# Calculate time from the equation with sin and cos

#### ncchieh

##### New member

I not sure I come to the correct place, but this is my assignment question and I really don't know how to solve it.

#### Subhotosh Khan

##### Super Moderator
Staff member
View attachment 13242
I not sure I come to the correct place, but this is my assignment question and I really don't know how to solve it.
Do you know how to calculate velocity vectors from given position vectors?

#### ncchieh

##### New member
Do you know how to calculate velocity vectors from given position vectors?
you mean differentiation?

#### Subhotosh Khan

##### Super Moderator
Staff member
you mean differentiation?
Yes!

Now tell us:

What are the velocity vectors of particles (1) and (2) at a time 't'?

What are their respective unit vectors?

#### ncchieh

##### New member
Yes!

Now tell us:

What are the velocity vectors of particles (1) and (2) at a time 't'?

What are their respective unit vectors?
Va =(-3sin t) i +(4cos t) j
Vb =(3cos t) i + (-4sin t) j

unit vector of A = ((-3sin t) / (9sin² t + 16cos² t)) i + ((4cos t) / (9sin² t + 16cos² t)) j
unit vector of B = ((3cos t) / (9cos² t + 16sin² t)) i + ((-4sin t) / (9cos² t + 16sin² t)) j

I cannot understand the logic of finding t with these information, what do i miss?

#### lev888

##### Full Member
Va =(-3sin t) i +(4cos t) j
Vb =(3cos t) i + (-4sin t) j

unit vector of A = ((-3sin t) / (9sin² t + 16cos² t)) i + ((4cos t) / (9sin² t + 16cos² t)) j
unit vector of B = ((3cos t) / (9cos² t + 16sin² t)) i + ((-4sin t) / (9cos² t + 16sin² t)) j

I cannot understand the logic of finding t with these information, what do i miss?
You need to use the information in the problem to constrain the vectors. Then find t from the resulting equation.

#### Subhotosh Khan

##### Super Moderator
Staff member
Va =(-3sin t) i +(4cos t) j
Vb =(3cos t) i + (-4sin t) j

unit vector of A = ((-3sin t) / (9sin² t + 16cos² t)) i + ((4cos t) / (9sin² t + 16cos² t)) j
unit vector of B = ((3cos t) / (9cos² t + 16sin² t)) i + ((-4sin t) / (9cos² t + 16sin² t)) j

I cannot understand the logic of finding t with these information, what do i miss?
You can see that for vectors Va and Vb, to have opposite directions, we must have

Va|x = Vb|x and Va|y = Vb|y

for that we need:

sin(t) = cos(t) ..........................................(1)

(1) has multiple solutions. The lowest positive value of 't' to satisfy (1) is

#### ncchieh

##### New member
You can see that for vectors Va and Vb, to have opposite directions, we must have

Va|x = Vb|x and Va|y = Vb|y

for that we need:

sin(t) = cos(t) ..........................................(1)

(1) has multiple solutions. The lowest positive value of 't' to satisfy (1) is

you mean like this?

-3 sint = 3 cost .......................(1)
4 cost = -4 sint ........................(2)

from (1)
(sin/cos) t = -1
tan t = -1
t = -45

from (2)
(sin/cos) t = -1
tan t = -1
t = -45

so my t is 45?

#### Subhotosh Khan

##### Super Moderator
Staff member
you mean like this?

-3 sint = 3 cost .......................(1)
4 cost = -4 sint ........................(2)

from (1)
(sin/cos) t = -1
tan t = -1
t = -45

from (2)
(sin/cos) t = -1
tan t = -1
t = -45

so my t is 45?
NO!

#### ncchieh

##### New member

NO!

I still cannot understand, i never done a question like this before, from my understanding in #7 is i use the x axis of vector A = x axis of vector B and same for y axis, from this, make a
sin(t) = cos(t) format of equation and use this equation to find t like i did it in #8

#### Subhotosh Khan

##### Super Moderator
Staff member

I still cannot understand, i never done a question like this before, from my understanding in #7 is i use the x axis of vector A = x axis of vector B and same for y axis, from this, make a
sin(t) = cos(t) format of equation and use this equation to find t like i did it in #8
Your work in #8 is very sloppy. You write:

from (1)
(sin/cos) t = -1

That is not a mathematical expression.

Moreover you write:

tan t = -1
t = -45

What does negative time (=-45) mean? That cannot be correct. Do the arithmetic carefully and correctly!

What is the unit of 't'?

What is the value of "t" - if you measure it in radians?

I was trying to make you realize that the question as written is flawed!

#### ncchieh

##### New member
Your work in #8 is very sloppy. You write:

from (1)
(sin/cos) t = -1

That is not a mathematical expression.

Moreover you write:

tan t = -1
t = -45

What does negative time (=-45) mean? That cannot be correct. Do the arithmetic carefully and correctly!

What is the unit of 't'?

What is the value of "t" - if you measure it in radians?

I was trying to make you realize that the question as written is flawed!

My friend tell me the formula should be Va = -Vb
So my answer should be t = 45

Then the question ask me to find t with seconds as unit
but the outcome from t = 45 is in radian unit

so if i use 45 as my answer for t, my t will be 45 radian but not second
Then I need to confirm with my teacher about the question want to to find t in second or in radian form

U mean this?

#### Dr.Peterson

##### Elite Member
I would expect t to be in radians (that is, the functions are to be evaluated taking their inputs in radians).

Therefore $$\displaystyle t = \tan^{-1}(1) = \frac{\pi}{4}$$ seconds, not 45 seconds.