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mathsAdvanced TrigonometryTrigonometric Ratio for Angles in All Quadrants

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?trigonometric ratios of triangles equivalently in 1st quadrant

  • 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?trigonometric ratios of angles equivalently in 1st quadrant

  • 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)` trigonometric ratios of angles equivalently in 1st quadrant 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. trigonometric ratios of angles equivalently 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. trigonometric ratio of all quadrants angle with x axis 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. trigonometric ratio of all quadrants angle with y axis 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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