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Thought-Process to Discover Knowledge

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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.

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mathsTrigonometryBasics of Trigonometry

Basics of Angles

In this page, the basics of Angles required to understand trigonometry is revised.



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Before starting with Trigonometry, let us revise the basics required to understand that.

What is an angle?definition of angle

  • measure of rotation between two lines
  • measure of rotation between two lines
  • length of a line

The answer is 'measure of rotation between two lines'

What is an angle for a given line in a coordinate system?angle of a line

  • angle is not defined for a single line
  • the angle from the x-axis to the line
  • the angle from the x-axis to the line
  • the angle from the y-axis to the line

The answer is 'the angle from the x-axis to the line'

Which of the following is a measure of an angle?

  • degree
  • radian
  • both the above
  • both the above

The answer is 'both the above'.

What is the definition of the measure "degree"?definition of degree

  • `1000` km = `1` degree
  • Rotation by a full circle is graded into `360` degrees
  • Rotation by a full circle is graded into `360` degrees

The answer is 'Rotation by a full circle is graded into `360` degrees'.

From the figure, one should understand.

 •  The angles are measured with respect to the +ve x-axis.

 •  The graded angles are represented for `20^@, 40^@, 60^@, cdots`.

 •  A line making an angle `30^@` is shown.

What is the definition of the measure "radian"?definition of radian

  • `1` radian = `1` degree
  • `1` radian = `10` degree
  • `1` radian = angle subtended by the arc of length equaling the radius
  • `1` radian = angle subtended by the arc of length equaling the radius

The answer is '`1` radian = angle subtended by the arc of length equaling radius'.

From the figure, one should understand

 •  radius of the circle is shown in red dotted line. An arc having length equal to radius of the circle is shown in red dotted arc.

 •  The radian measures `1, 2, 3` radians are marked

 •  The y-axis at `90^@` angle is at `pi/2` radian.

 •  the -ve x-axis at `180^@` angle is at `pi` radian.

 •  the full-circle subtends `2pi` radian or `360^@` angle.

 •  line making `30^@` angle is shown and the angle in radians is `pi/6`.

Given the magnitude of angle between x-axis and line `bar(OP)` is `30^@`, what is the given angle?positive negative angle of a line

  • `+ 30^@`
  • `+ 30^@`
  • `-30^@`

The answer is '`+30^@`'. An angle is positive if it is measured in counter-clockwise direction.

Given the magnitude of angle between x-axis and line `bar(OQ)` is `60^@`, what is the given angle?positive negative angle of a line

  • `+60^@`
  • `-60^@`
  • `-60^@`

The answer is '`-60^@`'.

The angles are measured in counter-clockwise direction. When a given angle is measured in clockwise direction, it is negative.

What is the measure of angle `/_PQR`?angles positive and negative

  • `+30^@`
  • `-30^@`
  • `-30^@`

The answer is '`-30^@`'. The angle is measured from `bar(PQ)` to `bar(QR)`, in clockwise direction.

What is a "right angle"?right angle

  • Angle at `90^@`
  • Angle at `pi/2` radian
  • both the above
  • both the above

The answer is 'both the above'.

`90^@ = pi/2` radian

What is a "straight angle"?straight angle

  • Angle at `180^@`
  • Angle at `pi` radian
  • both the above
  • both the above

The answer is 'both the above'.

What are "Complementary Angles"?complementary angles

  • Pair of angles having sum `90^@`
  • One angle that completes right angle with another angle
  • both the above
  • both the above

The answer is 'Both the Above'.

The word "complementary" is derived from the word "complete". When one angle completes the right angle with another angle, the angles are called complementary angles.

What are "Supplementary Angles"?supplementary angles

  • Pair of angles having sum `180^@`
  • One angle that supplements(comes as an addition) with another angle in a straight angle
  • both the above
  • both the above

The answer is 'Both the Above'.

The word "supplement" means "comes as an addition".

Basics of Angles to learn trigonometry :

 •  Angles measured in degrees and radians.

 •  Right angle and Straight Angle.

 •  Complementary and Supplementary angles.

Angle: For two line segments or rays with common initial point, the amount of rotation in counter clockwise direction from one to another is the angle.

Degree: Rotation by a full circle is graded as `360^@` and angles are measured in that graded scale.

Radian: Rotation by the length of radius along the arc of a circle, is `1` radian. Angles are measured in proportion to the `1` radian measure.

Right Angle: Angle of `90^@` or `pi/2` radian.

Straight Angle: Angle of `180^@` or `pi` radian.

Complementary Angles: Pair of angles having sum `90^@`.

Supplementary Angles: Pair of angles having sum `180^@`.

Solved Exercise Problem:

Which of the following is correct for the two angles `52^@` and `38^@`?

  • Right angle
  • Straight angles
  • Complementary angles
  • Complementary angles
  • Supplementary angles

The answer is 'Complementary angles'

                            
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