7.3: Radians
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Angles can be measured in units of degrees or radians.
Radians are a unit of angular measure defined such that
2π radians =360˚
For the crafty at heart, an angle of 1 radian can be created with a circle (use any lid from a jar) and a piece of string.


Recall that the circumference of a circle (which is the perimeter of a circle) has a formula:
Circumference = C=2πr where r= radius of the circle
How many strings of length r are needed to wrap around the entire circle? We divide the circumference r ways: 2πrr=2π≈6. About 6 strings of length r, or exactly 2π strings.
An angle’s counter-clockwise rotation can be measured as a fraction of a full rotation.
Once you get the hang of radians, your trigonometric calculations are quick and easy. Using radians for angles, rather than degrees, is the key to gain fluidity in trigonometry. They are simpler to use than degrees. Notice the shared denominators from one quadrant to the next. Also notice the denominator of an angle matches its own reference angle’s denominator.
30˚=π6 radians 45˚=π4 radians 60˚=π3 radians
Reference angles refer to QI, where your memorization comes into play. Memorize only QI and the unit circle’s ordered pairs in that quadrant (see Section 7.2). Avoid memorizing too much! Use intuitive computations to rehearse math concepts. If a radian value is not one of the above values, nor coterminal to the ones shown, then you will need a calculator for trigonometric computations.
Give the reference angle ˆθ (in radians) for the given standard angle θ. Then sketch both θ and ˆθ.
- θ=7π6
- θ=−2π3
- θ=11π4
- θ=6 radians
Solution
- θ=7π6. Decide whether 7π6 is closer to π or 2π.
Since π=6π6 and 2π=12π6, it is clear that 7π6>6π6 and closer to π.
The reference angle is ˆθ=π6.
7π6=π+π6.
The sketch is shown below.
- θ=−2π3. The angle rotates clockwise.
|−2π3|=2π3 which terminates in QII.
The clockwise rotation of −2π3 terminates in QIII.
Both 2π3 and −2π3 have the same reference angle.
ˆθ=π3.
The sketch is shown below.
- θ=11π4. The denominator is 4.
Since 8π4=2π and 11π4>8π4, this angle rotates beyond one full revolution. π is coterminal with 3π, 5π, 7π … while 2π is coterminal with 4π, 6π, 8π,...
Compare 11π4 to 3π=12π4. They are very close in value! 11π4<12π4 and 11π4 therefore terminates In QII. The reference angle ˆθ=π4.
The sketch of the angle is shown below.
- θ=6 radians. Decide whether 6 radians is closer to π or 2π.
Since π≈3.14 and 2π≈6.28, 6 radians is closer to 2π. 6<2π, therefore the angle terminates in QIV.
The reference angle ˆθ=2π−6≈0.28 radians.
The sketch of the angle is shown below.
The terminal side of a standard angle θ=3π4 radians intersects the unit circle. State the ordered pair of the intersection.
Solution
Since 3π4=4π4−π4, the angle θ=3π4 terminates in QII with reference angle ˆθ=π4.
The unit circle was introduced in Section 7.2 for degree-angles. The figure shown below includes radian-angles.
The ordered pair on the unit circle, intersecting the terminal side of π=3π4 is (−√22,√22).
The terminal side of a standard angle θ=5π2 radians intersects the unit circle. State the ordered pair of the intersection.
Solution
Since 5π2=π2+2π, the angles 5π2 and π2 are coterminal angles. Coterminal angles land in the same spot, so the ordered pair will be the same. The ordered pair on the unit circle, intersecting the terminal side of θ=5π2 is (0,1).
The terminal side of a standard angle θ=10π radians intersects the unit circle. State the ordered pair of the intersection.
Solution
Since 10π=2π(5), the angles 10π and 2π are coterminal angles. Therefore, the ordered pair on the unit circle, intersecting the terminal side of θ=10π is (1,0).
Try It! (Exercises)
For #1-10, A standard angle θ is given in radians. State the reference angle ˆθ in radians.
- θ=11π6
- θ=7π9
- θ=11π7
- θ=2 radians
- θ=−5π6
- θ=−5π4
- θ=9π4
- θ=17π6
- θ=10 radians
- θ=−6π5
For #11-20, the terminal side of the given standard angle, θ, intersects the unit circle at a point. State the ordered pair of the intersection.
- θ=7π6
- θ=4π3
- θ=11π3
- θ=7π2
- θ=5π
- θ=13π4
- θ=−7π6
- θ=−4π
- θ=−11π4
- θ=−5π2
For #21-27, a standard angle’s rotation is described in words. You are given several hints about its rotation. Note: a full revolution is 2π radians (if counter-clockwise) or −2π radians (if clockwise). Find the measure of the angle described using radians.
- An angle has a counter-clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QIII. The reference angle ˆθ=π3.
- An angle has a counter-clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QIV. The reference angle ˆθ=π4.
- An angle has a counter-clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QII. The reference angle ˆθ=π6.
- An angle has a clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QIV. The reference angle ˆθ=π3.
- An angle has a clockwise rotation. The angle does not make a full revolution. The terminal side of the angle intersects the unit circle at point (−1,0).
- An angle has a counter-clockwise rotation. The angle rotates more than one full revolution. The terminal side of the angle intersects the unit circle at point (−√22,−√22).
- An angle has a counter-clockwise rotation. The angle rotates more than two full revolutions. The terminal side of the angle intersects the unit circle at point (−12,√32).