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### Trigonometric Ratio for Angles in All Quadrants

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Conversion to 1st Quadrant

»  Conversion of angle omega to 1st quadrant
→  acute angle made by point P with x axis alpha
→  acute angle made by point P with y axis beta

»  Point R makes angle alpha in 1st quadrant.
→  R_x = P_x
→  R_y = P_y
→  x and y coordinates position is retained
→  sin omega = +- sin alpha
→  cos omega = +- cos alpha

»  Point Q makes angle beta in 1st quadrant.
→  Q_x = P_y
→  Q_y = P_x
→  x and y coordinates position is interchanged
→  sin omega = +- cos beta
→  cos omega = +- sin beta

Quickly follow the x and y projection in the quadrant to figure out the conversion. No need to memorize.

### First Quadrant Equivalent of Trigonometric Values

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

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simple steps to build the foundation

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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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Starting on learning "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.

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 Delta 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

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 Delta OQ′Q which is a similar triangle to Delta OP′P
• 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 Delta OR_xR.
For the angle beta the similar triangle in 1st quadrant is Delta 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?

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

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

For a triangle A B C, which of the following defines trigonometric ratios?
hypotenuse;sides
The hypotenuse and sides of the triangle define the ratios
unit;circle;similar;
The line at given angle theta on unit circle forms a similar triangle and ratios can be equivalently given in that.
both;above
Both the above
The answer is 'Both the above'
For a given angle omega, which of the following defines trigonometric ratios?
projection;x;ex;y;why
the projections on x and y axis
the line in 1st quadrant forming a triangle O Q prime Q which is a similar triangle to the triangle O P prime P
both;above
both the above
The answer is 'Both the above'
Consider the given angle omega. Note the following: for angle omega the point on unit circle is P and its x and y projections are P x and P y. ;; The angles line O P 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 triangle O R X R ;; For the angle beta the similar triangle in 1st quadrant is triangle O Q X Q ;; Note that line O R X equals line O P X and line O Q X equals line O P Y. ;; Which of these two triangles can be used to find the trigonometric ratios for point P?
match;point;p;pee
triangle O R X R ; as the x and y projections match with that of point P
interchanged;q;queue;knowing
triangle O Q X Q ; can be used knowing that the x and y projections are interchanged in the first quadrant.
both;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 angle 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 sine omega in terms of alpha?
sin;sine;sign
sine alpha
cos;cause
cos alpha
The answer is 'sine alpha'. the Trigonometric ratios are ;; sine omega equals sign of y projection into sine alpha. ;; cos omega equals sign of x projection into cos alpha. ;; tan omega equals sign of y projection by sign of x projection into tan alpha.
For the angle 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 sine omega in terms of beta?
sine;sign
sine beta
cos;cause
cos beta
The answer is 'cos beta'. the Trigonometric ratios are ;; sine omega equals sign of y projection into cos beta. ;; cos omega equals sign of x projection into sine beta. ;; tan omega equals sign of y projection by sign of x projection into cot beta.
The trigonometric ratios of angle 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 angle alpha maps to equivalent trigonometric ratios as the x and y projections are retained in x and y axes. The angle beta maps to complementary trigonometric ratios as the x and y projections are swapped in 1st quadrant.
The trigonometric ratios of omega can equivalently be represented with one of the complementary angles alpha and beta in 1st quadrant.

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