Question

Solve for the angle given the trig function and its value. (No Calculator)

32. tan Ѳ = -√3/3

33. cos^2 Ѳ = 1/2

34. sec Ѳ = undefined​

Answer 1

`32.\quad tan\theta=-(\sqrt3)/(3)\\\\.\qquad (sin\theta)/(cos\theta)=-(\sqrt3)/(3)\\\\\text{Since there is no`

`\text{Look on the Unit Circle (below) for coordinates }\bigg((-\sqrt3)/(2),(1)/(2)\bigg)\ and\ \bigg((\sqrt3)/(2),(-1)/(2)\bigg)\\\\\bold{Answer:}\large\boxed{150^o\ and\ 330^o\implies (5\pi)/(6)\ and\ (11\pi)/(6)}`

`33.\quad cos^2\theta=(1)/(2)\\\\\\.\qquad √(cos^2\theta)=\sqrt{(1)/(2)}\\\\\\.\qquad cos\theta=\pm(1)/(\sqrt2)\\\\\\\text{Rationalize the denominator:}\\.\qquad cos\theta=\pm(1)/(\sqrt2)\bigg((\sqrt2)/(\sqrt2)\bigg)\implies cos\theta=\pm(\sqrt2)/(2)\\\\\\\text{Look on the Unit Circle to find when }cos\theta=(\sqrt2)/(2)\ and\ (-\sqrt2)/(2)`

`\bold{Answer:}\large\boxed{45^o, 135^o, 225^o, 315^o\implies (\pi)/(4), (3\pi)/(4), (5\pi)/(4), (7\pi)/(4)}`

`34.\quad sec\theta=\text{unde fined}\implies sec\theta=(1)/(0)\\\\\\.\qquad (1)/(cos\theta)=(1)/(0)\implies cos\theta=0\\\\\\\text{Look on the Unit Circle to find when }cos\theta=0\\\\\bold{Answer:}\large\boxed{90^o\ and\ 270^o\implies(\pi)/(2)\ and\ (3\pi)/(2)}`

Answer 2

32. -30°

33. 45° or 135°

34. 90°

Explanation:

The table below shows a short list of trig function values.

32. -tan(x) = tan(-x)

33. -cos(x) = cos(180°-x)

34. sec(x) = 1/cos(x). 1/0 is "undefined".

The values shown above are the ones that are in the range of the inverse trig functions: arctan (-90°, 90°), arccos [0°, 180°], arcsec [0°, 90°)∪(90°, 180°].