Question

Find the value of trigonometric identity for the angle given$\sec (225)$?

Answer 1

The basic values that we already know are for zero, thirty, forty five, sixty and ninety degree. If any bigger angle asked in question then we have to break that angle in multiple of these known angle and if the given angle is not a perfect multiple of these angle then you can first make the highest multiple possible with the known angles and rest can be written as in summation of the product you obtained after multiplication.

Complete step by step solution:
Given question is $\sec 225$
The given trigonometric identity follows $2n\pi \,or\,\dfrac{{3n\pi }}{2}$ rule, this rule states that when any angle is exact multiple or greater than exact multiple then the given angle should be break down in this form and then further solve it for small angles.

Now we know $\cos \theta$ follows either $2n\pi \,or\,\dfrac{{3n\pi }}{2}$ and $\cos \theta = \dfrac{1}{{\sec \theta }}$and reciprocal is for our case

So we have change in equation in $\cos \theta$ so after conversion we get, $225$ can be written as
$(\pi + 45)$
Now put this angle in trigonometric identity we get
$\cos (\pi + 45)$
$\Rightarrow - \cos (45),\,[\cos (\pi + \theta ) = - \cos \theta ]$
$\Rightarrow - \dfrac{1}{{\sqrt 2 }}$
Now we know, $\cos \theta = \dfrac{1}{{\sec \theta }}$
So, $\sec (225)$ value is $- \sqrt 2 \,or\, - 1.414$

Formulae Used: $\cos \theta = \dfrac{1}{{\sec \theta }}$, $[\cos (\pi + \theta ) = - \cos \theta ]$,
$2n\pi \,or\,\dfrac{{3n\pi }}{2}$ rule for $\cos \theta$

Additional Information: While dealing with bigger angles you should always be aware of the breaking of angles in smaller forms because only smaller angles starting from zero, thirty, forty five, sixty and ninety can be easily remembered.

Note: Every trigonometric identity a rule while conversion to its smaller form and you should all convert the angles easily, rules state the plus and minus sign after conversion the angle.