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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**.

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

think,

figure-out, &

learn.

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

think,

figure-out, &

learn.

To make best use of nubtrek, understand what is available.

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

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exercise.

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Home

Origin of Trigonometric Ratios

» An angle `theta` specifies a class of similar-right-triangles

» a side narrows down to a specific right-triangle

» Given an angle and a side "How to compute other sides?"

» Ratio of sides of similar triangles for an angle `theta`

→ `text(side1) / text(side3) = text(constant1)`

→ `text(side2) / text(side3) = text(constant2)`

→ `text(side1) / text(side2) = text(constant3)`

» Any parameter of right-triangles (sides and angles) can be calculated using the ratios : **Trigonometric Ratios**.

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Angle defines set of similar right angled triangles. Providing one of the sides narrows it down to a specific right angled triangle.

Trigonometric Ratios for an angle are defined as ratio of sides of Right Angled Triangle.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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In this page, trigonometric ratios are defined for right angled triangles.

Starting on learning "Trigonometric Ratios". ;; In this page, trigonometric ratios are defined for right angled triangles.

We have learned

• Many applications are mathematically modeled using the sides of right angled triangles.

• the set of similar right angled triangles are defined by an angle.

What parameter of triangles are used in applications?

- measure of angles
- length of sides
- both the above

The answer is 'both the above'. Though the set of similar right angled triangles is specified by an angle, to narrow down to a specific right angled triangle, one of the sides is specified.

A right angled triangle is specified by

• an angle - that specifies the set of similar right angled triangles

• length of one of the three sides of the triangle

When using this, it may be required to find the length of other sides of the triangle as the specified side may not be the one required in the calculations.

In a right angled triangle, When an angle and one of the sides is specified, what does one require to compute the other sides of the triangle? `a` and `theta` are given. Need to compute `b` and `c`.

- Other sides cannot be computed
- Given one angle, the ratio of two sides is a constant

The answer is 'Given one angle, the ratio of two sides is a constant'

In the figure

• `a/c = m/o = p/r =` constant

• `b/c = n/o = q/r =` a different constant

• `a/b = m/n = p/q =` another constant

When specifying the ratio of the sides, two sides out of the three sides of the triangles are used. It can either be `a/c`, or `b/c` or `a/b`. How do you identify which sides are used in a given ratio?

- Give the sides specific names with respect to the specified angle
- Give the sides specific names with respect to a the right angle
- both the above

The answer is 'both the above'

Given set of right angled triangles defined by an angle `theta`.

• position of right angle is chosen.

• the position of the given angle is chosen • the side opposite to the right angle is fixed and called "hypotenuse"

• the side opposite to the given angle is fixed and called "opposite side"

• the side adjacent to the given angle is fixed and called "adjacent side"

If the given angle is chosen differently... • the side opposite to the right angle is "hypotenuse"

• the side opposite to the given angle is "opposite side"

• the side adjacent to the given angle is "adjacent side"

In a right angled triangle, When an angle and one of the sides is specified, the ratio of length of sides is constant.

• Ratio between opposite side and hypotenuse is required if hypotenuse is given and opposite side is to be computed (or vise versa).

• Ratio between adjacent side and hypotenuse is required if hypotenuse is given and adjacent side is to be computed (or vise versa).

• Ratio between opposite side and adjacent side is required if opposite side is given and adjacent side is to be computed (or vice versa).

How do we refer to which ratio we are using in the calculations?

- Give each of these a name and refer by the name.
- Describe each whenever the ratio is to be used.

The answer is 'Give each of these a name and refer by the name.'

For a set of similar right angled triangle specified by angle `theta`, the following ratios are defined. These are some of the trigonometric ratios Trigonometric Ratios:

`sin theta = text(opposite)/text(hypotenuse)`

`cos theta = text(adjacent)/text(hypotenuse)`

`tan theta = text(opposite)/text(adjacent)`

In the next set of pages, why the ratios are named as `sin`, `cos`, and `tan` is explained. Pl. continue...

What does 'trigonometry' mean?

- It is just a name.
- `color(coral)(text(trigon)) + color(deepskyblue)(text(metry)),``text( means) color(coral)(text(triangle)) + color(deepskyblue)(text(measurement procedures))`

Answer is '`color(coral)(text(trigon)) + color(deepskyblue)(text(metry)),``text( means) color(coral)(text(triangle)) + color(deepskyblue)(text(measurement procedures))`'

Note that when `sin A` is written, it does not mean four variables `s`, `i`, `n` and `A`. The same can be explained for `cos A` and `tan A`

`sin A` is a function of variable `A`.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Specifying Right Angled Triangle: ** Set of similar right angled triangles are defined by one parameter: "angle". And to narrow down to one specific triangle, one side is additionally specified.

