Question

Find value of $\cos ec{1305^o}$

Answer 1

Convert it in the convenient form of $\cos ec(n \cdot 360 + \theta )$ and use relevant identities and solve.
First, we try to minimize the angle value of it, by writing it in the form of $\cos ec(360 + \theta )$ and by using $\cos ec(n \cdot 360 + \theta ) = \cos ec\theta$ and then we will have smaller angle value, then we can write it in the form of . Then we will get the simplified value of it, which is the required solution for the problem. This problem is solved through recognizing what type of angle is positive and negative in the four quadrants

Complete step by step solution:
$\cos ec{\text{ }}(180^\circ {\text{ }} + \;\theta ){\text{ }}\; = {\text{ }}\; - {\text{ }}\cos ec{\text{ }}\theta$step by step solution:
First, we try to write it in the form of $\cos ec(n \cdot 360 + \theta )$.
We can write, such that we can use a trigonometric identity
So, on doing this, we will get
$\cos ec{1305^o} = \cos ec(3.360 + 225)$
With the help of identities, we will get
$\cos ec{1305^o} = \cos ec(225)$
Then, the obtained cosec value can be written in the form of $\cos ec{\text{ }}(180^\circ {\text{ }} + \;\theta ){\text{ }}\; = {\text{ }}\; - {\text{ }}\cos ec{\text{ }}\theta$
We can write $225 = 180 + 45$, such that we can use a trigonometric identity
So, we obtain the following
$\cos ec(225) = \cos ec(180 + 45)$
Then based on the identity known to us, this above expression gets converted to
$\cos ec(45)$
$\cos ec(180 + 45) = - \cos ec(45)$
Therefore, we have the direct value for $\cos ec(45)$, we can give it and we have the final value for the required angle.
$\cos ec{1305^o} = - \sqrt 2$.

Note: We have to be careful about the change in the nature of the angle based on the angle and the quadrant in which the angles, the nature of sign also changes accordingly. So, we have to be careful in understanding which quadrant the angle lies and whether the sign changes or not.