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Question: How do you prove \[\sin \left( {{180}^{\circ }}-a \right)=\sin a\]?...

How do you prove sin⁡(180∘−a)=sin⁡a\sin \left( {{180}^{\circ }}-a \right)=\sin a?

Explanation

Solution

In this type of question we have to use the basic simplification of trigonometric functions. Here, we have to use a simple trigonometric formula sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B. Also we have to use the values of the standard angles that are, sin⁡180∘=0\sin {{180}^{\circ }}=0 and cos⁡180∘=(−1)\cos {{180}^{\circ }}=\left( -1 \right). In this question we first consider the LHS and then by using above mentioned formula and values we can obtain the required result.

Complete step-by-step solution:
Now, we have to prove that sin⁡(180∘−a)=sin⁡a\sin \left( {{180}^{\circ }}-a \right)=\sin a
For this, let us take the LHS of the above equation
⇒LHS=sin⁡(180∘−a)\Rightarrow LHS=\sin \left( {{180}^{\circ }}-a \right)
Now, to simplify this trigonometric expression we have to use the compound angle formula of sine.
Also we know that, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B by applying it to LHS that is to sin⁡(180∘−a)\sin \left( {{180}^{\circ }}-a \right), we can write the LHS as
⇒LHS=sin⁡180∘cos⁡a−cos⁡180∘sin⁡a\Rightarrow LHS=\sin {{180}^{\circ }}\cos a-\cos {{180}^{\circ }}\sin a
Now, we know the values of sin⁡180∘\sin {{180}^{\circ }} and cos⁡180∘\cos {{180}^{\circ }} which are sin⁡180∘=0\sin {{180}^{\circ }}=0 and cos⁡180∘=(−1)\cos {{180}^{\circ }}=\left( -1 \right). Hence, by substituting these values in the above expression we can write our LHS as
⇒LHS=(0)cos⁡a−(−1)sin⁡a\Rightarrow LHS=\left( 0 \right)\cos a-\left( -1 \right)\sin a
By simplifying the trigonometric expression further we get,
⇒LHS=sin⁡a\Rightarrow LHS=\sin a
Also by given RHS=sin⁡aRHS=\sin a
⇒LHS=RHS\Rightarrow LHS=RHS
Hence we have proved, sin⁡(180∘−a)=sin⁡a\sin \left( {{180}^{\circ }}-a \right)=\sin a

Note: In this type of question students should remember the formulas of trigonometry. Students have to remember the basic algebraic rules and trigonometric identities which will help them in simplification of such trigonometric expressions. Students should have a good grip over the basic trigonometric identities and formulae. Also students have to note that the identity which we have proved in the given question is further used in many questions as a direct result and has a wide range of applications.