Question

2 sin 105°•cos 105°

Answer 1

$\displaystyle 2 sin 105^{\circ} \cdot cos105^{\circ}= 2 \cdot \frac{1}{2} (sin(105^{\circ}-105^{\circ})+sin(105^{\circ}+105^{\circ}))=$
$\displaystyle =sin0^{\circ}+sin210^{\circ}=sin(180^{\circ}+30^{\circ})=-sin30^{\circ}=-\frac{1}{2}$

Answer 2

To calculate the value of the expression 2 sin 105° • cos 105°, we first need to find the sine and cosine values of 105°. However, the exact values of sine and cosine for 105° are not commonly known angles, so we'll have to use trigonometric identities to find the answer.

We'll start by using the identity sin 2θ = 2 sin θ • cos θ. Rearranging this identity, we can write sin θ • cos θ = 1/2 • sin 2θ.

Now, let's apply this identity to our expression:

2 sin 105° • cos 105° = 2 • (1/2) • sin 2(105°)

To find sin 2(105°), we can use another trigonometric identity: sin 2θ = 2 sin θ • cos θ. In this case, θ = 105°, so we have sin 2(105°) = 2 sin 105° • cos 105°.

Now, we can substitute this back into our expression:

2 sin 105° • cos 105° = 2 • (1/2) • 2 sin 105° • cos 105°

Canceling out the 2's, we get:

2 sin 105° • cos 105° = sin 105° • cos 105°

So, the value of the expression 2 sin 105° • cos 105° is equal to sin 105° • cos 105°.

Note: Since sin 105° and cos 105° are not commonly known values, we can use a calculator or a trigonometric table to find their numerical approximations.