Let P be the point on the unit circle U that corresponds to t. Find the coordinates of P and the exact values of the trigonometric functions of t

Question

Let P be the point on the unit circle U that corresponds to t. Find the coordinates of P and the exact values of the trigonometric functions of t, whenever possible. (If an answer is undefined, enter UNDEFINED.)

(a)    t = 7𝜋/4

P(x,y)=

sin(7𝜋/4)=

cos(7𝜋/4)=

tan(7𝜋/4)=

csc(7𝜋/4)=

sec(7𝜋/4)=

cot(7𝜋/4)=

(b)  t = −3𝜋/4

P(x,y)=

sin(-3𝜋/4)=

cos(-3𝜋/4)=

tan(-3𝜋/4)=

csc(-3𝜋/4)=

sec(-3𝜋/4)=

cot(-3𝜋/4)=

Answer 1

Given: t = 7pi/4 or (-3)*pi/4

Find: The coordinates of the point, P, on the unit circle.

Also, find various trig functions.

SOLUTION. =====>. PART A -- With t = 7*pi/4

Note: On the unit circle, the radius is 1

So, for all trig functions, the hypotenuse is 1.

So, cos(t) = adjacent/1 and sin(t) = opposite/1

In simply...

x = cos(t) and y = sin(t)

Note: 8pi/4 = 2pi = all the way around.

In a positive (counter-clockwise) direction.

So, 7pi/4 is just pi/4 short of going all the way around.

So, the point is in Q4.

So, x will be positive and y will be negative.

P(t) = (x,y)

P(t) = ( cos(t), sin(t) )

With t = 7pi/4

P(t) = ( sqrt(2)/2, - sqrt(2)/2 )

cos(t) = x = sqrt(2)/2

sin(t) = y = - sqrt(2)/2

tan(t) = sin/cos = y/x = - 1

csc(t) = 1/sin = 1/y = - 2/sqrt(2) = - sqrt(2)

sec(t) = 1/cos = 1/x = 2/sqrt(2) = sqrt(2)

cot(t) = 1/tan = -1/1 = - 1

SOLUTION =====>. PART B -- With t = (-3)*pi/4

Note: If we went 4pi/4 or simply pi

in the negative (or clockwise) direction,

that would put us at 180 degrees or

at point (-1, 0) on the unit circle.

With t = (-3)*pi/4, going in negative direction,

that pus us in Q3. So, BOTH x and y are negative.

P(t) = (x,y)

P(t) = ( cos(t), sin(t) )

With t = (-3)*pi/4

P(t) = ( - sqrt(2)/2, - sqrt(2)/2 )

cos(t) = x = - sqrt(2)/2

sin(t) = y = - sqrt(2)/2

tan(t) = sin/cos = y/x = 1

csc(t) = 1/sin = 1/y = - 2/sqrt(2) = - sqrt(2)

sec(t) = 1/cos = 1/x = - 2/sqrt(2) = - sqrt(2)

cot(t) = 1/tan = 1/1 = 1

Answer 2