The Reference Angle

Section 10.3 The Reference Angle

Associated to every non-quadrantal angle is an acute angle called its reference angle. We will see that the reference angle and the quadrant completely determine its trigonometric function values.

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Definition 10.3.1. The Reference Angle.

Suppose \(\theta\) is a non-quadrantal angle in standard position. The reference angle of \(\theta\text{,}\) denoted by \(\reference{\theta}\text{,}\) is the smallest positive angle formed by the terminal ray of \(\theta\) and the \(x\)-axis.

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Study the Definition 10.3.1 above and each figure below to see how the reference angle is determine in each quadrant for \(0 \lt \theta \lt 2\pi\text{.}\)

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Figure 10.3.2. QII: \(\reference{\theta}=\pi-\theta\text{.}\)
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Figure 10.3.3. QI: \(\reference{\theta}=\theta\text{.}\)
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Figure 10.3.4. QIII: \(\reference{\theta}=\theta-\pi\text{.}\)
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Figure 10.3.5. QIV: \(\reference{\theta}=2\pi-\theta\text{.}\)
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Example 10.3.6. Using the Reference Angle.

Suppose \(\theta = 11\pi/3\text{.}\)

The reference angle is formed by the terminal ray and the positive \(x\)-axis. This smallest such positive angle is \(\reference{\theta} = \pi/3\text{.}\) Drawing and \(\theta\) and \(\reference{\theta}\) both in standard position highlights why its a useful to consider the reference angle.

If \((x,y)\) is a point on the terminal side of \(\reference{\theta}\text{,}\) then after adjusting for the quadrant, \((x,-y)\) is a point on the terminal side of \(\theta\text{.}\) Thus, all the trigonometric function values of \(\theta\) and \(\reference{\theta}\) are the same or opposite and the sign is determined entirely by the quadrant. To illustrate, we know sine is negative in Quadrant IV. So,

\begin{equation*} \sin(11\pi/3) = - \sin(\reference{\theta}) = - \sin(\pi/3) = - \frac{\sqrt{3}}{2}, \end{equation*}

where we obtained the last value by referring to the special triangle Figure 9.3.4. Since cosine is positive in Quadrant IV,

\begin{equation*} \cos(11\pi/3) = + \cos(\reference{\theta}) = \cos(\pi/3) = \frac{1}{2}. \end{equation*}

Since tangent is negative in Quadrant IV,

\begin{equation*} \tan(11\pi/3) = - \tan(\reference{\theta}) = -\tan(\pi/3) = -\sqrt{3}. \end{equation*}

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Thus, the use of the reference angle reduces all computations of trigonometric function values down to that of a positive acute angle.

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Principle 10.3.7. Reference Angle.

Suppose \(\theta\) is a non-quadrantal angle with reference angle \(\reference{\theta}\text{.}\) Then the trigonometric function values of \(\theta\) are the same or the opposite of those for \(\reference{\theta}\) with the sign determined by the quadrant. (See Theorem 10.2.2.)

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Example 10.3.8. Using the Reference Angle.

Use the reference angle together with the special triangles Figure 9.3.4 and Figure 9.3.2 to find the sine, cosine, and tangent of each angle.

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\(\displaystyle \theta = \frac{7\pi}{6}\)

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\(\displaystyle \theta = -\frac{5\pi}{4}\)

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\(\displaystyle \theta = 780^\circ\)

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Solution.

\(\theta\) is in Quadrant III with reference angle \(\reference{\theta} = \pi/6\text{.}\)

\begin{gather*} \sin(7\pi/6) = - \sin(\pi/6) = -1/2\\ \cos(7\pi/6) = - \cos(\pi/6) = -\sqrt{3}/2\\ \tan(7\pi/6) = + \tan(\pi/6) = 1/{\sqrt{3}} \end{gather*}

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\(\theta\) is in Quadrant II with reference angle \(\reference{\theta} = \pi/4\text{.}\)

\begin{gather*} \sin\left(-\frac{5\pi}{4}\right) = \sin(\pi/4) = \sqrt{2}/2\\ \cos\left(-\frac{5\pi}{4}\right) = -\cos(\pi/4) = -\sqrt{2}/2\\ \tan\left(-\frac{5\pi}{4}\right) = - \tan(\pi/4) = -1 \end{gather*}

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\(\theta = 780^\circ = (60+360+360)^\circ\) is in Quadrant I with reference angle \(\reference{\theta} = 60^\circ\text{.}\)

\begin{gather*} \sin(780^\circ) = \sin(60^\circ) = \sqrt{3}/2\\ \cos(780^\circ) = \cos(60^\circ) = 1/2\\ \tan(780^\circ) = \tan(60^\circ) = \sqrt{3} \end{gather*}

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