Simplify without using a calculator or maths table (sin(-330)+cos (330)/sin (660)
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).
To simplify the expression (sin(-330) + cos(330)) / sin(660) without using a calculator or math table, we can use the following trigonometric identities:
1. sin(-x) = -sin(x)
2. cos(-x) = cos(x)
3. sin(x + 360) = sin(x)
4. cos(x + 360) = cos(x)
Let's break down the expression step-by-step:
1. sin(-330) + cos(330)
Using identity 1: -sin(330) + cos(330)
2. sin(330) + cos(330)
We can rewrite 330 degrees as (330 - 360) since sin and cos functions are periodic with a period of 360.
Using identity 3: sin(-30) + cos(-30)
Using identity 1 and 2: -sin(30) + cos(30)
3. -sin(30) + cos(30)
We know the exact values of sine and cosine for 30 degrees.
sin(30) = 1/2
cos(30) = √3/2
Therefore, substituting these values:
-1/2 + √3/2
4. (sin(-330) + cos(330)) / sin(660)
Now, let's simplify the denominator sin(660) using the same approach as above:
sin(660) = sin(660 - 360) = sin(300)
sin(300) = -1/2
Substituting the values into the expression:
(-1/2 + √3/2) / (-1/2)
5. Simplifying the expression:
(-1/2 + √3/2) * (-2/1)
This cancels out the negative signs in the numerator and denominator:
(1/2 - √3/2) * 2
Distributing the multiplication:
1 - √3
Therefore, the simplified expression is 1 - √3.