Solveeit Logo
Login

Question

Question: If \[\sin \theta = - \dfrac{1}{{\sqrt 2 }}\] and \[\tan \theta = 1\] then \[\theta \] lies in which ...

If sin⁡θ=−12\sin \theta = - \dfrac{1}{{\sqrt 2 }} and tan⁡θ=1\tan \theta = 1 then θ\theta lies in which quadrant ?
A. First quadrant
B. Second quadrant
C. Third quadrant
D. Fourth quadrant

Explanation

Solution

To solve this problem, we need basics of trigonometry and the knowledge of trigonometric functions. As we should know in which quadrant which function is positive like sin is positive in the first and second quadrant but negative in the third and fourth quadrant. For tan also we can say it is positive in the first and third quadrant and negative in the remaining 2 quadrants. So applying that knowledge we will find the answer.

Complete step by step answer:
Given is sin⁡θ=−12\sin \theta = - \dfrac{1}{{\sqrt 2 }}
We know that sin is negative in the third and fourth quadrant.
Also we know that, sin⁡5π4=−12\sin \dfrac{{5\pi }}{4} = - \dfrac{1}{{\sqrt 2 }}
Second is tan⁡θ=1\tan \theta = 1. We know that tan is positive in the first and third quadrant.
Thus the common quadrant for sin and tan function is given as the third quadrant.

Thus the correct option is C.

Note: it is highly important for us to know in which quadrant which function is positive and which one is negative. For that we should rectify the ASTC rule. Where A stands for all. SS stands for sin, TT stands for tan and CC stands for cos. This shows that in the first quadrant all trigonometric functions are positive.In the second quadrant only sine and cosec functions are positive.In the third quadrant only tan and cot functions are positive.In the second quadrant only cos and sec functions are positive.