(a) If è is in standard position, then the reference angle è is the acute angle formed by the terminal side of è and the ---Select--- x-axis y-axis . So the reference angle for è = 120°

is è = ?°, and that for è = 210°is è = ? degrees.

(b) If è is any angle, the value of a trigonometric function of è is the same, except possibly for sign, as the value of the trigonometric function of
è. So sin(120°) = sin ?°, and sin(210°) = −sin ?°.

the reference angle is to the X (or -x) axis

for 120 it would be 180 - 120 = 60 degrees above x axis
cos 60 = 1/2
cos 120 = -1/2

(a) If è is in standard position, then the reference angle è is the acute angle formed by the terminal side of è and the x-axis.

So, for the angle è = 120°, the reference angle would be 180° - 120° = 60°. Therefore, the value of è is 60°.

Similarly, for the angle è = 210°, the reference angle would be 180° - 210° = -30°. However, since the reference angle should be positive, we take its supplement by adding 360°. Therefore, the value of è is 360° - 30° = 330°.

(b) If è is any angle, the value of a trigonometric function of è is the same, except possibly for sign, as the value of the trigonometric function of its reference angle.

Given that sin(120°) = sin ?°, we can conclude that the reference angle should also be 120°.

Similarly, sin(210°) = -sin ?° implies that the reference angle should also give a sine value of -sin 210°. Since sine is negative in the third quadrant, the reference angle should be in the second quadrant where sine is also negative. Therefore, the reference angle for sin(210°) would be 180° - 210° = -30°. However, since the reference angle should be positive, we add 360° to get its supplement. Hence, the value of the reference angle is 360° - 30° = 330°.

(a) To find the value of è given the reference angle, we need to consider the quadrant in which the angle is located.

For the reference angle of 120°, we know that it is an acute angle and lies in the second quadrant. In this quadrant, the terminal side of the angle intersects the negative x-axis.

Since the reference angle in standard position serves as the smallest positive angle with the same terminal side, we can find è by adding 180° to the reference angle.

Therefore, for the reference angle of 120°, è = 120° + 180° = 300°.

Similarly, for the reference angle of 210°, we know that it is an acute angle and lies in the third quadrant. The terminal side of this angle intersects the negative x-axis.

Using the same logic, we can find è by subtracting the reference angle from 180°.

Therefore, for the reference angle of 210°, è = 180° - 210° = -30°.

So, for the given values of reference angles 120° and 210°, the corresponding values of è are 300° and -30° degrees respectively.

(b) The statement in part (b) explains that the values of trigonometric functions (such as sin, cos, tan) for any angle è are the same, except for a possible difference in sign, as the values for the reference angle è.

For sin(120°) = sin ?°, we need to find the angle whose sine value is the same as sine of 120°.

Since sine is a periodic function with a period of 360°, we can subtract 360° from 120° to find an angle with the same sine value.

120° - 360° = -240°

Thus, sin(120°) = sin(-240°).

Similarly, for sin(210°) = -sin ?°, we need to find the angle whose sine value is the opposite (negative) of sine of 210°.

Since sine is an odd function, we can find the angle by negating the reference angle.

Therefore, sin(210°) = -sin(210°).

In summary, for the given values of 120° and 210°, the corresponding angles with the same sine value are -240° and 210° respectively.