Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 20, 2023

Find quadrant of $\theta$ when $\sin \theta > 0$ and $\sec \theta > 0$ .

STEP 1

Assumptions1. The functions $\sin \theta$ and $\sec \theta$ are defined for all real numbers $\theta$ except where $\cos \theta =0$ . . The quadrant of $\theta$ is determined by the signs of $\sin \theta$ and $\cos \theta$ .
3. $\sin \theta >0$ implies that $\theta$ is in either Quadrant I or Quadrant II.
4. $\sec \theta >0$ implies that $\cos \theta >0$ (since $\sec \theta = \frac{1}{\cos \theta}$ ), which means that $\theta$ is in either Quadrant I or Quadrant IV.

STEP 2

We need to find the quadrant where both $\sin \theta >0$ and $\sec \theta >0$ are true. This is the intersection of the quadrants determined in Assumption and Assumption4.

STEP 3

The only quadrant where both $\sin \theta >0$ and $\sec \theta >0$ (or equivalently $\cos \theta >0$ ) is Quadrant I.
So, $\theta$ lies in Quadrant I.

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