Section 9.4 Basic Identities
Let’s recap some familar relationships between the trigonometric functions.
Theorem 9.25. Reciprocal and Ratio Identities.
The reciprocal identities state:
\begin{gather*}
\csc(\theta) = \frac{1}{\sin(\theta)}\\
\sec(\theta) = \frac{1}{\cos(\theta)}\\
\cot(\theta) = \frac{1}{\tan(\theta)}
\end{gather*}
The ratio identities state:
\begin{gather*}
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\\
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\end{gather*}
The most famous identity satisfied by trigonometric functions is the so-called
Pythagorean Identity. From
Definition 9.1, suppose that
\((x,y)\) is a point on the terminal ray of
\(\theta\) and
\(r = \sqrt{x^2+y^2}\)
\begin{align*}
\left[\sin(\theta)\right]^2 + \left[\sin(\theta)\right]^2
&= \left[\frac{y}{r}\right]^2 + \left[\frac{x}{r}\right]^2 \\
&= \frac{y^2}{r^2} + \frac{x^2}{r^2}\\
&= \frac{y^2+x^2}{r^2}\\
&= \frac{r^2}{r^2}\\
&= 1.
\end{align*}
Thus,
\begin{equation*}
\left[\sin(\theta)\right]^2 + \left[\cos(\theta)\right]^2 = 1.
\end{equation*}
It’s customary to write positive integer powers of trigonometric function values like \(\left[\sin(\theta)\right]^2\) as \(\sin^2(\theta)\text{.}\) So, the identity becomes
\begin{equation*}
\sin^2(\theta) +\cos^2(\theta) =1.
\end{equation*}
Theorem 9.26. Pythagorean Identities.
\begin{gather*}
\sin^2(\theta) +\cos^2(\theta) =1\\
\tan^2(\theta) +1 = \sec^2(\theta)\\
1+ \cot^2(\theta) = \csc^2(\theta)
\end{gather*}
Proof.
We have already proved the first indentity. Starting from here and dividing every term by \(\cos^2(\theta)\text{,}\)
\begin{gather*}
\sin^2(\theta) +\cos^2(\theta) =1\\
\frac{\sin^2(\theta)}{\cos^2(\theta)} +\frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}\\
\left[\frac{\sin(\theta)}{\cos(\theta)}\right]^2 +1 = \left[\frac{1}{\cos(\theta)}\right]^2
\end{gather*}
\begin{equation*}
\tan^2(\theta) +1 = \sec^2(\theta).
\end{equation*}
The third identity is left to you to prove.
Example 9.27. Finding Trigonometric Function Values Using Identities.
Suppose \(\cos(\theta) = -0.7\text{.}\) Let’s find the remaining trigonometric function values of \(\theta\text{,}\) given that \(\theta\) is a Quadrant III angle.
By the Pythagorean Identity,
\begin{equation*}
\sin^2(\theta) = 1-\cos^2(\theta)
\end{equation*}
so that
\begin{equation*}
\sin(\theta) = \pm \sqrt{1-\cos^2(\theta)}.
\end{equation*}
But sine is negative in Quadrant II, so
\begin{equation*}
\sin(\theta) = {\color{red}-}\sqrt{1-\cos^2(\theta)} = \sqrt{1-(0.7)^2} \approx -0.51.
\end{equation*}
Everything falls into place now using ratio and reciprocal identities:
\begin{gather*}
\tan(\theta) \approx \frac{-0.51}{-0.7} \approx 0.73\\
\csc(\theta) \approx \frac{1}{-0.51} \approx -1.96\\
\sec(\theta) = \frac{1}{-0.7} \approx -1.43\\
\cot(\theta) \approx \frac{-0.7}{-0.51} \approx 1.37
\end{gather*}
