Question

Find the value of $sin{210}^{\circ }+cos{120}^{\circ }+tan{225}^{\circ }+cot{315}^{\circ }+sec{300}^{\circ }+cosec{150}^{\circ }$.

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Answer 1

We know the trigonometric ratios of $0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ },{90}^{\circ }$.

Let's try to find out whether the angles given in the question can be reduced to standard angles.

For eg. ${210}^{\circ }$ can be written as (180 + ${30}^{\circ }$)

${120}^{\circ }$ can be written as $({90}^{\circ }+{30}^{\circ })$

Similarly, other angles can also be brought down to standard angles.

From question,

$Sin{210}^{\circ }+cos{120}^{\circ }+tan{225}^{\circ }+cot{315}^{\circ }+sec{300}^{\circ }+cosec{150}^{\circ }$

=$sin({180}^{\circ }+{30}^{\circ })+cos({90}^{\circ }+{30}^{\circ })+tan({180}^{\circ }+{45}^{\circ })+cot({270}^{\circ }+{45}^{\circ })+sec({360}^{\circ }-{60}^{\circ })+cosec({180}^{\circ }-{30}^{\circ })$

= $-sin{30}^{\circ }-sin{30}^{\circ }+tan{45}^{\circ }-tan{45}^{\circ }+sec{60}^{\circ }+cosec{30}^{\circ }$

We get,

= $\frac{-1}{2}$ - $\frac{1}{2}$ + 1 - 1 + 2 + 2 (substituting the values of standard angles)

= 3