Trigonometric Functions of Any Angle The Reference Angle
- Slides: 8
Trigonometric Functions of Any Angle
The Reference Angle • A reference angle is the acute angle formed by the terminal side of an angle in standard position with the x-axis.
Trigonometric Function at θ • To find the value of a trigonometric function at θ, we simply need to find the value of that trigonometric function at the corresponding reference angle as long as we denote the lengths of the legs of the right triangle as “negative” when they exist either along the negative x-axis or the negative y-axis.
Think. . . • Which function is positive in each quadrant? ALL T C
You just need to know where the terminal side of the given angle lies. Sin θ Cos θ Tan θ First-quadrant Angle + + + Second-quadrant Angle + _ _ Third-quadrant Angle _ _ + Fourth-quadrant Angle _ + _
Example • If θ = 120 o, then its reference angle is 60 o. It follows that the trigonometric functions of 120 o have the same numerical values as those of 60 o. Since 120 o is a secondquadrant angle, then sin 120 o = sin 60 o cos 120 o = - cos 60 o tan 120 o =- tan 60 o
Another Example • If θ = 200 o, then the reference angle is 20 o. Now, 200 o is a third-quadrant angle. Therefore, • sin 200 o = - sin 20 o • cos 200 o = - cos 20 o • tan 200 o = tan 20 o
Your turn… • Give the reference angle for each given angle below. Then give the sin, cos, and tan of the angle in terms of the reference angle. a. 135 o b. 350 o c. - 210 o d. - 300 o e. 250 o
