Question

Find \[\sin 74^\circ \] using \[\sin (A + B) = \sin A\cos B + \cos A\sin B\]
Find \[\sin 16^\circ \] using \[\sin (A - B) = \sin A\cos B - \cos A\sin B\]

Answer 1

Hint:
Here we have to find the values of the given trigonometric ratios using the given trigonometric identities. We can find the trigonometric ratios using the sine and cosine table. Sine and cosine table gives us the trigonometric values at different angles. Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.

Formula used:
\[\sin (A + B) = \sin A\cos B + \cos A\sin B\]
\[\sin (A - B) = \sin A\cos B - \cos A\sin B\]

Complete step by step solution:
We have to find \[\sin 74^\circ \].
Now \[\sin 74^\circ \] can be written as \[\sin (37^\circ + 37^\circ )\] .
\[ \Rightarrow \sin \left( {74^\circ } \right) = \sin \left( {37^\circ + 37^\circ } \right)\]
By using the trigonometric identity, \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] in above equation, we get
\[ \Rightarrow \sin \left( {74^\circ } \right) = \sin 37^\circ \cos 37^\circ + \cos 37^\circ \sin 37^\circ \]
Adding the terms, we have
\[ \Rightarrow \sin \left( {74^\circ } \right) = 2\sin 37^\circ \cos 37^\circ \]
From the sine and cosine table, we have
\[ \Rightarrow \sin \left( {74^\circ } \right) = 2 \times 0.6018 \times 0.7986\]
By multiplying the terms, we get
\[ \Rightarrow \sin \left( {74^\circ } \right) = 0.9713\]
Now we have to find \[\sin 16^\circ \].
\[\sin 16^\circ \]can be written as \[\sin (30^\circ - 14^\circ )\]
\[ \Rightarrow \sin \left( {16^\circ } \right) = \sin \left( {30^\circ - 14^\circ } \right)\]
By using the trigonometric identity \[\sin (A - B) = \sin A\cos B - \cos A\sin B\] in the above equation, we get
\[ \Rightarrow \sin \left( {16^\circ } \right) = \sin 30^\circ \cos 14^\circ - \cos 30^\circ \sin 14^\circ \]
From the sine and cosine table, we have
\[ \Rightarrow \sin \left( {16^\circ } \right) = 0.5 \times 0.9703 - 0.8660 \times 0.2419\]
By multiplying the terms, we have
\[ \Rightarrow \sin \left( {16^\circ } \right) = 0.48515 - 0.2095\]
 Subtracting the terms, we get
\[ \Rightarrow \sin \left( {16^\circ } \right) = 0.27565\]

Therefore, \[\sin \left( {74^\circ } \right) = 0.9713\] and \[\sin \left( {16^\circ } \right) = 0.27565\].

Note:
We can find the value of sine and cosine using the table. Usually sine tables will accompany other tables of trigonometric functions. Hyperbolic trigonometric functions are logarithmic and are not natural sine, cosines or tangents. We can observe that the table of natural sine and natural cosines are generally divided into the following parts. They are the following:
(i) In the extreme left, vertical column of the table ,the angles are from \[0^\circ \] to \[90^\circ \] at intervals of \[1^\circ \].
(ii) In another vertical column, about the middle of the table, the angles are from \[89^\circ \] to \[0^\circ \] at intervals of \[1^\circ \].
(iii) In the horizontal row at the top of the table, the angles are from \[0'\] to \[60'\] at intervals of \[10'\].
(iv) In the horizontal row at the bottom of the table, the angles are from \[60'\] to \[0'\] at intervals of \[10'\].
(v) In the horizontal row at the extreme right of the table, the angles are from \[1'\] to \[9'\] at intervals of \[1'\]. This part of the table is known as Mean Difference Column.