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Thought-Process to Discover Knowledge

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### First Quadrant Equivalent of Trigonometric Values

For angles in 2nd, 3rd, and 4th quadrants, can trigonometric ratios be equivalently given as a trigonometric ratio of acute angle in 1st quadrant? In doing so, the sign and the complementary trigonometric ratios are to be appropriately matched.

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We studied about angles that are in 2nd, 3rd, or 4th quadrants. Before that, We have studied about trigonometric ratios for standard angles 0, 30, 45, 60, 90 degree, which are all in 1st quadrant.

Are there more standard angles in the 2nd, 3rd, or 4th quadrants? Should one learn the values of such angles?

Or can the trigonometric ratios of angles from 2nd, 3rd, and 4th quadrants be equivalently represented with trigonometric ratios of angles from 1st quadrant only. If such transformation is possible, then it simplifies.

Note: For any angle in 2nd, 3rd, and 4th quadrants, trigonometric ratios can be equivalently given as a trigonometric ratios in 1st quadrants. The objective of this page is to learn that.

For a given /_\ ABC, which of the following defines trigonometric ratios?

• The hypotenuse and sides of the triangle define the ratios
• The line at given angle theta on unit circle forms a similar triangle and ratios can be equivalently given in that.
• Both the above
• Both the above

The answer is 'Both the above'

For a given /_ omega, which of the following defines trigonometric ratios?

• the projections on x and y axis
• The line in 1st quadrant forming /_\ OQ′Q which is a similar triangle to /_\ OP′P
• both the above
• both the above

The answer is 'Both the above'

Consider the given /_ omega.

Note the following for /_omega the point on unit circle is P and its x and y projections are P_x and P_y.
The angle bar(OP) makes with x axis is alpha and with y axis is beta.
Two triangles are constructed in the 1st quadrant.
For the angle alpha the similar triangle in 1st quadrant is /_\ OR_xR.
For the angle beta the similar triangle in 1st quadrant is /_\ OQ_xQ
Note that bar(OR_x) = bar(OP_x) and bar(OQ_x) = bar(OP_y) Which of these two triangles can be used to find the trigonometric ratios for point P?

• /_\ OR_xR as the x and y projections match with that of P.
• /_\ OQ_xQ can be used knowing that the x and y projections are interchanged in the 1st quadrant.
• both the above.
• both the above.

The answer is 'both the above'

For any angle /_omega there are two possible similar triangles in 1st quadrant. First is based on angle made with x-axis, given as /_alpha, in which case the x and y projections are retained in magnitude in the 1st quadrant. So the trigonometric ratios are retained in the 1st quadrant.
Second is based on angle made with y-axis given as /_beta, in which case the x and y projections are interchanged in magnitude in the 1st quadrant. So the trigonometric ratios are given for complementary angles.
It is also noted that /_alpha and /_beta are complementary angles.

For the /_omega, the angle with x-axis /_alpha is used to construct the similar triangle in the 1st quadrant. The sign of the trigonometric ratio is derived from the signs of P_x and P_y. What is the magnitude of sin omega in terms of alpha?

• sin alpha
• sin alpha
• cos alpha

The answer is 'sin alpha'

The trigonometric ratios are

• sin omega
quad quad = text((sign of y proj.)) xx sin alpha

• cos omega
quad quad = text((sign of x proj.)) xx cos alpha

• tan omega
quad quad = text(sign of y proj.)/ text(sign of x proj.) xx tan alpha

For the /_omega, the angle with y-axis /_ beta is used to construct the similar triangle in the 1st quadrant. The sign of the trigonometric ratio is derived from the signs of P_x and P_y. What is the magnitude of sin omega in terms of beta?

• sin beta
• cos beta
• cos beta

The answer is 'cos beta'

The trigonometric ratios are

• sin omega
quad quad = text((sign of y proj.)) xx cos beta

• cos omega
quad quad = text((sign of x proj.)) xx sin beta

• tan omega
quad quad = text(sign of y proj.)/ text(sign of x proj.) xx cot beta

The trigonometric ratios of /_omega can equivalently be represented with one of the complementary angles /_alpha and /_beta in 1st quadrant. Where /_alpha is the angle with x-axis and /_beta is the angle with y-axis.

The sign of the trigonometric ratios are to be derived from the original quadrant of /_omega

The /_alpha maps to equivalent trigonometric ratios as the x and y projections are retained in x and y axes.

The /_beta maps to complementary trigonometric ratios as the x and y projections are swapped in 1st quadrant.

Equivalent Ratio in 1st Quadrant: Any angle /_omega is equivalently considered with either of
•  angle made with x-axis /_alpha
•  angle made with y-axis /_beta
The equivalent trigonometric value is given in first quadrant.

The trigonometric ratios are

• sin omega
quad quad = text((sign of y projection)) xx sin alpha
quad quad = text((sign of y projection)) xx cos beta

• cos omega
quad quad = text((sign of x projection)) xx cos alpha
quad quad = text((sign of x projection)) xx sin beta

• tan omega
quad quad = text(sign of y proj.)/ text(sign of x proj.) xx tan alpha
quad quad = text(sign of y proj.)/ text(sign of x proj.) xx cot beta

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