Question

In Exercises 5-7, find all the exact t-values for which the given statement is true,

5. cos t= 1

6. sin t - 1/2

7. tan t = negative square root of 3

Answer 1

Answer: See Below

Explanation:

NOTE: You need the Unit Circle to answer these (attached)

5) cos (t) = 1

Where on the Unit Circle does cos = 1?

Answer: at 0π (0°) and all rotations of 2π (360°)

In radians: t = 0π + 2πn

In degrees: t = 0° + 360n

******************************************************************************

`6)\quad sin (t) = (1)/(2)`

Where on the Unit Circle does
`sin = (1)/(2)`

Hint: sin is only positive in Quadrants I and II

`\text{Answer: at}\ (\pi)/(6)\ (30^o)\ \text{and at}\ (5\pi)/(6)\ (150^o)\ \text{and all rotations of}\ 2\pi \ (360^o)`

`\text{In radians:}\ t = (\pi)/(6) + 2\pi n \quad \text{and}\quad (5\pi)/(6) + 2\pi n`

In degrees: t = 30° + 360n and 150° + 360n

******************************************************************************

`7)\quad tan (t) = -\sqrt3`

Where on the Unit Circle does
`(sin)/(cos) = (-\sqrt3)/(1)\ or\ (\sqrt3)/(-1)\quad \rightarrow \quad (1,-\sqrt3)\ or\ (-1, \sqrt3)`

Hint: sin and cos are only opposite signs in Quadrants II and IV

`\text{Answer: at}\ (2\pi)/(3)\ (120^o)\ \text{and at}\ (5\pi)/(3)\ (300^o)\ \text{and all rotations of}\ 2\pi \ (360^o)`

`\text{In radians:}\ t = (2\pi)/(3) + 2\pi n \quad \text{and}\quad (5\pi)/(3) + 2\pi n`

In degrees: t = 120° + 360n and 300° + 360n