Question Video: Using the Pythagorean Identities to Evaluate a Trigonometric Function Involving Cofunction Identities given the Trigonometric Function and Quadrant of an Angle Mathematics

Video Transcript

Find the value of sec 90 degrees plus 𝜃 given that csc 𝜃 equals 17 over eight, where 𝜃 is between zero degrees and 90 degrees.

There are various ways that we can solve this question. We can use the fact that csc 𝜃 is 17 over eight and 𝜃 is between zero degrees and 90 degrees to find the value of 𝜃. And once we have this value of 𝜃, we can just substitute it into the expression sec 90 degrees plus 𝜃 that we wanted to evaluate.

So let’s do this. Csc 𝜃 is equal to 17 over eight; csc 𝜃 is a one over sin 𝜃, and so we have that one over sin 𝜃 is 17 over eight. Taking the reciprocal on both sides of the equation, or just rearranging the equation to make the subject sin 𝜃, we see that sin 𝜃 is equal to eight over 17, and hence that 𝜃 is the inverse sin or arcsin of eight over 17.

With our calculator in degree mode, we find that the 𝜃 is equal to 28.072 dot dot dot degrees. This value that are calculator gives us is in the range that we want; it’s between zero degrees and 90 degrees. And so we don’t have to modify it in any way. We just substitute this value of 𝜃 into sec 90 degrees plus 𝜃, remembering that sec 𝑥 is one over cos 𝑥 if our calculator doesn’t have a sec button to get a value of negative 17 over eight.

But perhaps your calculator doesn’t give you this exact value, negative 17 over eight, as a fraction or perhaps you can’t use a calculator at all. Luckily we can answer this question without using a calculator. We go back a few steps to where we discovered that sin 𝜃 is equal to eight over 17, and we turn our attention to sec 90 degrees plus 𝜃.

Sec 𝑥 is one over cos 𝑥 and so sec 90 degrees plus 𝜃 is one over cos 90 degrees plus 𝜃. We can use the angle sum formula for cosine, setting 𝑥 equal to 90 degrees and 𝑦 equal to 𝜃. 90 degrees is a special angle, so we know the values of cos 90 degrees and sin 90 degrees.

Cos 90 degrees is zero and sin 90 degrees is one. The zero cos 𝜃 term disappears, and we’re left with one over negative one sin 𝜃, which is negative one over sin 𝜃. And we know that one over sin 𝜃 is equal to csc 𝜃, so we find this is negative csc 𝜃. And we’re told in the question that csc 𝜃 is 17 over eight, so the value of sec 90 degrees plus 𝜃 is negative 17 over eight.

So that method preceded by simplifying sec 90 degrees plus 𝜃 using an angle sum identity. There is a third method which I’ll show you for fun, which involves doing something slightly nonobvious. We write down the expression we wish to evaluate again, sec 90 degrees plus 𝜃, and we notice that it looks very much like the left-hand side of the cofunction identity sec 90 degrees minus 𝑥 equals csc 𝑥.

The only difference is that we have a 90 degrees plus something and not 90 degrees minus something. But we can fix that by writing 90 degrees plus 𝜃 as 90 degrees minus negative 𝜃. We set 𝑥 equal to negative 𝜃 in the identity to get csc negative 𝜃. But csc is an odd function, and if you don’t believe me, we’ll prove this at the end of the video.

And so csc of negative 𝜃 is negative csc 𝜃. Using the value of csc 𝜃 we’re given in the question, we get a value of negative 17 over eight. That was a much quicker way of finding the value of sec 90 degrees plus 𝜃 using the cofunction identity sec 90 degrees minus 𝑥 equals csc 𝑥 and the fact that csc is an odd function.

This last fact is something we should prove csc negative 𝑥 is equal to one over sin negative 𝑥 by definition and certainly sine is an odd function. We can prove this, but we’re not going to in this video. And so sine of negative 𝑥 is negative of sine 𝑥. Multiplying the numerator and denominator of this fraction by negative one, we get that this is equal to negative one over sine 𝑥. And one over sine 𝑥 is csc 𝑥, so we get that this is equal to negative csc 𝑥.

We have proved that csc negative 𝑥 is equal to negative csc 𝑥, and hence that the csc function is odd.