Simplify without using a calculator or maths table(sin(-330 degrees)+cos(330)/sin (660))
sorry. wrong again
sin(-330°)+cos(330°)/sin(660°)
sin(30°)+cos(-30°)/sin(-60°)
= 1/2 - 1
= -1/2
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
To simplify the expression without using a calculator or math table, let's use the periodicity and symmetry properties of the trigonometric functions.
1. Start with sin(-330 degrees). Since the sin function has a period of 360 degrees, we can rewrite this as sin(-330 degrees + 360 degrees).
-330 degrees + 360 degrees gives us 30 degrees.
So, sin(-330 degrees) is equal to sin(30 degrees).
2. Next, let's simplify cos(330 degrees). Similar to sin, the cos function also has a period of 360 degrees. So, we can rewrite this as cos(330 degrees - 360 degrees).
330 degrees - 360 degrees gives us -30 degrees.
Therefore, cos(330 degrees) is equal to cos(-30 degrees).
3. Now, we have sin(30 degrees) + cos(-30 degrees) in the numerator. By using the symmetry property of the cosine function (cos(-θ) = cos(θ)), we can rewrite cos(-30 degrees) as cos(30 degrees). So, the numerator becomes sin(30 degrees) + cos(30 degrees).
4. In the denominator, we have sin(660 degrees). Since the sin function has a period of 360 degrees, we can rewrite this as sin(660 degrees - 360 degrees).
660 degrees - 360 degrees gives us 300 degrees.
Thus, sin(660 degrees) is equal to sin(300 degrees).
5. Finally, we have sin(30 degrees) + cos(30 degrees) in the numerator and sin(300 degrees) in the denominator.
Now, sin(30 degrees) and cos(30 degrees) are both well-known values. Remembering the unit circle, sin(30 degrees) = 1/2 and cos(30 degrees) = √3/2.
Additionally, sin(300 degrees) is equal to -1/2 based on the unit circle.
Now we can substitute these values into the expression:
(1/2 + √3/2) / (-1/2)
To simplify further, we divide each term in the numerator by the denominator:
(1/2) / (-1/2) + (√3/2) / (-1/2)
Simplifying further:
(1/2) * (-2/1) + (√3/2) * (-2/1)
This gives us:
-1 + (-√3) = -1 - √3
Therefore, the simplified form of the expression (sin(-330 degrees) + cos(330 degrees)) / sin(660 degrees) is -1 - √3.
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