What are the coordinates of the points that corresponds to -pi/4 on the unit circle ?

((√2)/2, -(√2)/2)

Step-by-step explanation:

The coordinates of the point on the unit circle for any angle θ are ...

... (x, y) = (cos(θ), sin(θ))

For the angle -π/4, the coordinates are ...

... (cos(-π/4), sin(-π/4)) = ((√2)/2, -(√2)/2)

( sqrt(2)/2,  -sqrt(2)/2 )

This is equivalent to ( 1/sqrt(2),  -1/sqrt(2) )

or you can also write ( sqrt(1/2),  -sqrt(1/2) )

in decimal form, the answer is approximately (0.7071, -0.7071) which is rounded to four decimal places.

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Explanation:

pi/4 radians = 45 degrees

The angle -pi/4 is the result of rotating 45 degrees clockwise starting off facing directly east. What results is a 45-45-90 triangle. The two legs are unknown. Call them p and q. In any 45-45-90 triangle, the two legs are the same length so p = q.

In the unit circle, the radius is 1. This means the hypotenuse of the right triangle is 1.

Use the pythagorean theorem to solve for p

a^2+b^2 = c^2

p^2+q^2 = 1^2

p^2+p^2 = 1

2p^2 = 1

p^2 = 1/2

p = sqrt(1/2) ... cosine is positive in Q4

p = sqrt(1)/sqrt(2)

p = 1/sqrt(2)

p = (1/sqrt(2))*(sqrt(2)/sqrt(2))

p = sqrt(2)/(sqrt(2*2))

p = sqrt(2)/2

Since p = q, this means the q is also sqrt(2)/2. We start at the origin and move p units to the right and q units down to arrive at the proper location. Check out the attached image for more visual info, and a diagram. Hopefully it clears up any confusion you may have.

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12