Solving and Simplifying using Trig IdentitiesDate: 12/5/95 at 23:9:4 From: Anonymous Subject: Trigonometry I am starting a new chapter and there are a couple of questions that I can not figure out. Could you please help? 3x 1. Solve for x: 20.0 = 85.0 2 sec @ 2. Simplify: 1 + ----- 2 cos @ 4 4 2 3. Show that tan y - sec y = 1 - 2sec y Thank you for your help. Date: 5/30/96 at 11:2:28 From: Doctor Alex Subject: Re: Trigonometry Solution 1 Lg both sides. Then you will get 3x*lg20 = lg85. I'm sure you can solve it now:-) Solution 2 I don't know how simple is simple :-) but since sec@=1/cos@, sec^2@/cos^2@ = sec^4@ => 1+sec^2@/cos^2@ = 1+sec^4@ Solution 3 Use the very useful identity A^2-B^2 = (A+B)(A-B) therefore tan^4y-sec^4y = (tan^2y+sec^2y)(tan^2y-sec^2y) ----(1) Remember the identity cos^2y + sin^2y = 1? Divide both sides by cos^2y to get 1+tan^2y = sec^2y Using this new identity, (tan^2y+sec^2y) becomes (2sec^2y-1) by adding sec^2y to both sides. Similarly (tan^2y-sec^2y) becomes -1 Substitute into (1) and you get (2sec^2y-1) (-1) = RHS I hope this helps. A^2-B^2=(A+B)(A-B) is a very very very useful identity. I would recommend that you remember it, because it can simplify many algebra and trig problems. -Doctor Alex, The Math Forum |
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