verify that the equation is an identity. \n cos left( \frac { 11 pi } { 6 } + x \right) + sin left( \frac {…
verify that the equation is an identity. \n cos left( \frac { 11 pi } { 6 } + x \right) + sin left( \frac { 4 pi } { 3 } + x \right) = 0 \napply sum identities to each term in the expression, ( cos left( \frac { 11 pi } { 6 } + x \right) + sin left( \frac { 4 pi } { 3 } + x \right) ). type the new expression below.\n(do not simplify. list the terms in the same order as they appear in the original list.)
Answer
Explanation:
Step1: Apply sum formula for cosine
The sum formula for cosine is (\cos(A + B)=\cos A\cos B-\sin A\sin B). For (\cos\left(\frac{11\pi}{6}+x\right)), we have (A = \frac{11\pi}{6}), (B=x). (\cos\left(\frac{11\pi}{6}+x\right)=\cos\frac{11\pi}{6}\cos x-\sin\frac{11\pi}{6}\sin x) Since (\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}), (\sin\frac{11\pi}{6}=-\frac{1}{2}), then (\cos\left(\frac{11\pi}{6}+x\right)=\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x)
Step2: Apply sum formula for sine
The sum formula for sine is (\sin(A + B)=\sin A\cos B+\cos A\sin B). For (\sin\left(\frac{4\pi}{3}+x\right)), we have (A=\frac{4\pi}{3}), (B = x) (\sin\left(\frac{4\pi}{3}+x\right)=\sin\frac{4\pi}{3}\cos x+\cos\frac{4\pi}{3}\sin x) Since (\sin\frac{4\pi}{3}=-\frac{\sqrt{3}}{2}), (\cos\frac{4\pi}{3}=-\frac{1}{2}), then (\sin\left(\frac{4\pi}{3}+x\right)=-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x)
Step3: Add the two expressions
[ \begin{align*} &\cos\left(\frac{11\pi}{6}+x\right)+\sin\left(\frac{4\pi}{3}+x\right)\ =&\left(\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x\right)+\left(-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x\right)\ =&\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x\ =&0 \end{align*} ]
Answer:
(\cos\left(\frac{11\pi}{6}+x\right)+\sin\left(\frac{4\pi}{3}+x\right)=0) is an identity.