Question

Find cos(2*ABC) 100POINTS

Answer 1

Answer:

$-\dfrac{7}{25}$

Step-by-step explanation:

Trigonometric Identities

$\cos(A \pm B)=\cos A \cos B \mp \sin A \sin B$

Trigonometric ratios

$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$

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:

• $\cos(\angle ABC)=\dfrac{3}{5}$

• $\sin(\angle ABC)=\dfrac{4}{5}$

Therefore, using the trig identities and ratios:

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

Answer 2

Answer:

- 0.28

Step-by-step explanation:

cos 2β = cos²β - sin²β

~~~~~~~

sin β = $\frac{4}{5}$ ⇒ sin² β = $\frac{16}{25}$

cos β = $\frac{3}{5}$ ⇒ cos² β = $\frac{9}{25}$

cos 2β = $\frac{9}{25}$ - $\frac{16}{25}$ = - $\frac{7}{25}$ = - 0.28