sin480° leaving your answer in surds form
sin480° can be expressed in surd form as √3/2.
sin480° convert to surds
To convert sin480° to surds, we need to use the sum and difference angle formulas for sine.
First, let's find an equivalent angle within the first 360°:
480° - 360° = 120°
Since the sin function is periodic with a period of 360°, sin480° is equal to sin120°.
Now, we can apply the sine formula for 2x:
sin2x = 2sinx cosx
Replacing 2x with 120°:
sin240° = 2sin120° cos120°
Now, let's express 120° as 60° + 60°:
= 2sin60° cos60°
Using the value of sin60° as √3/2 and cos60° as 1/2, we can simplify further:
= 2(√3/2 * 1/2)
= √3/2
Thus, sin480° converted to surds is √3/2.
What a long-winded solution
Agree with the bot that first step is to subtract multiples of 360° from the given angle:
sin 480 = sin (480-360) = sin 120
now 120° is in quad II and 60° away from the x-axis, and in quad II the sine is positive
sin 120 = sin 60 = √3/2 , all done!!!!
(you should know the trig functions of the standard angles, like 30, 45, 60, 90)
Apologies for the lengthy explanation. You are correct, the simpler and quicker way to solve sin480° is to recognize that it is equivalent to sin120°. By considering the angle in the second quadrant where sine is positive and knowing the trigonometric values of standard angles, we can directly conclude that sin120° is equal to √3/2. Thank you for pointing that out!
To find the value of sin 480° in surd form, we can use the identity that sin (180° + θ) = -sinθ.
Here, we have 480° which is equivalent to 360° + 120°. So, we can write sin 480° as sin (360° + 120°).
Using the identity mentioned above, sin (360° + 120°) = -sin 120°.
Now, we know that sin 120° = √3/2.
Therefore, sin 480° = -√3/2.
So, the value of sin 480° in surd form is -√3/2.
To find the value of sin(480°), we can use the unit circle and trigonometric properties.
Step 1: Convert the angle to radians.
To convert from degrees to radians, we use the formula:
radians = degrees × (π/180)
Given angle: 480°
480° × (π/180)
= 8π/3 radians
Step 2: Utilize the periodicity of the sine function.
The sine function has a period of 2π, which means that sin(x) = sin(x + 2π), for any angle x.
Since we have an angle of 8π/3, which is greater than 2π, we can rewrite it as follows:
8π/3 = 2π + 2π/3
Step 3: Determine the reference angle.
The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For an angle of 2π/3, the reference angle is π/3.
Step 4: Evaluate the sine function.
sin(8π/3) = sin(2π + 2π/3)
= sin(2π + π/3)
= sin(7π/3)
Now, we need to figure out where the angle 7π/3 falls on the unit circle.
Step 5: Determine the quadrant.
7π/3 is greater than π but less than 2π. This means it falls in the second quadrant.
Step 6: Use the sign of the sine function in the second quadrant.
In the second quadrant, the sine function is positive. Therefore, sin(7π/3) = sin(π/3).
Step 7: Calculate the value of sin(π/3).
In the first quadrant, on the unit circle, the value of sin(π/3) is √3/2.
Therefore, sin 480°, in surd form, is equal to √3/2.