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Question: What if the inverse cosine of \( 0.55 \) ?...

What if the inverse cosine of 0.550.55 ?

Explanation

Solution

Hint : In order to find the inverse cosine of 0.550.55 , we need to know about the inverse function. Inverse function is also known as anti-function. Basically, the inverse of a function is the reverse of a function. The inverse is not a function. The inverse of a function is denoted by a function with power of −1- 1 , as f−1(x){f^{ - 1}}\left( x \right) .

Complete step-by-step answer :
We are given 0.550.55 , to find the cosine inverse.
Since, we know that the inverse of a function is the reverse of a function and it’s denoted by f−1(x){f^{ - 1}}\left( x \right) .
So, for the cosine inverse it can be represented or written as cos⁡−1(x){\cos ^{ - 1}}\left( x \right) .Substituting the value of xx as 0.550.55 .
Now, the cosine inverse of 0.550.55 is written as cos⁡−1(0.55){\cos ^{ - 1}}\left( {0.55} \right) .
Consider a function to be cos⁡θ=0.55\cos \theta = 0.55 , where θ\theta is an angle.
Multiplying cos⁡−1{\cos ^{ - 1}} both sides, we get:
cos⁡−1(cos⁡θ)=cos⁡−1(0.55){\cos ^{ - 1}}\left( {\cos \theta } \right) = {\cos ^{ - 1}}\left( {0.55} \right)
θ=cos⁡−1(0.55)\theta = {\cos ^{ - 1}}\left( {0.55} \right)
So, we can see we need to find the angle whose value is 0.550.55 .
But, we only the values of the angles such as 30∘{30^ \circ } , 45∘{45^ \circ } , 60∘{60^ \circ } and 90∘{90^ \circ } , and for no cosine angle, it gives 0.550.55 .
So, we need to use scientific calculator and using that we get:
cos⁡−1(0.55)=56.6329870308{\cos ^{ - 1}}\left( {0.55} \right) = 56.6329870308
Since, it is an angle so it’s in degrees.
Also, written as:
cos⁡−1(0.55)=56.6329870308∘{\cos ^{ - 1}}\left( {0.55} \right) = {56.6329870308^ \circ }
cos⁡−1(0.55)≈57∘{\cos ^{ - 1}}\left( {0.55} \right) \approx {57^ \circ }
Therefore, the inverse cosine of 0.550.55 is 56.6329870308∘{56.6329870308^ \circ } or cos⁡−1(0.55)≈57∘{\cos ^{ - 1}}\left( {0.55} \right) \approx {57^ \circ } .

Note : If needed we can change the value of inverse cosine from degrees to radian.
We can also use an approximation method to find the angle for the value by finding the values at other angles, then deciding in which range it could lie.
It’s important to use a scientific calculator when there is no value or angle is known.