Question

Find Sin (BAC+30) 80POINTS

Answer 1

sin<BAC

• P/H
• 16/20

cos<BAC

• B/H
• 12/20

NOw

• sin(<BAC+30)
• sin<BAC×cos30+cos<BACsin30
• 16/20(√3/2)+12/20(1/2)
• 2/5√3+3/10

Answer 2

`(2)/(5)√(3)+(3)/(10)`

Explanation:

Trigonometric Identities

`\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B`

Trigonometric ratios

`\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)`

where:

• 
  `\theta` is the angle
• O is the side opposite the angle
• A is the side adjacent the angle
• H is the hypotenuse (the side opposite the right angle)

Using the trig ratio formulas for cosine and sine:

• 
  `\cos (\angle BAC)=(12)/(20)`

• 
  `\sin (\angle BAC)=(16)/(20)`

Angles

`\sin (30^(\circ))=(1)/(2)`

`\cos (30^(\circ))=(√(3))/(2)`

Therefore, using the trig identities and ratios:

`\begin{aligned}\sin (\angle BAC + 30^(\circ)) & = \sin (\angle BAC) \cos (30^(\circ))+\cos (\angle BAC) \sin (30^(\circ))\\\\& = (16)/(20) \cdot (√(3))/(2) + (12)/(20) \cdot (1)/(2)\\\\& = (16)/(40)√(3)+(12)/(40)\\\\& = (2)/(5)√(3)+(3)/(10)\end{aligned}`