Simplify without using a calculator or maths table (sin(-330)+cos (330)/sin (660)

Answer 1

First, let's simplify the expression step by step:

1. sin(-330):

We know that sin(-x) = -sin(x), so sin(-330) = -sin(330). Since sin(330) lies in the fourth quadrant (where sine is negative), we can determine that sin(330) < 0. Therefore, -sin(330) = sin(-330).

So, sin(-330) = sin(330).

2. cos(330):

We know that cos(-x) = cos(x), so cos(330) = cos(-330). Since cos(330) lies in the fourth quadrant (where cosine is positive), we can determine that cos(330) > 0. Therefore, cos(330) = cos(-330).

So, cos(330) = cos(-330).

3. sin(660):

Since the sine function has a periodicity of 360 degrees, we can add or subtract multiples of 360 to the angle while keeping the sine value unchanged. Therefore, sin(660) = sin(660 - 2 * 360) = sin(-60).

4. Now we can simplify the expression:

(sin(-330) + cos(330)) / sin(660) = (sin(330) + cos(330)) / sin(-60)

Using the identities sin(x) = cos(90 - x) and cos(x) = sin(90 - x), we can rewrite the expression further:

= (cos(90 - 330) + sin(90 - 330)) / sin(-60)
= (cos(-60) + sin(-60)) / sin(-60)

Since cos(-x) = cos(x) and sin(-x) = -sin(x), we can simplify it even more:

= (cos(60) - sin(60)) / (-sin(60))
= (1/2 - √3/2) / (-√3/2)
= -(1/2 - √3/2) / (√3/2)

To rationalize the denominator, multiply the expression by (√3)/(√3):

= -((1/2 - √3/2) / (√3/2)) * (√3/√3)
= -((√3 - 3) / 2) / √3
= -(√3 - 3) / (2√3)

Therefore, the simplified form of the expression (sin(-330) + cos(330)) / sin(660) is -(√3 - 3) / (2√3).