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Q: Trigonometric Functions ( Answered ,   2 Comments ) Question
 Subject: Trigonometric Functions Category: Science > Math Asked by: 3eb-ga List Price: \$35.00 Posted: 22 Nov 2005 18:26 PST Expires: 22 Dec 2005 18:26 PST Question ID: 596522
 ```I would like to know when to use specific trig functions in word problems and why. For example, let's say I have a tree, from a point 115 feet from the base of the tree, the angle of elevation to the top of the tree is 64.3 degrees. Find the height of the tree to the nearest foot. In my precalculus class we used the Tangent function, to get a result of approx. 239 feet. Now why did we use this function? I would like examples of different problems, which trig functions used, and why. Thank you very much for your help.``` Clarification of Question by 3eb-ga on 22 Nov 2005 18:41 PST ```Just a clarification, to start me in the right direction all I need now is Trig functions of Acute angles.``` Subject: Re: Trigonometric Functions Answered By: answerguru-ga on 22 Nov 2005 19:05 PST Rated: ```Hi 3eb-ga, Thanks for your question - the first step when looking at this type of problem is to draw out what is being described. Since trigonometry is the calculation of distance and angles of right triangles, we can always classify each side of the triangle relative to a given angle. Remember that each right triangle contains two legs and a hypotenuse. In your example you are given an angle of 64.3 degrees. After drawing this out you can see that this angle is opposite to the tree. Therefore the leg of the right triangle represented by the tree is called the "opposite" side. Since we know the opposite side, we can then label the "adjacent side", which is the other leg of the triangle. This is the side of the triangle represented by the ground (we know the value of this side is 115 feet). The tangent of an angle is a ratio of (opposite side / adjacent side), or for short: tan(x) = opp/adj tan(64.3 deg) = opp/115 115*tan(64.3) = opp opp = 238.95 feet Now in addition to the tangent, there are two other common trig functions called sine and cosine. Sine of an angle is the ratio of the opposite side to the hypotenuse. The formula is: sin(x) = (opposite side / hypotenuse) = opp/hyp We would use the sine function when we are given an angle and need to determine either the opposite side or the hypotenuse (where we are given the other side in the question). Cosine of an angle is the ratio of the adjacent side to the hypotenuse. The formula is: cos(x) = (adjacent side / hypotenuse) = adj/hyp We would use the cosine function when we are given an angle and need to determine either the adjacent side or the hypotenuse (where we are given the other side in the question). Here are the steps to following general: 1. Draw out a diagram of the problem using the information in the question 2. Based on the angle that you are given, label the sides of the right triangle (opposite, adjacent, and hypotenuse). With some practice you will be able to identify the sides without labelling them. 3. Identify which side you want to calculate, and then select the correct trig function to use based on the information you've been provided. So if you have an angle and an adjacent side and need to calculate the hypotenuse, the only equation that works is cosine. 4. Plug the information you have into the appropriate equation and solve for the desired variable. As a rule for this type of question, you will always be given one angle and the distance of one leg. I believe what confuses most people is the identification of the different sides. Once you can do that, it is just a matter of identifying what you have, what you are looking for, and solving the equation that uses all three pieces of information. I hope this has given you a deeper understanding of the basic trigonometric functions. If you have problems understanding any of the information above, please post a clarification and I will respond promptly. Thanks for using Google Answers! Cheers, answerguru-ga``` Request for Answer Clarification by 3eb-ga on 23 Nov 2005 08:34 PST ```Thank you for the 3 main functions, but could you please include Cosecant, Secant, and Cotangent. Please include examples of each, and I'll include a tip. Thanks again for your prompt attention.``` Clarification of Answer by answerguru-ga on 23 Nov 2005 11:18 PST ```Hi 3eb-ga, In response to your clarification I am including details on Cosecant, Secant, and Cotangent. These were not originally included because they are simply alternate formulas. The cotangent of an angle is a ratio of (adjacent side / opposite side), or for short: cot(x) = adj/opp We would use the cotangent function when given an angle and need to determine either the opposite or adjacent side (where we are given the other side in the question). This is the same condition as tangent, so you could use either one. Using your original tree example, you could easily solve the problem using cotangent: tan(64.3) = opp/115 1/tan(64.3) = 115/opp cot(64.3) = 115/opp opp = cot(64.3)/115 opp = 238.95 feet Cosecant of an angle is the ratio of the hypotenuse to the opposite side. The formula is: csc(x) = (hypotenuse / opposite side) = hyp/opp The relationship between sine and cosecant is: csc(x) = 1/sin(x) We would use the cosecant function when we are given an angle and need to determine either the opposite side or the hypotenuse (where we are given the other side in the question). This is the same condition as cosine, so you could use either one. Secant of an angle is the ratio of the hypotenuse to the adjacent side. The formula is: sec(x) = (hypotenuse / adjacent side) = hyp/adj The relationship between cosine and secant is: sec(x) = 1/cos(x) We would use the secant function when we are given an angle and need to determine either the adjacent side or the hypotenuse (where we are given the other side in the question). This is the same condition as cosine, so you could use either one. Cheers, answerguru-ga```
 3eb-ga rated this answer: and gave an additional tip of: \$5.00 ```Thank you very much for the quick response. The information given will be of much help!``` ```If 3eb-ga is at high school level, I can accept the definition of sin(x), cos(x) and tan(x) only if the right-angle triangle is used to define the three sides. If 3eb-ga is at college level, I would suggest using the unit-circle to define the trigonometic function.```
 ```I have found a small error in the answer to the trig function... Look at step 4 tan(64.3) = opp/115 1/tan(64.3) = 115/opp cot(64.3) = 115/opp opp = cot(64.3)/115 opp = 238.95 feet It should be opp=115/cot(64.3) But somehow 238.95 feet is the right answer. Here is just a quick formula sum up. (cosX)^(-1)=secX (sinX)^(-1)=cscX (tanX)^(-1)=cotX For more useful information go here... http://en.wikipedia.org/wiki/Trigonometric_functions``` 