Use a sum or difference identity to find the exact value.
-Two voltages, $\mathrm { V } _ { 1 } = 80 \sin ( 110 \pi \mathrm { t } )$ and $\mathrm { V } _ { 2 } = 150 \cos ( 110 \pi \mathrm { t } )$, are added in a circuit to produce the sum $V = V _ { 1 } + V _ { 2 }$. Graph $V$ in $[ - 0.02,0.02,0.002 ]$ by $[ - 250,250,20 ]$ and use the graph to estimate values for $\mathrm { a }$ and $\mathrm { k }$ so that $\mathrm { V } = \mathrm { a } \sin ( 110 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.
A) $a = 17 ; k = 1.0808$
B) $a = 170 ; k = 169.317$
C) $a = 170 ; \mathrm { k } = 1.0808$
D) $a = 170 ; k = 373.5108$

Question

Quizplus · Accepted Answer

The sum of the two voltages can be represented as a single sinusoidal function using trigonometric identities. The amplitude $a$ can be found using the Pythagorean theorem, $a = \sqrt{80^2 + 150^2} = 170$. The phase shift $k$ can be calculated by finding the angle whose tangent is the ratio of the coefficients of the sine and cosine functions, which gives $k = \arctan\left(\frac{80}{150}\right)$, resulting in $k \approx 1.0808$ radians.

Use a Sum or Difference Identity to Find the Exact V1=80sin⁡(110πt)\mathrm { V } _ { 1 } = 80 \sin ( 110 \pi \mathrm { t } )V1​=80sin(110πt)

Use a Sum or Difference Identity to Find the Exact $\mathrm { V } _ { 1 } = 80 \sin ( 110 \pi \mathrm { t } )$