Question

Find cos(2*ABC) 100POINTS

Answer 1

- 0.28

Explanation:

cos 2β = cos²β - sin²β

~~~~~~~

sin β =
`(4)/(5)` ⇒ sin² β =
`(16)/(25)`

cos β =
`(3)/(5)` ⇒ cos² β =
`(9)/(25)`

cos 2β =
`(9)/(25)` -
`(16)/(25)` = -
`(7)/(25)` = - 0.28

Answer 2

`-(7)/(25)`

Explanation:

Trigonometric Identities

`\cos(A \pm B)=\cos A \cos B \mp \sin 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 ABC)=(3)/(5)`

• 
  `\sin(\angle ABC)=(4)/(5)`

Therefore, using the trig identities and ratios:

`\begin{aligned}\implies \cos(2 \cdot \angle ABC) & = \cos(\angle ABC + \angle ABC)\\\\& = \cos (\angle ABC) \cos (\angle ABC) - \sin(\angle ABC) \sin (\angle ABC)\\\\& = \cos^2(\angle ABC)-\sin^2(\angle ABC)\\\\& = \left((3)/(5)\right)^2-\left((4)/(5)\right)^2\\\\& = (3^2)/(5^2)-(4^2)/(5^2)\\\\& = (9)/(25)-(16)/(25)\\\\& = (9-16)/(25)\\\\& = -(7)/(25) \end{aligned}`