Angles in Standard Position :
An angle is said to be in standard position, if its vertex is at the origin and its initial side is along the positive x-axis. An angle is said to be in the first quadrant, if in the standard position, its terminal side falls in the first quadrant.
Similarly, we can define for the other three quadrants.
Angles in standard position having their terminal sides along the x-axis or y-axis are called quadrantal angles.
Thus, 0°, 90°, 180°, 270° and 360° are quadrantal angles.
The degree measurement of a quadrantal angle is a multiple of 90°.
(90° - θ) -------> I st Quadrant
(90° + θ) and (180° - θ) -------> II nd Quadrant
(180° + θ) and (270° - θ) -------> III rd Quadrant
(270° + θ), (360° - θ) and (- θ) -------> IV th Quadrant
The trigonometric ratios sinθ, cosθ, tanθ, cscθ, secθ and cotθ will have different signs (positive or negative) based the quadrant where the the angle θ falls.
It can be easily remembered by ASTC rule.
This is nothing but "all sin tan cos" rule in trigonometry.
The "all sin tan cos" rule can be remembered easily using the following phrases.
"All Sliver Tea Cups"
"All Students Take Calculus"
ASTC formla has been explained clearly in the figure shown below.
In the first quadrant (0° to 90°), all trigonometric ratios are positive.
In the second quadrant (90° to 180°), sin and csc are positive and other trigonometric ratios are negative.
In the third quadrant (180° to 270°), tan and cot are positive and other trigonometric ratios are negative.
In the fourth quadrant (270° to 360°), cos and sec are positive and other trigonometric ratios are negative.
When we have the angles 90° and 270° in the trigonometric ratios in the form of
(90° + θ)
(90° - θ)
(270° + θ)
(270° - θ)
We have to do the following conversions,
sin θ <------> cos θ
tan θ <------> cot θ
csc θ <------> sec θ
sin (270° + θ) = - cos θ
cos (90° - θ) = sin θ
For the angles 0° or 360° and 180°, we should not make the above conversions.
After having gone through the stuff given above, we hope that the students would have understood angles in standard position. ,
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