Find the following trig functions?

Question

1. tan (2pi/3)

2. cos^2 (-135 degrees)

3. cot (690 degrees)

4. sin (-5pi/2)

Can you please tell me how to solve it by showing me the steps?
I obviously know how to plug them into my calculator but I have to find the exact answers. No demical approximation is allowed.. help me! I'll be waiting in front of my computer.

Answer 1

On a unit circle, 2pi/3 has the exact same y-coordinate but exact opposite x-coordinate as pi/3. Since tan pi/3 = sqrt 3, tan 2pi/3 = - sqrt 3

690 degrees is once around (360) plus 330 more. 330 has the same x-coordinate as 30 but opposite y; and cot = reciprocal of tan, and tan 30 = sqrt3 over 3, so cot 690 = - 3 over sqrt 3 which simplifies to - sqrt 3


-5 pi/2 if you count clockwise around the unit circle is all the way around once (4pi/2) plus pi/2 more so you are straight down. sin = y and y there is -1 so sin -5pi/2 = -1

sorry missed #2 -135 is in quadrant 3 and has the exact opposite x value as 45, although opposite doesn't matter because it's squared. Since cos = x and for 45, x = sqrt2 over 2, and it's squared you get 2 / 4 or 1/2

Answer 2

Π = 180°

1) tan (2*180/3) = tan 120

2)cos 2A = 2cos^2 A -1
(cos 2A + 1)/2 = cos^2 A
cos^2 (-135) = (cos 2(-135) + 1)/2
cos^2 (-135) = (cos 2(-135))/2 + 1/2
cos^2 (-135) = cos -270 + 1/2
cos^2 (-135) = 0 + 1/2

3)cot(690°) = 1/tan 690° = 1/tan 330°
1 circle = 360°
2 circle = 720
720 - 690 = 30
360 - 330 = 30

4) sin (-5(180)/2) = sin -450 = sin 90

Answer 3

You should know the value of the sin,cos and tan functions for 0 deg, 30 deg(pi/6), 45 deg(pi/4), 60 deg(pi/3), and 90 deg(pi/2). Then you need to know the sign of the function in the quadrant that includes the arguement angle. You need to know the correspondence between angles in different quadrants.
1. Tan(2pi/3) is negative. Magnitude=that of tan(pi/3) or sqrt(3).
2. Cos2(-135). Cos is negative in this quadrant, but square of cosine is positive. Magnitude= than of Cos pi/4 or sqrt(2)/2, so Cos^2(-135)=1/2
3. Cot(690) is also Cot(330). Cot is negative in 4th quadrant. Cot(330) magnitude= Cot(30) and Cot(690)= -sqrt(3)
4. Sin(-5pi/2)= Sin(-pi/2). Sin is negative in 4th quadrant, and in magnitude =Sin(pi/2). So result is -1.

Answer 4

sure. all of them carry genuine, offered none of their denominators exchange into 0. be conscious: arccos ? = x/r, arctan ? = y/x are incorrect. arccos x/r = ?, arctan y/x = ? are the right equations. d: