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Question: How do you evaluate \(\sin \left( -\dfrac{\pi }{6} \right)\) ? (a) Using trigonometric angle iden...

How do you evaluate sin⁡(−π6)\sin \left( -\dfrac{\pi }{6} \right) ?
(a) Using trigonometric angle identities
(b) Using linear formulas
(c) a and b both
(d) none of the above

Explanation

Solution

In this problem we are to find the value of sin⁡(−π6)\sin \left( -\dfrac{\pi }{6} \right). We will try to use the trigonometric double angle identities to find and simplify the value of our needed problem. We can start with the fact that sin⁡(−θ)=−sin⁡(θ)\sin \left( -\theta \right)=-\sin \left( \theta \right) and consider θ=π6\theta =\dfrac{\pi }{6} to get ahead with the problem and evaluate the value.

Complete step by step solution:
We start with,
sin⁡(−π6)\sin \left( -\dfrac{\pi }{6} \right)
As the value of sin⁡(−θ)=−sin⁡(θ)\sin \left( -\theta \right)=-\sin \left( \theta \right), the term sin⁡(−π6)\sin \left( -\dfrac{\pi }{6} \right)can be written as −sin⁡(π6)-\sin \left( \dfrac{\pi }{6} \right).
⇒−sin⁡(π6)\Rightarrow -\sin \left( \dfrac{\pi }{6} \right)
Now, we get to see, π6\dfrac{\pi }{6} lies in the first quadrant where the value of all functions are always positive.
Now, say, θ=π6\theta =\dfrac{\pi }{6} ,
So, sin⁡θ=sin⁡π6\sin \theta =\sin \dfrac{\pi }{6}
Also, we know, sin⁡(−x)=−sin⁡x\sin \left( -x \right)=-\sin x ,
So,
sin⁡(−π6)=−sin⁡π6\sin \left( -\dfrac{\pi }{6} \right)=-\sin \dfrac{\pi }{6}
As, sin⁡π6=12\sin \dfrac{\pi }{6}=\dfrac{1}{2} ,
We can conclude, sin⁡(−π6)=−12\sin \left( -\dfrac{\pi }{6} \right)=-\dfrac{1}{2}
Hence the solution is, (a) Using trigonometric angle identities

Note: To understand how the values of trigonometric ratios like sin⁡(−π6)\sin \left( -\dfrac{\pi }{6} \right) change in different quadrants, first we have to understand ASTC rule. The ASTC rule is nothing but the "all sin tan cos" rule in trigonometry. The angles which lie between 0° and 90° are said to lie in the first quadrant. The angles between 90° and 180° are in the second quadrant, angles between 180° and 270° are in the third quadrant and angles between 270° and 360° are in the fourth quadrant. In the first quadrant, the values for sin, cos and tan are positive. In the second quadrant, the values for sin are positive only. In the third quadrant, the values for tan are positive only. In the fourth quadrant, the values for cos are positive only.