a Here you need to use a calculator. Make sure the calculator is set to radians.
B
b 4π3 corresponds to a rotation of 240^(∘). What is the reference angle to 240^(∘)?
A
a - 0.76
B
b -sqrt(3)/2
Practice makes perfect
a A radian of 4 does not have an exact answer. Therefore, to determine sin(4), we have to use a calculator. Let's make sure the calculator is set to radians. Push MODE and select Radian on the third row.
The calculator is now set to radians, which means if we write sin(4) on the calculator, the argument will be interpreted as radians.
Rounded to two decimals, we see that sin 4 has a value of - 0.76.
b To figure out if 4π3 radians gives an exact measurement, we have to determine what rotation it corresponds to. Half a lap around the unit circle corresponds to an angle of 180^(∘) and an arc length of π. With this information, we can write the following equation.
180^(∘)=π
If we multiply the equation by 4 and divide by 3, we can determine the angle of rotation.
A 240^(∘) rotation around the unit circle corresponds to a reference angle of 60^(∘).
In a 30^(∘)-60^(∘)-90^(∘) triangle, the shorter leg is half the length of the hypotenuse, and the longer leg is sqrt(3) times longer than the shorter leg.
Notice that sin( 4π3) is on the negative side of the y-axis. Therefore, the value of sin( 4π3) must be the opposite number of sin 60^(∘).
