If [tex]$\sin \theta\ \textgreater \ 0$[/tex] and [tex]$\cos \theta\ \textgreater \ 0$[/tex], then the terminal point determined by [tex]$\theta$[/tex] is in:
A. quadrant 4.
B. quadrant 2.
C. quadrant 1.
D. quadrant 3.
Answer (1)
Sine is positive and cosine is positive.
Recall the signs of sine and cosine in each quadrant.
Determine the quadrant where both sine and cosine are positive.
The terminal point determined by θ is in quadrant 1. C
Explanation
Analyze the problem We are given that 0"> sin θ > 0 and 0"> cos θ > 0 . We need to determine which quadrant the terminal point determined by θ lies in. Let's recall the signs of sine and cosine in each quadrant.
Recall the signs of trigonometric functions in each quadrant In quadrant 1, both sin θ and cos θ are positive. In quadrant 2, sin θ is positive and cos θ is negative. In quadrant 3, both sin θ and cos θ are negative. In quadrant 4, sin θ is negative and cos θ is positive.
Determine the quadrant Since we are given that 0"> sin θ > 0 and 0"> cos θ > 0 , the terminal point determined by θ must be in quadrant 1.
Examples
Understanding the signs of trigonometric functions in different quadrants is crucial in various fields, such as navigation and physics. For example, when analyzing the motion of a projectile, knowing the quadrant of the angle of launch helps determine the signs of the horizontal and vertical components of the initial velocity. If a projectile is launched at an angle in the first quadrant, both the horizontal and vertical components of its initial velocity are positive, indicating that the projectile is moving both upwards and to the right.
