We study six Trigonometric Functions namely Tangent, sine, cosine, cotangent, cosecant and secant. Last three Functions are multiplicative inverse of other three functions. Here we will see how to find secant of 3pi by 4 or 1350, first we need to understand four quadrants. 1350 angle lies in second quadrant where only sine function is positive. Cosine and secant functions possess negative and positive values in second and the third quadrants respectively. So to make our angle represent first quadrant angle where all functions are positive, we need to subtract desired angle from 1800 or pi. So our angle i.e. 1350 can be written as (pi – 45)0.
Next we need to understand is multiplicative inverse relationship between Trigonometric Functions given as:
sec a * cos a = 1............. equation (1).
cosec a * sin a = 1,
cot a * tan a = 1,
Let us now calculate the value of secant of 3pi / 4. For this we first need to evaluate cosine of 3pi / 4 as:
cos 135 = cos (pi – 45) [As 1350 can be written as 1800 – 450]. Applying the formula for cos (A – B) as: cos (A – B) = cos A cos B + sin A sin B.
So, cos (pi – 45) can be written as: cos pi cos 45 + sin pi sin 45 = - cos 45 = -1 / (√ 2).
Using the property stated in equation 1 we can write:
sec 135 * cos 135 = 1,
or sec 135 = 1 / cos 135,
or sec 135 = -√ 2,
Thus value of secant of 3pi by 4 comes out to be -√2 which lies in Domain of Secant function.