Question

How do you find the value of $\sec (315)?$

Answer 1

Hint: In the question, it is not given that the argument of the secant function is either in degrees or in radians, so better solve by taking both the units. In degree unit case, subtract the given argument with ${360^0}$ and in radians case, divide the argument with $2\pi $ and take the remainder as the new argument and then find the value.

Complete step by step answer:
In order to find the value of $\sec (315)$, we will divide it into two cases on the basis of its unit, i.e. degrees and radians (since unit of angle is not mentioned in the question, so we will take both of them).
Case I: Taking degree as the unit of the angle,
In this case, since degree is the unit so we can also write it as $\sec \left( {{{315}^0}} \right)$
Now to find the value, since the angle is lying in the fourth quadrant so we will subtract it from ${360^0}$ and take the result as the new argument,
$
  \sec \left( {{{315}^0}} \right) = \sec \left( {{{360}^0} - {{315}^0}} \right) \\
   = \sec \left( {{{45}^0}} \right) \\
   = \sqrt 2 \\
 $
Case II: Taking radians as the unit of angle,
Here to find the principal argument of the given argument we will divide the given angle with $2\pi $ and take the remainder as the new argument,
$315 \div 2\pi = 50\pi ,\;{\text{and}}\;0.84$ as the remainder,
Now, we will take $0.84$ as the new argument,
We will get $\sec (0.84)$
But we cannot calculate this value without a calculator, but we can find an approximate value,
Since $0.84$ lies somewhere between $\dfrac{\pi }{6}\;{\text{and}}\;\dfrac{\pi }{3}$ so we can take its approximate value as $1.49$

Note: In both the cases we have calculated the value of secant according to the quadrant in which the argument is lying, so that we have given it a positive or negative sign according to its argument. So you should also take care of the quadrant.