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

Answer 1

To simplify the expression without using a calculator or mathematical tables, we can start by utilizing the periodicity of the trigonometric functions.

First, let's consider the value of sin(-330 degrees). Since the sine function has a period of 360 degrees, we can subtract multiples of 360 from -330 to bring it to a value within the range of -360 to 0:

-330 degrees = -360 degrees + 30 degrees

Now, we can simplify the expression using this equivalent value:

sin(-330 degrees) = sin(-360 degrees + 30 degrees)

Since the sine function is odd, we have:

sin(-360 degrees + 30 degrees) = -sin(30 degrees)

Now let's simplify cos(330 degrees) using the periodicity of cosine:

cos(330 degrees) = cos(360 degrees - 30 degrees)

Since the cosine function is even, we have:

cos(360 degrees - 30 degrees) = cos(30 degrees)

Finally, let's simplify the sin(660) term by using the periodicity of sine:

sin(660 degrees) = sin(2 * 360 degrees + 60 degrees)

Since the sine function is odd, we have:

sin(2 * 360 degrees + 60 degrees) = sin(60 degrees)

Now, let's substitute these simplified values back into the original expression:

sin(-330 degrees) + cos(330 degrees) / sin(660 degrees)
= -sin(30 degrees) + cos(30 degrees) / sin(60 degrees)

At this point, we can utilize the trigonometric identity:

sin(30 degrees) = 1/2, and
cos(30 degrees) = √3/2

Applying these values to the expression:

-1/2 + (√3/2) / sin(60 degrees)
= -1/2 + (√3/2) / (√3/2)
= -1/2 + 1
= 1/2

Answer 2

sorry. wrong again
sin(-330°)+cos(330°)/sin(660°)
sin(30°)+cos(-30°)/sin(-60°)
= 1/2 - 1
= -1/2

Answer 3

Apologies for the mistake. You are correct.

sin(-330°) is equal to sin(30°) since the sine function is periodic with a period of 360°.
cos(330°) is equal to cos(-30°) since the cosine function is also periodic with a period of 360°.
Similarly, sin(660°) is equal to sin(-60°).

Therefore, the expression can be simplified as follows:

sin(-330°) + cos(330°) / sin(660°)
= sin(30°) + cos(-30°) / sin(-60°)
= 1/2 + (-√3/2) / (-√3/2)
= 1/2 - 1
= -1/2