Trigonometry123

Question

With..... sin²Θ + Cos²Θ = 1

When on a test or doing problems what would I see or need to recognize to know i need to use this trig identity to get an answer or the answer? I understand how to solve it if its in front of me but i do not know when to apply it.

Answer 1

This is not as easy a question as others in math. Using trig identities is like using tools to solve a puzzle. You have the identities in your tool box and you look for ways to simplify expressions using the tools in your toolbox. Sometimes you might use a reciprocal identity [cos(x) = 1/sec(x)]. Other times you might use a Pythagorean Identity [cos²(x) + sin²(x) = 1]. And there are lots of others that you may learn in the near future.

Look for clues. If you see expressions with squares in them, there's a good chance you will want to consider how to use a Pythagorean Identity to simplify because they have squares in them. If there aren't any squares, you'll probably need to think of other identities to simplify.

Example: Simplify cos²(x) + cos²(x)tan²(x)

You see a trig function squared so there's a really good chance you're going to need to use one of the Pythagorean Identities. But there's no "cos²(x) + sin²(x) = 1". So, the question becomes, "Can I use some other trig identity first, to convert this into a "cos²(x) + sin²(x) = 1" or other Pythagorean Identity?"

Method #1: Realize that tan(x) = sin(x)/cos(x). That means tan²(x) = sin²(x)/cos²(x) so I can replace the tan²(x) with "sin²(x)/cos²(x)". That gives me cos²(x) + cos²(x)[sin²(x)/cos²(x)] which allows me to cancel a cos²(x) from the top and bottom on the far right of this expression. That leaves me cos²(x) + sin²(x) and since cos²(x) + sin²(x) = 1, the entire expression simplifies to 1.

Method #2: I see tan²(x) and I think about the Pythagorean Identity 1 + tan²(x) = sec²(x). That means tan²(x) = sec²(x).- 1 so I replace tan²(x) with "sec²(x).- 1" giving me:

cos²(x) + cos²(x)[sec²(x).- 1]

Now, distributing across the parenthesis gives me:

cos²(x) + cos²(x)sec²(x).- cos²(x)

and since the cos²(x) at the beginning is wiped out by the "- cos²(x)" at the end, I get cos²(x)sec²(x). Since sec(x) = 1/cos(x), this simplifies to cos²(x)/cos²(x) which equals 1

These are just a couple of examples. I wish it was as easy as just saying something like "add 4 to each side of the equation" but these require creativity. You REALLY need to have a sheet of trig identities that you use that you can consider your "toolbox" and, when you get a problem, look in your toolbox for possible tools that might work in this particular situation. A screwdriver is a great tool, but if you're hammering in a nail, it's not going to be of much good. You'll need a different tool. Eventually, you'll need to memorize all the identities so you can become proficient at these types of problems.

Hope this helps a little bit.