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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 the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 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 coordinate 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 = 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 90+theta,

•  the angle is in 2nd quadrant, so x projection is negative and y projection is positive.

•  the /_theta is measured with y-axis, and so the similar triangle in 1st quadrant will have /_theta with x-axis, thereby swapping the x and y projections.

the trigonometric ratios are:  • sin (90+theta) = cos theta

• cos (90+theta) = - sin theta

• tan (90+theta) = - cot theta

The angle omega = 180-theta is shown in the figure as point P. The similar triangle with /_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

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