Question 6.3.7: Using Reference Angles to Find Trigonometric Function Values...
Using Reference Angles to Find Trigonometric Function Values
OBJECTIVE
Find the value of any trigonometric function of a nonquadrantal angle θ.
Step 1 Assuming that θ > 360° or θ < 0°, find a coterminal angle for θ with degree measure between 0° and 360°. Otherwise, go to Step 2.
Step 2 Find the reference angle θ′ for the angle resulting from Step 1.Write the trigonometric function of θ′.
Step 3 Choose the correct sign for the trigonometric function value of θ based on the quadrant in which it lies. Write the given trigonometric function of θ in terms of the same trigonometric function of θ′ with the appropriate sign.
Find sin 1320°.
Because 1320^{\circ}=3\left(360^{\circ}\right)+240^{\circ}, \text { the angle } 240^{\circ} is coterminal with the angle 1320°.
Because 240° is in quadrant III, its reference angle θ’ is
\theta^{\prime}=240^{\circ}-180^{\circ}=60^{\circ}
and
\sin \theta^{\prime}=\sin 60^{\circ}=\frac{\sqrt{3}}{2}
The angle 1320° and its coterminal angle, 240°, lie in quadrant III, where the sine is negative. So
\begin {matrix}\underset{\uparrow }{\sin 1320°} = \underset{\uparrow }{\sin 240°} & = \underset{\uparrow }{-\sin 60°} & = -\frac{\sqrt{3}}{2}. \\ \fbox {coterminal angles } & \fbox {reference angle} & \end {matrix}