The Value (5p)/4 is a solution for the equation 3v2csc?+5=-1 true or false?

Question

The Value (5π)/4 is a solution for the equation 3√2cscθ+5=-1

 

true or false 

 

Please explain this to me?

 

there is no solution to the equation secx=0

true or false? 

Answer 1

Okay-- they’re both true.

 

The first problem is kinda long, so let’s save that for last. Here’s the second question:

 

Sec(x) = 0 True or false? Let’s see.

 

• A secant is 1/cos x.
• Ask yourself: what could (cos x) be equal to? Between -1 and 1, right? That means a sec is really 1, over a number between -1 and 1.
• So start plugging in some numbers: you’ll find that the closer the denominator is to 0, the BIGGER the secant gets. For example, put ½ in the denominator, and the answer is 2, put ¼ in and the answer is 4!
• So to find the closest secant to 0, we want to find the denominator which is as far from 0 as possible. Well, cos(x) can’t be bigger than 1 or smaller than -1, like we said, so let’s take those and use them as our denominator. We get: 1/1 = 1 OR 1/-1 = -1

 

So there you have it – sec( x) cannot be smaller than 1 or bigger than -1. It never gets any closer to 0 than that. So sec(x) = 0 is impossible.

 

 

Okay, next:

 

3sqrt(2) * csc(x)+5 = -1 True?

 

First let’s do some algebra, getting csc(x) on one side, like you would any other algebra problem.
Move the 5 over by subtracting it from both sides:

 

3sqrt(2) * csc(x)+5 = -1
                         -5    -5
3sqrt(2) * csc(x) = -6

 

Then, move the 3sqrt2 over by dividing it from both sides:

 

Csc(x) = -6/3sqrt2

 

Simplify this:
-2/sqrt 2.

 

A number divided by its square root is its square root, so simplify again:

 

Csc(x) = -sqrt(2)

 

So far so good?

 

Okay, now let's read the question again: is csc(5pi/4)=- sqrt2?

Any calculator would tell you that this is true, and we can technically stop here. But why is that so?

 

We know that Csc = 1/sin

 

So you can exchange csc with 1/sin, in the equation:

 

1/sin (5pi/4) = -sqrt(2)

 

Flip both sides to get the following:

 

Sin 5pi/4 = -1/sqrt(2)

 

We can stop here, consult a sin graph and verify. Or we can go further:

 

-1/sqrt (2) may sound more familiar to you if you rephrase it by multiplying the whole thing by sqrt(2)/sqrt(2), to get Sqrt(2)/2.

 

Sqrt(2)/2 ! This is one of the basic, memorized values of sin or cos! It is the result for sin pi/4.

Now let’s think unit circle: pi/4 is in the 1st quadrant, so sin(pi/4) is positive. When you take the corresponding angle in the 2nd quadrant, you have 3pi/4, which also gets you sqrt(2)/2.

Now the final step: flip the angle across the 2nd quadrant into the 3rd quadrant – you get 5pi/4 ! But since we are in the 3rd quadrant, now, sin is negative.

THEREFORE, sin(5pi/4)=-sqrt(2). Put all the numbers back where they belong, and you end up with your original equation.