SP #7: Unit Q Concept 2: Finding all trig functions using idenitites

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 This is our problem. We are given that cotangent is 7/8 and sine is negative. Also we know that the problem lies in the third quadrant since tangent is positive and cosine is negative according to ASTC. We use what we know about identities to solve for the rest of the trig functions. We will then check our answers by solving with the unit circle, which we learned in Unit N or by using SOH CAH TOA from Unit O.

 One of the easiest things to do is find tangent which is the inverse of cotangent. Knowing this, when we use the reciprocal identities we find that tangent is equal to 8/7.

 Now that we know tangent we can use a Pythagorean identity to find secant. Secant ends up being equal to radical113 divided by 7.

 Cosine is the inverse of secant, which we just found, so we are able to use a reciprocal identity using secant to find cosine. Cosine will be 7 rad 113/ 113.

 First here we find sine using a Pythagorean identity and we can do this since we have found the value of cosine. We then find cosecant by doing the inverse of sine, also known as using a reciprocal identity.

 We have now found all the trig functions using identities. Note that all the answers except tangent and cotangent are negative because the problem lies in quadrant III.

Here is the same problem done using SOH CAH TOA. Note that the answers are identical with the answers found using identities.

1).  It is important to know several things in doing this problem. One must pay attention to what identity is being used to solve for each function. There are different ways to find each function with different identities but one must make sure that there is only one unknown trig function when plugging into an identity. Also, one must make sure to get their signs correct by understanding how we found what quadrant the answer was in and what effect that quadrant has on the answers.