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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.

The results of representing trigonometric ratios of angles in 4th quadrant equivalently as trigonometric ratios of acute angle in 1st quadrant are explained.

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The angle omega = 90-theta is shown in the figure as point P. The similar triangle with angle theta is given by Q(x,y) in 1st quadrant. Note: Given Q(x,y), the coordinates of P is (color(coral)(P_x),color(deepskyblue)(P_y)):
color(coral)(P_x=y) and color(deepskyblue)(P_y=x)

What is tan (90-theta)?

• y/x = tan theta
• y/(-x) = - tan theta
• x/y= cot theta
• x/y= cot theta
• (-x)/y = - cot theta

The answer is 'x/y = cot theta'

tan (90-theta) (tan of the given angle)
quad = (P_y)/(P_x) (by definition of tan)
quad = y/x (substituting the values)
quad = cot theta (equivalently in 1st quadrant.)

Note: learners can work out this for sin and cos.

For angles given as 90-theta, the trigonometric ratios are:  • sin (90-theta) = cos theta

• cos (90-theta) = sin theta

• tan (90-theta) = cot theta

For angles measured in reference to x axis, the trigonometric ratios remain the same when equivalently represented in the 1st quadrant.

For angles measured in reference to y axis, the trigonometric ratios are equivalently complementary ratios in the 1st quadrant.

For Angles in reference to x-axis:
sin(180+-theta) = -+sin theta
cos(180+-theta) = -cos theta
tan(180+-theta) = +-tan theta

sin(-theta) = -sin theta
cos(-theta) = cos theta
tan(-theta) = -tan theta

For Angles in reference to y-axis:
sin(90+-theta) = cos theta
cos(90+-theta) = -+sin theta
tan(90+-theta) = -+cot theta

sin(270+-theta) = -cos theta
cos(270+-theta) = sin theta
tan(270+-theta) = -tan theta

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