Question

Let $x = -16\pi/3$. Part a: Determine the reference angle of $x$. Part b: Find the exact values of $\sin x$, $\tan x$, and $\sec x$ in simplest form.

A) a) $5\pi/3$, b) $\sin x = -\sqrt{3}/2, \tan x = \sqrt{3}/3, \sec x = -2$
B) a) $\pi/3$, b) $\sin x = \sqrt{3}/2, \tan x = -\sqrt{3}/3, \sec x = -2$
C) a) $5\pi/3$, b) $\sin x = -\sqrt{3}/2, \tan x = -\sqrt{3}/3, \sec x = -2$
D) a) $\pi/3$, b) $\sin x = \sqrt{3}/2, \tan x = \sqrt{3}/3, \sec x = -2$

Answer 1

The reference angle of
`$x = -16\pi/3$` is
`$\pi/3$`. The exact values of
`$\sin x$`,
`$\tan x$`, and
`$\sec x$` are
`$\sin x = √(3)/2$`,
`$\tan x = -√(3)/3$`, and
`$\sec x = -2$`.

Explanation:

To find the reference angle of
`$x = -16\pi/3$`, we need to determine the angle in the first quadrant that has the same trigonometric values as
`$x$`. Since the reference angle is always positive and less than or equal to
`$\pi/2$`, we can find it by finding the equivalent angle within the range of
`$0$` to
`$\pi/2$`.

In this case, if we add
`$2\pi$` repeatedly to
`$x = -16\pi/3$`, we get
`$x = -10\pi/3`,
`-4\pi/3`,
`2\pi/3$`, which is equivalent to $\pi/3$. Therefore, the reference angle of
`$x = -16\pi/3$` is
`$\pi/3$`.

Now, let's find the exact values of
`$\sin x$`,
`$\tan x$`, and
`$\sec x$` using the reference angle
`$\pi/3$`.

Since
`$\sin$` is positive in the first and second quadrants, and negative in the third and fourth quadrants, the value of $\sin x$ is negative.

Using the reference angle
`$\pi/3$`, we know that the point on the unit circle corresponding to
`$\pi/3$`is
`$(1/2, √(3)/2)$`. Therefore,
`$\sin x = -√(3)/2$`.

Since
`$\tan$` is positive in the first and third quadrants, and negative in the second and fourth quadrants, the value of
`$\tan x$` is negative.

Using the reference angle
`$\pi/3$`, we know that the point on the unit circle corresponding to
`$\pi/3$` is
`$(1/2, √(3)/2)$`. Therefore,
`$\tan x = -√(3)/3$`.

Finally, the value of
`$\sec x$` can be found using the reciprocal of
`$\cos$`which is
`$(1)/(\cos x)$`. Since
`$\cos x$` is negative in the second and third quadrants, the value of
`$\sec x$` is negative.

Using the reference angle
`$\pi/3$`, we know that the point on the unit circle corresponding to
`$\pi/3$` is
`$(1/2, √(3)/2)$`. Therefore,
`$\sec x = -2$`.

In conclusion, the exact values of
`$\sin x$`,
`$\tan x$`, and
`$\sec x$` for
`$x = -16\pi/3$` in simplest form are $\sin x =
`-√(3)/2$, $\tan x = -√(3)/3$`, and
`$\sec x = -2$`.

Answer 2

a)
`$5\pi/3$, b) $\sin x = -√(3)/2, \tan x = -√(3)/3, \sec x = -2$`

Explanation:

For part a), the reference angle of $x = -16\pi/3$ is $5\pi/3$. This is determined by finding the positive acute angle formed between the terminal side of the angle and the x-axis when considering the negative coterminal angle.

For part b), evaluating the trigonometric functions:

-
`$\sin x = -√(3)/2$:` This is obtained from the unit circle corresponding to the sine value at $5\pi/3$.

-
`$\tan x = -√(3)/3$`: Obtained by dividing
`$\sin x$ by $\cos x$ ($\tan x = \sin x / \cos x$).`

- $\sec x = -2$: This is the reciprocal of $\cos x$, which is $-1/2$ on the unit circle, giving
`$\sec x = -2$.`

These values are derived from the unit circle, understanding the relationship between trigonometric functions and the coordinates on the circle at the given angle $5\pi/3$, ensuring the signs are correctly represented for each function.

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