An angle theta satisfies the relation (csc theta)(cos theta)=-1
Give an example of information needed to determine a unique value for theta
If sec theta is known to be negative, what is the csc theta value of theta?
cscØ cosØ = -1
(1/sinØ)(cosØ) = -1
cotØ = -1
tanØ = -1, I know tan45° = +1
so Ø is in quadrants II or IV
Ø = 180-45 = 135°
or
Ø = 360-45 = 315°
so the additional information would have to be:
in which quadrant is the angle in.
sinØ = 1/√2 if Ø = 135
sinØ = -1/√2 if Ø = 315
so cscØ = √2 in II and cscØ = -√2 in IV
An angle theta satisfies the relation (csc theta)(cos theta)=-1
Give an example of information needed to determine a unique value for theta
If sec theta is known to be negative, what is the csc theta value of theta?
To determine a unique value for theta, we need to know the quadrant in which it lies. The quadrant can be determined by the signs of the trigonometric functions involved in the relation.
If sec theta is known to be negative, it means that theta lies in either the second or third quadrant. In these quadrants, cos theta is also negative.
Using the relation (csc theta)(cos theta) = -1, we can rearrange it to solve for csc theta:
csc theta = -1 / cos theta
Since cos theta is negative, we have:
csc theta = -1 / (-cos theta)
csc theta = 1 / cos theta
Therefore, if sec theta is known to be negative, the csc theta value of theta is equal to 1 / cos theta.
To find a unique value for theta that satisfies the given relation, we can start by manipulating the equation:
(csc theta)(cos theta) = -1
First, we can rewrite csc theta in terms of sine function:
(1/sin theta)(cos theta) = -1
Next, we can multiply both sides by sin theta to eliminate the fraction:
cos theta = -sin theta
Now, we have an equation relating cosine and sine. To find a unique value for theta, we need to have additional information.
Example of information needed to determine a unique value for theta:
1. The interval in which theta lies: Since both sine and cosine functions are periodic, providing the interval would restrict the possible values of theta and make it unique. For example, if we are given that theta is between 0 and 360 degrees or between 0 and 2π radians, we can find the exact value of theta that satisfies the equation.
Now, let's move on to the second part of your question:
If sec theta is known to be negative, what is the csc theta value of theta?
If sec theta is negative, it means that cos theta is negative. From the equation cos theta = -sin theta, if cos theta is negative, then sin theta must also be negative.
Since csc theta is the reciprocal of sin theta, which is negative in this case, csc theta would also be negative.
So, if sec theta is known to be negative, the csc theta value of theta would also be negative.