Question

Consider the following

t=11π/6
(a) Find the reference number t for the value of t. t= π /6
(b) Find the terminal point determined by t
(x,y)=
t. (x, y) = ([ x) =

Answer 1

`(a) The reference number for the value of \( t = (11\pi)/(6) \) is \( (\pi)/(6) \).(b) The terminal point determined by \( t = (11\pi)/(6) \) is \( \left(-(√(3))/(2), -(1)/(2)\right) \).`

Step-by-step explanation:

(a) To find the reference number for
`\( t = (11\pi)/(6) \), we need to determine an equivalent angle between 0 and \( 2\pi \) (or 0 and \( 360^\circ \)). As \( (11\pi)/(6) \) exceeds \( 2\pi \) by \( (\pi)/(6) \), the reference angle for \( (11\pi)/(6) \) is \( (\pi)/(6) \).`

(b) To determine the terminal point given
`\( t = (11\pi)/(6) \), we use the unit circle and consider the angle \( (11\pi)/(6) \). In standard position, this angle corresponds to \( (\pi)/(6) \) past \( 2\pi \), or \( (\pi)/(6) \) clockwise from the positive x-axis. For this angle, the coordinates of the terminal point are \( \left(-(√(3))/(2), -(1)/(2)\right) \), as cosine of \( (\pi)/(6) \) is \( (√(3))/(2) \) and sine of \( (\pi)/(6) \) is \( (1)/(2) \).`

Understanding reference angles and terminal points on the unit circle is crucial in trigonometry, aiding in determining equivalent angles and corresponding coordinates on the circle for given values of \( t \).

Answer 2

The reference angle for t=11π/6 is -π/6. The terminal point determined by t=11π/6 in the Cartesian coordinate system is (√3/2, -1/2).

The given value of t is 11π/6. To find the reference angle t' for this value, you can subtract the nearest multiple of π from t:

t' = t - nπ where n is the integer that makes t' fall within the range 0 ≤ t' < π.

t' = 11π/6 - 2π = -π/6

So, the reference angle t' for t = 11π/6 is -π/6. Now, to find the terminal point determined by t = 11π/6 in the Cartesian coordinate system, you can use the polar coordinates conversion formulas:

x = r cos(t)

y = r sin(t)

Since t = 11π/6, let's use r = 1 (assuming the radius is 1 for simplicity):

x = cos(11π/6)

y = sin(11π/6)

Now, let's calculate these values:

x = cos(11π/6) = √3/2

y = sin(11π/6) = -1/2

So, the terminal point (x, y) determined by t = 11π/6 is (√3/2, -1/2).