91Ó°ÊÓ

The reference angle is a powerful tool used in trigonometry to simplify the process of finding trigonometric values. It represents the smallest angle that a given angle makes with the x-axis. To find the reference angle, you adjust the given angle until it falls within the first quadrant of the unit circle, where trigonometric functions take their simplest forms.
For example, if you have 285° as in the original problem, you subtract it from the closest axis angle that's larger, such as 360°, to get an acute angle in the first quadrant or subtract from the angle representing the next axis in a clockwise direction, like 180°, leading to 105°. A good rule of thumb is:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• Second quadrant: Subtract the angle from 180°.
• Third quadrant: Subtract 180° from the angle.
• Fourth quadrant: Subtract the angle from 360°.

Using reference angles helps to determine the behavior of the sine, cosine, and tangent values based on the quadrant the angle is in.

The sine function, often denoted as \( \sin \), relates an angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. In the unit circle, sine gives the y-coordinate of a point corresponding to an angle.For the angle 285°, we use its reference angle, 105°. In the second quadrant, the sine value is positive. Thus, the sine of 285°, like the sine of 105°, remains the same as the trigonometric functions in this part of the circle are consistent.
The exact value of the sine of 105° is \( \frac{\sqrt{3}}{2} \). Remembering the signs and values of trigonometric functions in each quadrant will enable you to quickly determine exact values.

Cosine, represented by \( \cos \), is another fundamental trigonometric function, which corresponds to the x-coordinate on the unit circle. It is the ratio of the adjacent side to the hypotenuse in a right triangle.For 285°, we use the reference angle of 105° again, which lies in the second quadrant. Here, cosine values are negative due to the negative x-direction.
Therefore, the cosine of 285° is the negative cosine of 105°, which results in an exact value of \(-\frac{1}{2}\). Understanding the quadrant rules helps keep track of the positivity or negativity of the trigonometric values used.

The tangent function, or \( \tan \), is derived from dividing the sine of an angle by the cosine of the same angle. It reflects the relationship between the opposite and adjacent sides of a right triangle, For an angle like 285°, knowing its reference angle helps us find its tangent as well. Since tangent is the sine divided by the cosine, it takes on the negativity or positivity from their respective signs in the quadrant.
In the second quadrant, where sine is positive and cosine is negative, tangent becomes negative. Hence, the tangent of 285°, calculated from \( \frac{\sqrt{3}/2}{-1/2} \), results in \(-\sqrt{3}\). Understanding how these relationships update based on the quadrant helps in determining the behavior of tangent efficiently.