Fundamental Identities
For any angle or value \(\theta\), the following relationships are always true.
Fundamental Identities
Reciprocal
Quotient
Pythagorean
Eve-Odd
\[\begin{align*} \sin(\theta) = \frac{1}{\csc(\theta)} &\qquad \csc(\theta) = \frac{1}{\sin(\theta)} \\[3pt] \cos(\theta) = \frac{1}{\sec(\theta)} &\qquad \sec(\theta) = \frac{1}{\cos(\theta)} \\[3pt] \tan(\theta) = \frac{1}{\cot(\theta)} &\qquad \cot(\theta) = \frac{1}{\tan(\theta)} \end{align*}\]
\[\begin{align*} \tan(\theta) &= \frac{\sin(\theta)}{\cos(\theta)} \\[3pt] \cot(\theta) &= \frac{\cos(\theta)}{\sin(\theta)} \end{align*}\]
\[\begin{align*} \sin^2(\theta) + \cos^2(\theta) = 1 \\[6pt] \tan^2(\theta) + 1 = \sec^2(\theta) \\[6pt] 1 + \cot^2(\theta) = \csc^2(\theta) \end{align*}\]
\[\begin{align*} \sin(-\theta) &= -\sin(\theta) \\[8pt] \cos(-\theta) &= \cos(\theta) \\[8pt] \tan(-\theta) &= -\tan(\theta) \end{align*}\]
We have already seen the sets of identities given above when we learned about the unit circle. We now want to explore using them or other contexts. Here are a couple additonal sets of identities that we should familiarize ourselves with.
More Fundamental Identities
Complementary
Supplementary
\[\begin{align*} \sin\left(\frac{\pi}{2}-\theta\right) = \cos(\theta) &\qquad \csc\left(\frac{\pi}{2}-\theta\right) = \sec(\theta) \\[3pt] \cos\left(\frac{\pi}{2}-\theta\right) = \sin(\theta) &\qquad \sec\left(\frac{\pi}{2}-\theta\right) = \csc(\theta) \\[3pt] \tan\left(\frac{\pi}{2}-\theta\right) = \cot(\theta) &\qquad \cot\left(\frac{\pi}{2}-\theta\right) = \tan(\theta) \end{align*}\]
\[\begin{align*} \sin(\pi-\theta) &= \sin(\theta) \\[3pt] \cos(\pi-\theta) &= -\sin(\theta) \\[3pt] \tan(\pi-\theta) &= -\tan(\theta) \end{align*}\]
Simplifying Expressions
When simplfying trigonometric expressions, each step in the process will involve either some algebraic operation or some trigonometric identity substitution. So ask yourself each step along the way whether there is some algebra you can do or an identity you can use. A general strategy is described below.
- Can I use an identity? Look to substitute an identity to rewrite part of the expression. Look for common expressions or parts of the identities listed above. Look for relationships between a trig function in the given problem and another trig function that might be helpful. Sometimes rewriting an expression in terms of sine or cosine can be helpful.
- Can I do some algebra? Use algebraic operations such as factoring, combining like terms, FOILing or expanding. If you have fractions, try finding an LCD and combine the fractions, or divide fractions by multiplying the reciprocal. Cancel common terms in fractions.
- Try something. If you are not sure what to do, just try something. If it doesn't work or if the problem seems to be getting worse, then take a step back and try something different. Don't be afraid to experiment. It is common to require more than one attempt to work through a problem, so don't beat yourself up. If needed, ask for help. You are not alone.
Let's look at some examples using some of these Fundamental Identities to simplify other trigonometric expressions.
Verify Identities
When trying to verify or prove an identity, are goal is to show that a given equation is actually an identity. We follow the same process as simplifying expressions, but now our goal is not to completely simplify, but rather to rewrite one side of the equation to match the other side. There is one key restriction when verifying an identity. We cannot treat the potential identity as an equation and perform an operation to both sides of the equal sign with the thought that it "balances" the equation. Instead, treat each side of the potential identity as a separate expression that we must use algebra or other identities to rewrite, hopefully closer to matching the other side. Let's look at some examples.
