1.
Use the sum of angles for sine formula, \(\sin(\alpha +\beta) = \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)\text{,}\) to evaluate and simplify the expression \(\sin(\theta + \theta)\)
Section 7.4 Double Angle Identities
In this section, we will learn to use double-angle and half-angle formulas to find exact trigonometric values and verify identities, and use reduction identities to simplify an expression.
Subsection Textbook Reference
This relates to content in §9.3 of Algebra and Trigonometry 2e.
Exercises Preparation Exercises
Answer the following without using a calculator. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.
2.
In which quadrants are the function values negative?
(a)
\(\sin(\theta)\)
(b)
\(\cos(\theta)\)
(c)
\(\tan(\theta)\)
(d)
\(\sec(\theta)\)
3.
True or false:
\begin{equation*}
\sin(60^\circ + 60^\circ) = 2\sin(60^\circ)
\end{equation*}
Explain your reasoning.
4.
Verify the identity algebraically:
\begin{equation*}
\frac{\cos(\tau)}{1 + \sin(\tau)} = \sec(\tau)-\tan(\tau)
\end{equation*}
Exercises Practice Exercises
1.
Find the values using the half angle identities.
(a)
\(\sin(22.5^\circ)\)
(b)
\(\cos(15^\circ)\)
2.
(a)
\(\cos(\phi)\)
(b)
\(\tan(\phi)\)
(c)
\(\sin(2\phi)\)
(d)
\(\cos(2\phi)\)
(e)
\(\tan(2\phi)\)
(f)
\(\sin(\frac{\phi}{2})\)
(g)
\(\cos(\frac{\phi}{2})\)
3.
Verify the identities algebraically.
(a)
\(\sec^2(\theta)=\frac{2}{1 + \cos(2\theta)}\)
(b)
\(\left(2\sin(\theta) - 3\cos(\theta)\right)^2\sec^2(\theta)=4 - 6\sin(2\theta) + 5\cos^2(\theta)\)
4.
Using a graphing utility, find the value of \(C\) that makes the equation an identity:
\begin{equation*}
\left(\cos(x) + \sec(x)\right)^2 = C + \tan^2(x) - \sin^2(x)
\end{equation*}
Subsection Definitions
Definition 7.4.1. Double-Angle Identities.
Each of the trigonometric functions has a formula that accounts for an angle that is twice as large as some starting angle, the double-angle identities. We will focus only on those for sine, cosine, and tangent. The formulas are summarized below.
\begin{equation*}
\sin(2\theta)=2\sin(\theta)\cos(\theta)
\end{equation*}
\begin{align*}
\cos(2\theta)\amp=\cos^2(\theta)-\sin^2(\theta)\\
\amp=1-2\sin^2(\theta)\\
\amp=2\cos^2(\theta)-1
\end{align*}
\begin{equation*}
\tan(2\theta) = \dfrac{2\tan(\theta)}{1-\tan^2(\theta)}
\end{equation*}
Definition 7.4.2. Half-Angle Identities.
\begin{equation*}
\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\dfrac{1-\cos(\theta)}{2}}
\end{equation*}
\begin{equation*}
\cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\dfrac{1+\cos(\theta)}{2}}
\end{equation*}
\begin{equation*}
\tan\left(\frac{\theta}{2}\right)=\pm\sqrt{\dfrac{1-\cos(\theta)}{1+\cos(\theta)}}
\end{equation*}
Exercises Exit Exercises
1.
(a)
\(\cos(\rho)\)
(b)
\(\sin(2\rho)\)
(c)
\(\sin(\dfrac{\rho}{2}\)
2.
Algebraically verify the identity
\begin{equation*}
\sin(4\alpha) = 8 \sin(\alpha)\cos^3(\alpha) - 4\sin(\alpha)\cos(\alpha)
\end{equation*}
Reflection Reflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
