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Thought-Process to Discover Knowledge

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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.

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The results of representing trigonometric ratios of angles in 3rd quadrant equivalently as trigonometric ratios of acute angle in 1st quadrant are explained.

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The angle omega = 180+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=-x) and color(deepskyblue)(P_y=-y)

What is tan (180+theta)?

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

The answer is '-y/-x = tan theta'

tan (180+theta) (tan of the given angle)
quad = (P_y)/(P_x) (by definition of tan)
quad = (-y)/(-x) (substituting the values)
quad = tan theta (equivalently in 1st quadrant.)

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

For angles given as 180+theta, the trigonometric ratios are:  • sin (180+theta) = - sin theta

• cos (180+theta) = - cos theta

• tan (180+theta) = tan theta

The angle omega = 270-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 (270-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 (270-theta) (tan of the given angle)
quad = (P_y)/(P_x) (by definition of tan)
quad = (-x)/(-y) (substituting the values)
quad = cot theta (equivalently in 1st quadrant.)

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

For angles given as 270-theta, the trigonometric ratios are:  • sin (270-theta) = - cos theta

• cos (270-theta) = - sin theta

• tan (270-theta) = cot theta

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