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Degrees and radians

Trigonometry, grade 10 (5 units) Lesson 2 of 21Learning

Degrees and radians

The radian

An angle with its vertex at the centre of the unit circle cuts an arc from it. The measure of the angle in radians is the length of that arc.

  • An angle of one radian cuts an arc of length 1 from the unit circle – about 57.3∘\displaystyle 57.3^{\circ}.
  • A full turn is 360∘\displaystyle 360^{\circ}, and its arc is the whole circumference, 2π\displaystyle 2\pi. So 2π\displaystyle 2\pi radians are 360∘\displaystyle 360^{\circ}:
π\displaystyle \pi =\displaystyle = 180∘\displaystyle 180^{\circ}

Winding the number t\displaystyle t onto the unit circle (the previous lesson) makes an angle of t\displaystyle t radians between the segment from the origin to (1,0)\displaystyle (1,0) and the segment from the origin to P(t)\displaystyle P(t). A negative number – an angle measured clockwise.

An angle in radians is written with no unit: π2\displaystyle \frac{\pi}{2} is an angle of π2\displaystyle \frac{\pi}{2} radians, that is 90∘\displaystyle 90^{\circ}.

Converting between degrees and radians

1∘\displaystyle 1^{\circ} is π180\displaystyle \frac{\pi}{180} radians, and one radian is 180∘π\displaystyle \frac{180^{\circ}}{\pi}. So:

  • from degrees to radians – multiply by π180\displaystyle \frac{\pi}{180}:  60∘=60⋅π180=π3\displaystyle {\ 60^{\circ}}={60\cdot\frac{\pi}{180}}={\frac{\pi}{3}};
  • from radians to degrees – multiply by 180∘π\displaystyle \frac{180^{\circ}}{\pi}:  3π4=3π4⋅180∘π=135∘\displaystyle {\ \frac{3\pi}{4}}={\frac{3\pi}{4}\cdot\frac{180^{\circ}}{\pi}}={135^{\circ}}.

When the angle is written with π\displaystyle \pi, you can also substitute π=180∘\displaystyle {\pi}={180^{\circ}}: 3π4=3⋅180∘4=135∘\displaystyle {\frac{3\pi}{4}}={\frac{3\cdot180^{\circ}}{4}}={135^{\circ}}.

Angles worth remembering
Degrees30∘\displaystyle 30^{\circ}45∘\displaystyle 45^{\circ}60∘\displaystyle 60^{\circ}90∘\displaystyle 90^{\circ}180∘\displaystyle 180^{\circ}270∘\displaystyle 270^{\circ}360∘\displaystyle 360^{\circ}
Radiansπ6\displaystyle \frac{\pi}{6}π4\displaystyle \frac{\pi}{4}π3\displaystyle \frac{\pi}{3}π2\displaystyle \frac{\pi}{2}π\displaystyle \pi3π2\displaystyle \frac{3\pi}{2}2π\displaystyle 2\pi

All the others are built from them: 2π3\displaystyle \frac{2\pi}{3} is twice 60∘\displaystyle 60^{\circ}, that is 120∘\displaystyle 120^{\circ}; 5π6\displaystyle \frac{5\pi}{6} is five times 30∘\displaystyle 30^{\circ}, that is 150∘\displaystyle 150^{\circ}.

The length of an arc and the area of a sector

In a circle of radius R\displaystyle R, a central angle of α\displaystyle \alpha radians is α2π\displaystyle \frac{\alpha}{2\pi} of a full turn. So its arc is the same part of the circumference 2πR\displaystyle 2\pi R, and its sector the same part of the disc’s area πR2\displaystyle \pi R^2:

ℓ\displaystyle \ell =\displaystyle = αR\displaystyle \alpha R
S\displaystyle S =\displaystyle = 12αR2\displaystyle \frac{1}{2}\alpha R^2

The example below works out the arc and the sector of a central angle α=π3\displaystyle {\alpha}={\frac{\pi}{3}} in a circle of radius R=6\displaystyle {R}={6}:

  • ℓ=π3⋅6=2π≈6.28\displaystyle {\ell}={\frac{\pi}{3}\cdot6}={2\pi\approx6.28};
  • S=12⋅π3⋅36=6π≈18.85\displaystyle {S}={\frac{1}{2}\cdot\frac{\pi}{3}\cdot36}={6\pi\approx18.85}.

The formulas hold only when the angle is in radians.

Exercise 1

Write the angle in radians, with π\displaystyle \pi.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 2

          Write the angle in degrees.

          a

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              c

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 3

                In a circle of radius 10\displaystyle 10 cm there is a central angle of 0.6\displaystyle 0.6 radians.

                a
                The length of the arc

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  b
                  The area of the sector

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Exercise 4

                    The minute hand of a clock is 12\displaystyle 12 cm long.
                    Its tip moves for 20\displaystyle 20 minutes.

                    a

                    Through what angle did the hand turn? Write it in radians.

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      b

                      How many centimetres did the tip of the hand travel?
                      Round to two decimal places (the last line starts with ≈\displaystyle \approx).

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                        Exercise 5

                        A sector of a circle of radius 4\displaystyle 4 has area 6\displaystyle 6.
                        Its central angle is x\displaystyle x radians.
                        Write an equation in x\displaystyle x and solve it.