To switch between radians and degrees, push MODE on your calculator.
B
b
The circumference of a circle corresponds to an angle of 360^(∘).
A
a
sin 60^(∘) ≈ 0.866
sin π3 ≈ 0.866 Explanation: See solution.
B
b
sin π4 ≈ 0.7071
Angle: 45^(∘).
Practice makes perfect
a
To calculate sin 60^(∘), we have to make sure the calculator is set to degrees. Push MODE and check the third row.
The calculator is now set to degrees, which means if we write sin(60) on the calculator, the argument will be interpreted as degrees.
Let's switch to radians. Push MODE once more and select Radian.
The calculator is now set to radians, which means if we write sin(Ï€/3) on the calculator, the argument will be interpreted as radians.
As we can see from our calculations, we got the same answer. Both radians and degrees measure the length of the opposite side of a right triangle in the unit circle. The difference is that when set to degrees, the calculator uses an angle to calculate the length.
When switched to radians, the calculator uses the arc length an angle makes on the unit circle to measure the length of the opposite side of the angle.
b
In Part A we set the calculator to radians. Let's calculate sin π4. The argument will be interpreted as radians.
To figure out which angle this corresponds to, we should start by finding the reference angle. Notice that a full lap around the unit circle corresponds to an angle of 360^(∘) and a circumference of 2π.
This means we can equate 360^(∘) with 2π radians.
360^(∘)=2π
To determine how many degrees corresponds to π4 radians, we have to divide both sides by 8 so that the right-hand side equals π4.
As we can see, π4 radians corresponds to a reference angle of 45^(∘).
However, this is just one angle which has the same measure as sin π4. In the diagram below we see two more angles that also give the same measure as sin π4. The first angle, we obtain by subtracting the reference angle from 180^(∘). The second angle, we get by adding 360^(∘) to the reference angle.
θ_2 &= 180^(∘)-45^(∘)=135^(∘)
θ_3 &= 45^(∘)+360^(∘)=405^(∘)
Let's illustrate the angles.
