Find the corresponding angle by using the relationship 180^(∘)=π.
B
b
Find the corresponding angle by using the relationship 360^(∘)=2π.
A
a
Exact value: 1/sqrt(2)
Approximate value: 0.707
B
b
Exact value: sqrt(3)/2
Approximate value: 0.866
Practice makes perfect
a
Half a lap around the unit circle corresponds to an angle of 180^(∘) and a circumference of π. Therefore, we can write the following equation to relate degrees and radians.
180^(∘)=π
To determine how many degrees corresponds to π4 radians, we have to divide both sides by 4.
As we can see, a radian of π4 corresponds to a reference angle of 45^(∘). To find the exact answer, we should draw the right triangle that our reference angle makes with the horizontal axis and the unit circle.
Notice that this is a 45-45-90 triangle, which is an isosceles triangle. In this triangle the hypotenuse is always sqrt(2) times longer than the legs. Since the hypotenuse in the unit circle is 1, we can calculate the length of the legs.
The corresponding angle to 2π3 radians is 120^(∘). The reference angle to 120^(∘) is 60^(∘).
Notice that this is a 30^(∘)-60^(∘)-90^(∘) triangle. In such a triangle the hypotenuse is always twice the length of the short leg, and the long leg is sqrt(3) times the length of the short leg.
Now we can determine the exact and approximate value of the expression.
Exact value:& sin 2Ï€/3= 1/2sqrt(3) [0.6em]
Approximate value:& sin 2π/3 ≈ 0.866
