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CanadaON
Grade 12

Find relative acute angle (rad)

Lesson

In the unit circle centred at the origin of a coordinate system, we measure angles of any magnitude between the positive axis and the radius to a point that moves on the circle. The trigonometric functions of those angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant.

For example, by reference to a unit circle diagram, it can be checked that  $\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}$sin3π4=sinπ4. That is, the second quadrant angle $\frac{3\pi}{4}$3π4 is related to the acute angle $\frac{\pi}{4}$π4. Similarly, $\cos\frac{5\pi}{4}=-\cos\frac{\pi}{4}$cos5π4=cosπ4 and we say  $\frac{5\pi}{4}$5π4 in the third quadrant is related to  $\frac{\pi}{4}$π4 in the first quadrant. 

The first quadrant or acute angle to which a particular angle is related in this way may be called a reference angle. The reference angle is between $0$0 and $\frac{pi}{2}$pi2.  (In some countries we also refer to the reference angle as the relative acute angle)

To find a reference angle, first, if necessary, add or subtract multiples of $2\pi$2π radians to obtain an angle between $0$0 and $2\pi$2π. Then, decide what quadrant the angle is in.

If the resulting angle is in the first quadrant, it is the reference angle. If it is in the second quadrant, subtract it from $\pi$π to obtain the reference angle. If it is in the third quadrant, subtract $\pi$π from the angle. In the fourth quadrant, subtract the angle from $2\pi$2π.

Examples

Question 1

Find the reference angle for $\frac{2\pi}{3}$2π3.

Question 2

Consider the angle $\frac{4\pi}{3}$4π3.

  1. In which quadrant is the terminal side of this angle?

    $IV$IV

    A

    $I$I

    B

    $III$III

    C

    $II$II

    D
  2. Determine the acute angle between the terminal side of $\frac{4\pi}{3}$4π3 and the $x$x-axis.

  3. Hence what acute angle in the first quadrant can $\frac{4\pi}{3}$4π3 be related to?

Question 3

We want to evaluate $\sin\frac{7\pi}{6}$sin7π6 by first rewriting it in terms of the related acute angle. What is the related acute angle of $\frac{7\pi}{6}$7π6?

 

Outcomes

12F.B.1.3

Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure

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