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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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.continue

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.continue

Welcome to **nub****trek**.

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The content is presented in small-focused learning units to enable you to

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learn.

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

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*summary of this topic*

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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`

*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.

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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.

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

*your progress details*

Progress

*About you*

Progress

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 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

quadrant;triangle;similar;prime

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.