Question

Without using tables, give the value of each of the following:
(i) \[\sin 120^\circ \]
(ii) \[\cot 330^\circ \]
(iii) \[\sec 210^\circ \]
(iv) \[\cos 315^\circ \]
(v) \[{\rm{cosec}}675^\circ \]
(vi) \[\cos 855^\circ \]
(vii) \[\sin 4530^\circ \]

Answer 1

Hint:
Here, we will write each of the angle values as a sum or difference of known angle values. Then, we will apply trigonometric identities to the resulting expressions to obtain the required values.

Formulae used:
We will use the following formulas:
1) \[\sin (90^\circ + \theta ) = \cos \theta \]
2) \[\cot (360^\circ - \theta ) = - \cot \theta \]
3) \[\sec (180^\circ + \theta ) = - \sec \theta \]
4) \[\cos (360^\circ - \theta ) = \cos \theta \]
5) \[{\rm{cosec(}}360^\circ k - \theta ) = - {\rm{cosec}}\theta \]
6) \[\cos (180^\circ k - \theta ) = - \cos \theta \]
7) \[\sin (180^\circ k + \theta ) = - \sin \theta \]

Complete Step by step Solution:
(i) We can write \[120^\circ \]as \[120^\circ = 90^\circ + 30^\circ \].
So, \[\sin 120^\circ = \sin (90^\circ + 30^\circ )\]
Using the formula \[\sin (90^\circ + \theta ) = \cos \theta \], we get
\[ \Rightarrow \sin 120^\circ = \cos 30^\circ \]
We know that \[\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}\]. Thus, we have,
\[ \Rightarrow \sin 120^\circ = \dfrac{{\sqrt 3 }}{2}\]

(ii)We can write \[330^\circ {\rm{ as }}330^\circ = 360^\circ - 30^\circ \]. So,
\[\cot 330^\circ = \cot (360^\circ - 30^\circ )\]

Using the formula \[\cot (360^\circ - \theta ) = - \cot \theta \], we get
\[ \Rightarrow \cot 330^\circ = - \cot 30^\circ \]
We know that \[\cot 30^\circ = \sqrt 3 \]. Thus, we have,
\[ \Rightarrow \cot 330^\circ = - \sqrt 3 \]


(iii) We can write \[210^\circ \] as \[210^\circ = 180^\circ + 30^\circ \]. So,
\[\sec 210^\circ = \sec (180^\circ + 30^\circ )\]
Using the formula \[\sec (180^\circ + \theta ) = - \sec \theta \], we get
\[ \Rightarrow \sec 210^\circ = - \sec 30^\circ \]
We know that \[\sec 30^\circ = \dfrac{2}{{\sqrt 3 }}\]. Thus, we have,
\[ \Rightarrow \sec 210^\circ = - \dfrac{2}{{\sqrt 3 }}\]

(iv) We can write \[315^\circ {\rm{ as }}315^\circ = 360^\circ - 45^\circ \]. So,
\[\cos 315^\circ = \cos (360^\circ - 45^\circ )\]
Using the formula \[\cos (360^\circ - \theta ) = \cos \theta \], we get
\[ \Rightarrow \cos 315^\circ = \cos 45^\circ \]
We know that \[\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}\]. Thus, we have,
\[ \Rightarrow \cos 315^\circ = \dfrac{1}{{\sqrt 2 }}\]

(v) We can write \[675^\circ {\rm{ as }}675^\circ = 720^\circ - 45^\circ \].
We can write \[720^\circ = 360^\circ \times 2\].
\[ \Rightarrow 675^\circ = (360^\circ \times 2) - 45^\circ \]

Thus, \[\cos ec 675^\circ = \cos ec((360^\circ \times 2) - 45^\circ )\]
Using the formula \[\cos ec(360^\circ k - \theta ) = - \cos ec\theta \], we get
\[ \Rightarrow \cos ec675^\circ = - \cos ec45^\circ \]
We know that \[{\rm{cosec}}45^\circ = \sqrt 2 \]. Thus, we have,

\[ \Rightarrow \cos ec675^\circ = - \sqrt 2 \]

(vi) We can write \[855^\circ {\rm{ as }}855^\circ = 900^\circ - 45^\circ .\]
As we can write \[900^\circ = 180^\circ \times 5\], so,
\[ \Rightarrow 855^\circ = (180^\circ \times 5) - 45^\circ \]
Thus, \[\cos 855^\circ = \cos ((180^\circ \times 5) - 45^\circ )\]
Using the formula \[\cos (180^\circ k - \theta ) = - \cos \theta \], we get
\[ \Rightarrow \cos 855^\circ = - \cos 45^\circ \]
We know that \[\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}\]. Thus, we have,
\[ \Rightarrow \cos 855^\circ = - \dfrac{1}{{\sqrt 2 }}\]
We can write \[4530^\circ {\rm{ as }}4530^\circ = 4500^\circ + 30^\circ \].
As we can write \[4500^\circ = 180^\circ \times 25\], so
\[ \Rightarrow 4530^\circ = (180^\circ \times 25) + 30^\circ \]

(vii) Thus, \[\sin 4530^\circ = \sin ((180^\circ \times 25) + 30^\circ )\]
Using the formula \[\sin (180^\circ k + \theta ) = - \sin \theta \], we get
\[ \Rightarrow \sin 4530^\circ = - \sin 30^\circ \]
We know that \[\sin 30^\circ = \dfrac{1}{2}\]. Thus, we have,
\[ \Rightarrow \sin 4530^\circ = - \dfrac{1}{2}\]

Note:
To resolve \[\theta \] into solvable terms, we have adopted the following procedure:
For example, let us take \[855^\circ \].
We will check which multiple of \[180^\circ \] is nearest to \[855^\circ \].
We see that \[900^\circ = 180^\circ \times 5\].
But since we need \[855^\circ \], we can subtract \[45^\circ {\rm{ from }}900^\circ \] to get \[855^\circ \]. Therefore, \[855^\circ = (180^\circ \times 5) - 45^\circ \].
Similarly, we can resolve other angles into solvable terms.