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Question: How do you find the exact value of \[cos{\text{ }}\left( { - 30} \right){\text{ }}\]?...

How do you find the exact value of cos (−30) cos{\text{ }}\left( { - 30} \right){\text{ }}?

Explanation

Solution

To find the exact value of the given expression, we need to use the property of the cosine function which is cos (−x) = cos xcos{\text{ }}\left( { - x} \right){\text{ }} = {\text{ }}cos{\text{ }}x. Here, we need to note that the given value is in the range of 0 to 900{\text{ }}to{\text{ }}90, so we will directly imply the cosine property to attain the answer. Real functions which relate any angle of a right angled triangle to the ratio of any two of its sides are Trigonometric functions.

Complete step-by-step answer:
According to the given data, we need to simplify cos (−30) cos{\text{ }}\left( { - 30} \right){\text{ }}
We need to use the property of the cosine function which is cos (−x) = cos xcos{\text{ }}\left( { - x} \right){\text{ }} = {\text{ }}cos{\text{ }}x
For any angle, cos (−x) = cos xcos{\text{ }}\left( { - x} \right){\text{ }} = {\text{ }}cos{\text{ }}x
So,
∴cos (−30) = cos 30\therefore cos{\text{ }}\left( { - 30} \right){\text{ }} = {\text{ }}cos{\text{ 30}}
We know,
 ⇒cos 30 = 32=123{\text{ }} \Rightarrow cos{\text{ 30 = }}\dfrac{{\sqrt 3 }}{2} = \dfrac{1}{2}\sqrt 3
Therefore,
cos (−30)=cos 30 = 32=123cos{\text{ }}\left( { - 30} \right) = cos{\text{ 30 = }}\dfrac{{\sqrt 3 }}{2} = \dfrac{1}{2}\sqrt 3
⇒cos ( - 30) = 123\Rightarrow cos{\text{ ( - 30) = }}\dfrac{1}{2}\sqrt 3

Additional Information:
Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. The widely used ones are sin, cos and tan. While the rest can be referred to as the reciprocal of the others, i.e., cosec, sec and cot respectively.
If the given value is in the range of 0 to 900{\text{ }}to{\text{ }}90, then directly apply this formula. Otherwise, if it is not in the range of 0 to 900{\text{ }}to{\text{ }}90, divide the value by 180180 to do so. Use the function of cosine which is cos⁡((2n+1)π±x)=−cos⁡x\cos ((2n + 1)\pi \pm x) = - \cos x and cos⁡(2nπ±x)=cos⁡x\cos (2n\pi \pm x) = \cos xto find the exact value of the given angle.

Note: One should be careful while evaluating trigonometric values and rearranging the terms to convert from one function to the other. Also it’s very important to remember and apply this property, cos (−x) = cos xcos{\text{ }}\left( { - x} \right){\text{ }} = {\text{ }}cos{\text{ }}x, correctly to avoid any mistake.