Question

What is the exact value of \[\cos -330?\]

Answer 1

Hint: To solve this question we should have the basic knowledge of trigonometric functions. In this question we have to find the value of cosine function with the angle \[-{{330}^{0}}\]. For this we should know the value of trigonometric functions in all the four quadrants. And the values of some angles of cosine should be known.

Complete step-by-step solution:
As we are aware of the fact that in total there are four quadrants and they are first quadrant, second quadrant, third quadrant and fourth quadrant. For the first quadrant angle lies in the range from \[{{0}^{0}}\]to \[{{90}^{0}}\]. For the second quadrant angle lies in the range from \[{{90}^{0}}\] to \[{{180}^{0}}\]. The third quadrant angle is in the range from \[{{180}^{0}}\]to \[{{270}^{0}}\]. And for the fourth quadrant angle lies from \[{{270}^{0}}\]to \[{{360}^{0}}\]. But the sign of all the trigonometric functions in all the four quadrants are different. So there is a trick to remember the sign of trigonometric functions in all the quadrants.
The best way to remember the signs is to remember the statement All Should Take Coffee.
Now let us try to get this statement what does this mean. Here All represents that for the first quadrant all the trigonometric functions are positive in nature. Should represent that only sine function and the reciprocal of the sine function i.e. cosecant is positive in nature in the second quadrant, all other functions are negative. Take represent that only tangent function and the reciprocal of the tangent function i.e. cotangent is positive in nature in the third quadrant. Coffee represents that only the cosine function and the reciprocal of the cosine function i.e. secant is positive in nature in the fourth quadrant.
For the given question we have to find the value for \[\cos (-330)\]. And we know that cosine is an even function that means \[\cos (\theta )=\cos (-\theta )\].
\[\Rightarrow \cos (-{{330}^{0}})=\cos ({{330}^{0}})\]
And we can express \[{{330}^{0}}\]as \[{{360}^{0}}-{{30}^{0}}\]
\[\Rightarrow \cos ({{360}^{0}}-{{30}^{0}})\]
And this value lies in the fourth quadrant so cosine will be positive and \[\cos ({{360}^{0}}-\theta )=\cos \theta \].
\[= \cos {{30}^{0}}\]
By the trigonometric table we get the value
\[= \dfrac{\sqrt{3}}{2}\]
Hence we can conclude that \[\cos (-{{330}^{0}})\] is \[\dfrac{\sqrt{3}}{2}\].

Note: There are many applications of trigonometry in real life. For example, if we know the distance from where we observe the building аnd thе аngle оf elevаtiоn we саn eаsily find the height оf the building. Similаrly, if we hаve the vаlue оf one side аnd the аngle оf depression from the tор оf the building we find аnоther side in the triаngle.