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Taking online classes and trying to find the angle in degrees of csc when given a decimal

Example in text not helping

Question is csc (theta) = 1.245 and they want an angle in degrees

Thank you in advance Lisa

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    Calculator is allowed? – 2017-02-16
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    Yes using a calculator is how I found sin cos and tan angles in other questions – 2017-02-16

2 Answers 2

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Almost all calculators are programmed with just the sine function ($\sin$), the cosine function ($\cos$), and the tangent function ($\tan$). To work with the secant, cosecant and cotangent functions, we need to use some trig identities and some algebra.

Your problem asks us to solve for $\theta$ where $\csc\theta=1.245$.

We know that $\csc\theta=\frac{1}{\sin\theta}$. Using this identity, we solve for $\theta$.

\begin{align} \csc\theta&=1.245\\ \frac{1}{\sin\theta}&=1.245\\ \frac{1}{1.245}&=\sin\theta\\ \sin^{-1}\frac{1}{1.245}&=\theta \end{align}

Remember that $\sin^{-1}$ is the inverse sine, also denoted as $\arcsin$. Using the calculator, we find that $\sin^{-1}(1/1.245)=53.44^{\circ}\text{ or }0.933\text{ radians}$.

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    THANK YOU !!!!! That makes total sense now and examples in text are not similar at all ! Can this example be used for cot and sec too. I really appreciate the help – 2017-02-16
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    Yes, you can use the same method for $\sec$ and $\cot$. If you are happy with this solution, please mark it as your accepted solution. – 2017-02-16
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    Sorry. Total newby. On my iPhone. Couldn't sleep til I figured it out or got the answer :) where do I mark an accepted solution :) thanks. Is it the green check mark :) – 2017-02-16
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    Looks like you marked the solution. Thanks! – 2017-02-16
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    Thank you and to everyone who helped :) – 2017-02-16
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All you need to do is take the inverse function to return the original angle. We know that $\csc (\theta)= 1.256$, so solving for $\theta$

$\csc^{-1}(\theta)=\csc^{-1}(1.245)=\ \approx53.44$ degrees

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    Sorry it was 1.245. But would you use csc or sin? I did sin and answer was wrong. – 2017-02-16
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    Notice that the cosecant function is defined as $\frac{1}{\sin}$. Because the problem references cosecant and not sine, you would want to use inverse cosecant and **not** inverse sine. – 2017-02-16
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    Just checked calculator has no csc function. I'll need to buy a new calculator Thank you very much. – 2017-02-16
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    You won't have to. Just use the "$\sin$" button, and then put $1$ over your answer or raise it to the $-1$ power, whichever is easier. – 2017-02-16
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    So I could do 1 / sin 1.245 – 2017-02-16
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    Yes! In fact you can use this for the trigonometric ratios sine, cosine, and tangent. For cosecant $(\csc)$, use $\frac{1}{\sin}$, for secant($\sec$), use $\frac{1}{\cos}$ and for cotangent ($\cot$) use $\frac{1}{\tan}$ – 2017-02-16