raph each angle and identify its reference angle.
a. 140°
b. 240°
c. 380°
Solution:
Trigonometric Functions of Any Angle Example 1.svg
a. 140° makes a 40° angle with the x-axis. Therefore the reference angle is 40°.
b. 240° makes a 60° with the x-axis. Therefore the reference angle is 60°.
c. 380° is a full rotation of 360°, plus an additional 20°. So this angle is co-terminal with 20°, and 20° is its reference angle.
If an angle has a reference angle of 30°, 45°, or 60°, we can identify its ordered pair on the unit circle, and so we can find the values of the six trig functions of that angle. For example, above we stated that 150° has a reference angle of 30°. Because of its relationship to 30°, the ordered pair for is 150° is \left ( -\tfrac{\sqrt{3}}{2},\tfrac{1}{2} \right ). Now we can find the values of the six trig functions of 150°:
\cos (150^\circ) = x = \frac{-\sqrt{3}}{2} \sec (150^\circ) = \frac{1}{x} = \frac{1}{\frac{-\sqrt{3}}{2}} = \frac{-2}{\sqrt{3}}
\sin (150^\circ) = y = \frac{1}{2} \csc (150^\circ) = \frac{1}{y} = \frac{1}{\frac{1}{2}} = 2
\tan (150^\circ) = \frac{y}{x} = \frac{\frac{1}{2}}{\frac{-\sqrt{3}}{2}} = \frac{1}{-\sqrt{3}}
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