suppose \(k\) is a number such that \(\cos (t+k)=\cos t\) for every number \(t .\) Show that \(\cos k=1 .\) [Hint: Consider \(t=0.1\) (c) Use parts (a) and (b) to show that there is no positive number \(k\) less than \(2 \pi\) with the property that \(\cos (t+k)=\cos t\) for every number \(t .\) Therefore, \(k=2 \pi\) is the smallest such number, and the cosine function has period \(2 \pi\)

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. $$\tan 1.5$$

(a) Judging from their graphs, which of the functions \(f(t)=\sin t, g(t)=\cos t,\) and \(h(t)=\tan t\) appear to be even functions? Which appear to be odd functions? (b) Confirm your answers in part (a) algebraically by using appropriate identities from Section 6.3

Here is proof that the sine function has period \(2 \pi .\) We saw in the text that \(\sin (t+2 \pi)=\sin t\) for every \(t .\) We must show that there is no positive number smaller than \(2 \pi\) with this property. Do this as follows: (a) Find a number \(t\) such that \(\sin (t+\pi) \neq \sin t\) (b) Find all numbers \(k\) such that \(0

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. $$\cos 3+\sin 3$$

In Exercises \(71-76,\) find all the solutions of the equation. $$\sin t=1$$

In Exercises \(71-76,\) find all the solutions of the equation. $$\cos t=-1$$

In Exercises \(71-76,\) find all the solutions of the equation. $$\tan t=0$$

In Exercises \(71-76,\) find all the solutions of the equation. $$\sin t=-1$$

In Exercises \(71-76,\) find all the solutions of the equation. $$|\cos t|=1$$

Using only the definition and no calculator, determine which number is larger: \(\sin (\cos 0)\) or \(\cos (\sin 0)\)