Tag: angle and their measurement

Questions Related to angle and their measurement

The value of $\tan\left(7\dfrac{1}{2}\right)^o$ is 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\dfrac {2\sqrt 2 -(1+\sqrt 3)}{\sqrt 3-1}$

  2. $\dfrac {1+\sqrt 3}{1-\sqrt 3}$

  3. $\dfrac {1}{\sqrt 3}+\sqrt 3$

  4. $\sqrt 2 +\sqrt 3$

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Correct Option: A
Explanation:

$\tan\left(7\dfrac{1}{2}\right)^o=\tan\left(\dfrac{15}{2}\right)^o$


                       $=\dfrac{\sin\dfrac{15^o}{2}}{\cos\dfrac{15^o}{2}}$

                       $=\dfrac{2\sin\dfrac{15^o}{2}\times \sin\dfrac{15^o}{2}}{2\sin\dfrac{15^o}{2}\times \cos\dfrac{15^o}{2}}$

                        $=\dfrac{2\sin^2\dfrac{15^o}{2}}{\sin\left(2\times\dfrac{15^o}{2}\right)}$

                        $=\dfrac{1-\cos\left(2\times\dfrac{15^o}{2}\right)}{\sin 15^o}$

                        $=\dfrac{1-\cos 15^o}{\sin 15^o}$

Now,
$\cos 15^o=\cos(45^o-30^o)$
              $=\cos 45^o\cos30^o+\sin 45^o\sin 30^o$
              $=\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{3}}{2}+\dfrac{1}{\sqrt{2}}\times\dfrac {1}{2}$
              $=\dfrac{\sqrt{3}+1}{2\sqrt{2}}$

$\sin 15^o=\sin(45^o-30^o)$
             $=\sin 45^o\cos30^o-\cos 45^o\sin 30^o$
             $=\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{3}}{2}-\dfrac{1}{\sqrt{2}}\times\dfrac {1}{2}$
             $=\dfrac{\sqrt{3}-1}{2\sqrt{2}}$


$\tan\left(7\dfrac{1}{2}\right)^o=\dfrac{1-\dfrac{\sqrt{3}+1}{2\sqrt{2}}}{\dfrac{\sqrt{3}-1}{2\sqrt{2}}}$

                      $=\dfrac{2\sqrt{2}-(\sqrt{3}+1)}{\sqrt{3}-1}$

Value of $ \displaystyle \sin 45^{\circ} \cos 45 \left ( \tan 45^{\circ}+\cot 45^{\circ} \right )^{2}   $  is 

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: B
Explanation:

$ \displaystyle \sin 45^{\circ} \cos 45 \left ( \tan 45^{\circ}+\cot 45^{\circ} \right )^{2}   $

$=\dfrac{1}{\sqrt2} \times \dfrac{1}{\sqrt2} (1+1)^2 $


$=\dfrac42$

$=2$

Two angles are called adjacent if

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. they lie in the same plane and have a common vertex

  2. they have a ray in common

  3. the intersection of their interiors is empty

  4. all the above

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Correct Option: D
Explanation:

Two angles are called adjacent if
they lie in same plane and have a common vertex,they have a ray in common and the intersection of their interiors is empty.
do (D) all the above is correct

In a cyclic quadrilateral ABCD, $\displaystyle \angle ABC=60^{\circ}$ and if O be the centre of the circle then the measure of $\displaystyle \angle OAC$ is

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. $\displaystyle 20^{\circ}$

  2. $\displaystyle 30^{\circ}$

  3. $\displaystyle 40^{\circ}$

  4. $\displaystyle 50^{\circ}$

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Correct Option: B

Triangle measurement is called as _______.

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. pythagoras

  2. trigonometry

  3. calculus

  4. area of square

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Correct Option: B
Explanation:
Trigonometry is the study of triangle measurement.
Hence, the answer is trigonometry.

In the early 9th century AD, _________ produced accurate sine and cosine tables, and the first table of tangents.

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. Habash al-Hasib al-Marwazi

  2. Muhammad ibn Jabir al-Harrani al-Battani

  3. Muhammad ibn Musa al-Khwarizmi

  4. Abu al-Wafa al-Buzjani

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Correct Option: C
Explanation:

In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry.

Who is the founder of trigonometry?

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. Euclid

  2. Issac Newton

  3. William Rowan Hamilton

  4. Hipparchus

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Correct Option: D
Explanation:

Hipparchus the founder of trigonometry. 

So, option D is correct.

The history of trigonometry goes back to the earliest recorded mathematics in Egypt and _____.

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. German

  2. Indian

  3. Babylon

  4. Japanese

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Correct Option: C
Explanation:

The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon.

So, option C is correct.

Trigonometry is used mainly due to the purpose of time keeping and _____.

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. space

  2. stars

  3. astronomy

  4. planets

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Correct Option: C
Explanation:

Trigonometry is used mainly due to the purpose of time keeping and astronomy.

So, option C is correct.

The first recorded use of trigonometry came from the Hellenistic mathematician ________________.

mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

  1. William Rowan Hamilton

  2. Hipparchus

  3. Newton

  4. Bartholomaeus Pitiscus

Show answer
Correct Option: B
Explanation:

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus.

So, option B is correct.