Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ,where $C = \arctan ( b / a ) , a = 1 , b = 3 , B = 2$ ,to rewrite the trigonometric expression in the following form. $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ $a \sin B \theta + b \cos B \theta$
A) $\sqrt { 10 }$ $\sin ( 2 \theta + 1.249 )$
B) $\sqrt { 10 }$ $\sin ( \theta - 1.249 )$
C) $\sqrt { 10 }$ $\sin ( 2 \theta - 1.249 )$
D) $\sqrt { 10 }$ $\sin ( \theta + 1.249 )$
E) $\sin ( 2 \theta + 1.249 )$

Question

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ,where $C = \arctan ( b / a ) , a = 1 , b = 3 , B = 2$ ,to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ​ $a \sin B \theta + b \cos B \theta$ ​
A) $\sqrt { 10 }$ $\sin ( 2 \theta + 1.249 )$
B) $\sqrt { 10 }$ $\sin ( \theta - 1.249 )$
C) $\sqrt { 10 }$ $\sin ( 2 \theta - 1.249 )$
D) $\sqrt { 10 }$ $\sin ( \theta + 1.249 )$
E) $\sin ( 2 \theta + 1.249 )$

Quizplus · Accepted Answer

Given $a = 1$, $b = 3$, and $B = 2$, we calculate $\sqrt{a^2 + b^2} = \sqrt{1^2 + 3^2} = \sqrt{10}$ and $C = \arctan(b/a) = \arctan(3/1)$. The value of $\arctan(3)$ is approximately 1.249 radians. Thus, the expression becomes $\sqrt{10} \sin(2\theta + 1.249)$.

Use the Formula asin⁡Bθ+bcos⁡Bθ=a2+b2sin⁡(Bθ+C)a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )asinBθ+bcosBθ=a2+b2​sin(Bθ+C)

Use the Formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$