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Section 7.3 Sum And Difference
In this section, we will learn to use sum and difference formulas for sine, cosine, and tangent, and to use sum and difference formulas to verify identities.
Subsection Textbook Reference
Exercises Preparation Exercises
1.
Answer the following without using a calculator. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.
(a)
Write down the memorized values from the unit circle.
\(\displaystyle \sin(\pi)\)
\(\displaystyle \sin(\frac{\pi}{3})\)
\(\displaystyle \sin(\frac{5\pi}{3})\)
\(\displaystyle \sin(\frac{3\pi}{2})\)
\(\displaystyle \sin(\frac{7\pi}{6})\)
\(\displaystyle \sin(-\frac{3\pi}{4})\)
(b)
Count from \(0\) to \(\pi\) by multiples of \(\frac{\pi}{12}\text{,}\) and then reduce those fractions. Here’s how to start out:
\begin{equation*}
\frac{0\pi}{12}, \frac{\pi}{12}, \frac{2\pi}{12}, \frac{3\pi}{12}...
\end{equation*}
(c) Write down the three Pythagorean Identities.
(d)
True or False:
\begin{equation*}
\sin(45^\circ+60^\circ) = \sin(45^\circ) + \sin(60^\circ)
\end{equation*}
Explain your reasoning.
(e)
Verify the identity:
\begin{equation*}
\frac{\sec(\phi)\sin(\phi)}{\tan(\phi)+\cot(\phi)} = \sin^2(\phi)
\end{equation*}
Exercises Practice Exercises
1.
Use the sum and difference formulas and your memorized unit circle values to evaluate the following expressions without a calculator.
(a)
\(\sin(60^\circ - 45^\circ)\)
(b)
\(\sin\left(\frac{2\pi}{3} + \frac{\pi}{4}\right)\)
(c) (d)
\(\cos\left(\frac{17\pi}{12}\right)\)
(e) (f)
\(\tan\left(\frac{\pi}{3} - \frac{\pi}{4}\right)\)
2.
Imagine that there are two angles,
\(\alpha\) and
\(\beta\text{,}\) not in the same triangle, such that
\(\sin(\alpha) = \frac{2}{7}\) where
\(\frac{\pi}{2}\lt \alpha \lt \pi\) and
\(\cos(\beta) = -\frac{5}{6}\) where
\(\pi \lt \beta \lt \frac{3\pi}{2}\text{.}\)
(a) Find the value of
\(\cos(\alpha)\)
(b) Find the value of
\(\sin(\beta)\)
(c) Find the value of
\(\sin(\alpha +\beta)\)
(d) Find the value of
\(\cos(\alpha -\beta)\)
3.
Verify the identities algebraically.
(a)
\(\cos(\lambda + \nu) \cos(\lambda - \nu) = \cos^2(\lambda) - \sin^2(\nu)\)
(b)
\(\tan(\rho + \frac{\pi}{4}) = \frac{\cos(\rho)+\sin(\rho)}{\cos(\rho)-\sin(\rho)}\)
Subsection Definitions
Definition 7.3.1 . Sum And Difference Identities.
There are sum and difference identities for each of the trigonometric functions, but we only focus on those for sine, cosine, and tangent.
\(\displaystyle \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)
\(\displaystyle \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\)
\(\displaystyle \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\)
\(\displaystyle \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\)
\(\displaystyle \tan(\alpha + \beta) = \dfrac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)\)
\(\displaystyle \tan(\alpha - \beta) = \dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)\)
Exercises Exit Exercises
1. What is the difference between a zero and an
\(x\) -intercept of a polynomial function?
2. Evaluate the expression
\(\cos(285^\circ)\\) without a calculator.
3.
Imagine that there are two angles,
\(\kappa\) and
\(\tau\text{,}\) not necessarily in the same triangle, such that
\(\sin(\kappa) = - \frac{3}{4}\) where
\(\frac{3\pi}{2}\lt \kappa\ lt \pi\) and
\(\cos(\tau) = \frac{3}{8}\) where
\(0 \lt \tau \lt \frac{\pi}{2}\text{.}\)
(a) Find the values of
\(\cos(\kappa)\) and
\(\sin(\tau)\text{.}\)
(b) Find the value of
\(\sin(\kappa - \tau)\text{.}\)
(c) Find the value of
\(\tan(\kappa + \tau)\text{.}\)
Reflection Reflection
1. On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