**Trigonometric Ratios: **

`sin theta = text(opposite)/text(hypotenuse)`

`cos theta = text(adjacent)/text(hypotenuse)`

`tan theta = text(opposite)/text(adjacent)`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Say `sin` to practice the pronunciation.

- Practice Saying the Answer

The `sin` is pronounced as sine.

Say `cos` to practice the pronunciation.

- Practice Saying the Answer

The `cos` is pronounced as cos.

Say `tan` to practice the pronunciation.

- Practice Saying the Answer

The `tan` is pronounced as tan.

*your progress details*

Progress

*About you*

Progress

In the previous pages, We have learned the following. ;; Many applications are mathematically modeled using the sides of right angled triangles. ;; the set of similar right angled triangles are defined by an angle.

What parameter of triangles are used in applications?

measure;angles

measure of angles

length;sides

length of sides

both;above

both the above

The answer is 'both the above'. Though the set of similar right angled triangles is specified by an angle, to narrow down to a specific right angled triangle, one of the sides is specified.

A right angled triangle is specified by ;; an angle - that specifies the set of similar right angled triangles. ;; length of one of the three sides of the triangle. When using this, it may be required to find the length of other sides of the triangle as the specified side may not be the one required in the calculations.

Angle defines set of similar right angled triangles. Providing one of the sides narrows it down to a specific right angled triangle.

Set of similar right angled triangles are defined by one parameter: "angle". And to narrow down to one specific triangle, one side is additionally specified.

In a right angled triangle, When an angle and one of the sides is specified, what does one require to compute the other sides of the triangle?

cannot

Other sides cannot be computed

ratio;constant;one angle

Given one angle, the ratio of two sides is a constant

The answer is 'Given one angle, the ratio of two sides is a constant' ;; In the figure ;; a by c = m by o = p by r = constant ;; b by c = n by o = q by r = a different constant ;; a by b = m by n = p by q = another constant

When specifying the ratio of the sides, two sides out of the three sides of the triangles are used. It can either be a by c, ;; or b by c ;; or a by b . How do you identify which sides are used in a given ratio?

specified angle;names

Give the sides specific names with respect to the specified angle

right angle; respect

Give the sides specific names with respect to a the right angle

both;above

both the above

The answer is ""

Given set of right angled triangles defined by an angle theta. ;; position of right angle is chosen. ;; the position of the given angle is chosen. ;; the side opposite to the right angle is fixed and called "hypotenuse". ;; the side opposite to the given angle is fixed and called "opposite side". ;; the side adjacent to the given angle is fixed and called "adjacent side".

If the given angle is chosen differently... ;; the side opposite to the right angle is "hypotenuse". ;; the side opposite to the given angle is "opposite side". ;; the side adjacent to the given angle is "adjacent side".

In a right angled triangle, When an angle and one of the sides is specified, the ratio of length of sides is constant. ;; Ratio between opposite side and hypotenuse is required if hypotenuse is given and opposite side is to be computed or vise versa. ;; Ratio between adjacent side and hypotenuse is required if hypotenuse is given and adjacent side is to be computed or vise versa. ;; Ratio between opposite side and adjacent side is required if opposite side is given and adjacent side is to be computed or vice versa. How do we refer to which ratio we are using in the calculations?

name;refer

Give each of these a name and refer by the name.

describe;ratio

Describe each whenever the ratio is to be used.

The answer is 'Give each of these a name and refer by the name.'

For a set of similar right angled triangle specified by angle theta, the following ratios are defined. These are some of the trigonometric ratios. ;; sine theta equals opposite by hypotenuse. ;; cause theta equals adjacent by hypotenue. ;; tan theta equals opposite by adjacent. In the next set of pages, why the ratios are named as sin , cos , and tan is explained.

What does 'trigonometry' mean?

just;name

It is just a name.

means;triangle;measurement

color(coral)(text(trigon)) + color(deepskyblue)(text(metry)), text( means) color(coral)(text(triangle)) + color(deepskyblue)(text(measurement procedures))

Answer is triangle and measurement procedures.

Trigonometric Ratios for an angle are defined as ratio of sides of Right Angled Triangle.

trigonometric ratios. ;; sine theta equals opposite by hypotenuse. ;; cause theta equals adjacent by hypotenue. ;; tan theta equals opposite by adjacent

Say sine to practice the pronunciation.

sine

The sine is pronounced as sine.

Say cos to practice the pronunciation.

cause;cos

The cos is pronounced as cos.

Say tan to practice the pronunciation.

tan

The tan is pronounced as tan.

Note that when sine A is written, it does not mean four variables s, i, n and A. The same can be explained for cos A and tan A. ;; sine A is a function of variable A.