Question

How do you find the exact value of sine, cosine, and tangent of the number \[-\dfrac{10\pi }{3}\], without using the calculator?

Answer 1

Hint: The given question is the trigonometric expression and in order to solve this solve we have to use the properties of trigonometric functions. First we need to remove the full rotation of \[2\pi \]until the angle is between 0 to \[2\pi \]. Then using the trigonometric ratios table which is also known as a log table, we will find the exact value of the given expression.

Complete step by step solution:
We have given that,
Sin, cos, and tan of \[-\dfrac{10\pi }{3}\],
On trigonometric unit circle;
\[\Rightarrow \sin \left( -\dfrac{10\pi }{3} \right)=\sin \left( \dfrac{-4\pi }{3}-2\pi \right)=\sin \dfrac{2\pi }{3}\]
Using the trigonometric table of special arc, we obtained the sine value,
Therefore,
\[\Rightarrow \sin \left( -\dfrac{10\pi }{3} \right)=\sin \left( \dfrac{-4\pi }{3}-2\pi \right)=\sin \dfrac{2\pi }{3}=\dfrac{\sqrt{3}}{2}\]
Thus,
\[\therefore \sin \left( -\dfrac{10\pi }{3} \right)=\dfrac{\sqrt{3}}{2}\]
Now,
On trigonometric unit circle;
\[\Rightarrow \cos \left( -\dfrac{10\pi }{3} \right)=\cos \left( \dfrac{-4\pi }{3}-2\pi \right)=\cos \dfrac{2\pi }{3}\]
Using the trigonometric table of special arc, we obtained the sine value,
Therefore,
\[\Rightarrow \cos \left( -\dfrac{10\pi }{3} \right)=\cos \left( \dfrac{-4\pi }{3}-2\pi \right)=\cos \dfrac{2\pi }{3}=-\dfrac{1}{2}\]
Thus,
\[\therefore \cos \left( -\dfrac{10\pi }{3} \right)=-\dfrac{1}{2}\]
Now,
On trigonometric unit circle;
Removing the full rotation of \[2\pi \] as the angle is between 0 to\[2\pi \].
\[\Rightarrow \tan \left( -\dfrac{10\pi }{3} \right)=\tan \left( \dfrac{-4\pi }{3}-2\pi \right)=\tan \dfrac{2\pi }{3}\]
On simplifying using the tangent formula, we have
\[\Rightarrow \tan \dfrac{2\pi }{3}=\dfrac{\sin \left( \dfrac{2\pi }{3} \right)}{\cos \left( \dfrac{2\pi }{3} \right)}=\dfrac{\sqrt{3}}{2}\times -\dfrac{2}{1}=-\sqrt{3}\]
\[\therefore \tan \left( -\dfrac{10\pi }{3} \right)=-\sqrt{3}\]
Therefore,
Without using the calculator, the exact value of;
\[\Rightarrow \sin \left( -\dfrac{10\pi }{3} \right)=\dfrac{\sqrt{3}}{2}\]
\[\Rightarrow \cos \left( -\dfrac{10\pi }{3} \right)=-\dfrac{1}{2}\]
\[\Rightarrow \tan \left( -\dfrac{10\pi }{3} \right)=-\sqrt{3}\]

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.