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Question: How do you find \(\sec x = - 5\) using a calculator?...

How do you find sec⁡x=−5\sec x = - 5 using a calculator?

Explanation

Solution

First we use the formula. The formula is:
sec⁡x=1cos⁡x\sec x = \dfrac{1}{{\cos x}}
After that we substitute the xx value in the formula. We find the value ofsec⁡x=−5\sec x = - 5.Use the calculator, we get the value of sec⁡x=−5\sec x = - 5. After that we use the division method.
We just use the substitution and use the calculator.
And we convert into the inverse trigonometric function in the given trigonometric function.
Finally we get the answer.

Complete step by step answer:
The given trigonometry is sec⁡x=−5\sec x = - 5
We use the calculator.
First we change the given equation
Let, sec⁡x=−5\sec x = - 5
Apply the formula for sec⁡x=1cos⁡x\sec x = \dfrac{1}{{\cos x}}
We substitute in the given equation, hence we get
⇒1cos⁡x=−5\Rightarrow \dfrac{1}{{\cos x}} = - 5
We rewrite the function, hence we get
⇒cos⁡x=−15\Rightarrow \cos x = - \dfrac{1}{5}
Divide11by55
⇒cos⁡x=−0.20\Rightarrow \cos x = - 0.20
Interchange the cosine function, hence we get
⇒x=cos⁡−1(−0.20)\Rightarrow x = {\cos ^{ - 1}}( - 0.20)
Now we use the scientific calculator
First we change the mode of degrees in the calculator.
We push the shift+cos⁡{\text{shift}} + \cos button, its show the function of cos⁡−1{\cos ^{ - 1}}
⇒cos⁡−1\Rightarrow {\cos ^{ - 1}}
We enter the input−0.20 - 0.20, we get in the calculator
⇒cos⁡−1−0.20\Rightarrow {\cos ^{ - 1}} - 0.20
And then push the (==) is equal to button, we get the result
⇒cos⁡−1−0.20=±101.54\Rightarrow {\cos ^{ - 1}} - 0.20 = \pm 101.54
Finally we get the answer.
If you have a calculator such as Casio you can type sec⁡x=−5\sec x = - 5
Directly press == and get the answer immediately without using the reciprocal key.

Note:
The trigonometric ratios are defined with reference to a right triangle.
sin⁡(θ)=opposite sidehypotenuse;cos⁡(θ)=adjacent sidehypotenuse\sin (\theta ) = \dfrac{{{\text{opposite side}}}}{{{\text{hypotenuse}}}};\cos (\theta ) = \dfrac{{{\text{adjacent side}}}}{{{\text{hypotenuse}}}}
With the help of sine and cosine, the remaining trigonometric ratios tangent, cotangent, cosecant and secant are determined by using the relations.
tan⁡θ=sin⁡θcos⁡θ,csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }},\csc\theta = \dfrac{1}{{\sin \theta }},\sec \theta = \dfrac{1}{{\cos \theta }},\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}
The secant and cosecant are inverses of cosine and sine respectively.