Question

Why trigonometric ratios of Standard Angles can't be negative?

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Answer 1

Answer:

The only difference is that now x or y (or both) can be negative because our angle can now be in any quadrant. It follows that the trigonometric ratios can turn out to be negative or positive. In the earlier section, the angles involved were always less than 90° so all 6 ratios were positive.

Answer 2

Answer:

First we need to understand what we mean by standard angles. Generally, angles of 0°, 30°, 45°, 60° and 90° are called standard angles. these angles fall in the first quadrant of the circle. In fact trigonometrical ratios for any angle between 0° and 90° will be positive because of the following reason

trigonometrical ratios are defined as ratios of sides of a right angle triangle.

sine is defined as ratio of side opposite to angle (called perpendicular also) to hypotenuse

cosine is defined as ratio of side adjacent to angle (called base also) to hypotenuse

tangent is defined as ratio of side opposite to angle (perpendicular) to side adjacent to angle (base)

now, this right angle triangle may fall in any of the four quadrants depending upon the angle.

the angle is generally measured with positive x axis anticlockwise.

now, in first quadrant, base lies along positive x axis, perpendicular lies along positive y axis and hypotenuse which is sum of square of the two perpendicular sides is always positive no matter in which quadrant it is.

since, all the sides used to determine trigonometrical ratios in the first quadrant are positive, the ratios are also positive.

this way we can easily decide the sign of a trigonometrical ratio in any quadrant.