Having trouble with the fundamental identities of trig.

[Al72579]

I just started studying my trig for my summer class which will start in a few weeks but I'am going crazy over a a certain indenity problem. It seems simple enough: "Express sin t in terms of sec t".I just can't get it right though I come close. I have the answer in the book but I can't figure out how to get it! I hope that one of you would be able to show me the steps so I can stop fretting over it and resume my studies.

 

[soroban]

Hello, Al72579!

Express $\displaystyle \sin\theta$ in terms of $\displaystyle \sec\theta$

You're expected to know the basic identities and their variations.

The reciprocal identities:

. . $\displaystyle \sin\theta\:=\:\frac{1}{\csc\theta}\;\;\;\;\csc\theta \:=\:\frac{1}{\sin\theta}$

. . $\displaystyle \cos\theta\:=\:\frac{1}{\sec\theta}\;\;\;\;\sec\theta\:=\:\frac{1}{\cos\theta}$

. . $\displaystyle \tan\theta\:=\:\frac{1}{\cot\theta}\;\;\;\;\cot\theta\:=\:\frac{1}{\tan\theta}$


The "Pythagorean Identities":

. . $\displaystyle \sin^2\theta\,+\,\cos^2\theta\:=\:1\;$ [1]

. . $\displaystyle \sec^2\theta\:=\:\tan^2\theta\,+\,1\;$ [2]


These two have a variety of variations.

For example, [1] can be rearranged like this:

. . . .$\displaystyle \sin^2\theta\,+\,\cos^2\theta\:=\:1\;\;\Rightarrow\;\;\sin^2\theta \:=\:1\,-\,\cos^2\theta\;\;\Rightarrow\;\;\sin\theta\:=\:\sqrt{1\,-\,\cos^2\theta}$

or: $\displaystyle \:\sin^2\theta\,+\,\cos^2\theta\:=\:1\;\;\Rightarrow\;\;\cos^2\theta\:=\:1\,-\,\sin^2\theta\;\;\Rightarrow\;\;\cos\theta\:=\:\sqrt{1\,-\,\sin^2\theta}$

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We are given: $\displaystyle \,\sin\theta$ . . . and we want it in terms of $\displaystyle sec\theta$

We can write: \(\displaystyle \L\:\sin\theta\;=\;\sqrt{1\,-\,\cos^2\theta}\) . . . right?

. . and we know that: $\displaystyle \,\cos\theta\:=\:\frac{1}{\sec\theta}$

Substitute: \(\displaystyle \L\:\sin\theta\;=\;\sqrt{1\,-\,\left(\frac{1}{\sec\theta}\right)^2}\;\) . . . There!

 

[Mrspi]

I just started studying my trig for my summer class which will start in a few weeks but I'am going crazy over a a certain indenity problem. It seems simple enough: "Express sin t in terms of sec t".I just can't get it right though I come close. I have the answer in the book but I can't figure out how to get it! I hope that one of you would be able to show me the steps so I can stop fretting over it and resume my studies.

You are expected to know the basic Pythagorean identity

sin<SUP>2</SUP>t + cos<SUP>2</SUP>t = 1

You're also expected to know that sec t = 1/cos t, so cos t = 1 / sec t

sin<SUP>2</SUP>t + (1 / sec t)<SUP>2</SUP> = 1

Now....solve that for sin t.....


(you're too fast for me, Soroban)

 

[Al72579]

Fundmentals

Thanks soroban. It was the variations that I was shakey about. I understand it now.