Section 4.4: Reference Angles Reference Angles An angle’s reference angle is the measure of the smallest, positive, acute angle formed by the terminal side of the angle and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 1 for examples of reference angles for angles in different quadrants.

A GENERAL NOTE: REFERENCE ANGLES An angle’s reference angle is the size of the smallest acute angle, , formed by the terminal side of the angle and the horizontal axis.

Using Reference Angles Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies. We can find the exact trig value of any angle in any quadrant if we apply the trig function to the reference angle. The sign depends on the quadrant of the original angle. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x– and y-values in the original quadrant. Figure 3 shows which functions are positive in which quadrant. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “All Students Take Calculus” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” all of the six trigonometric functions are positive. In quadrant II, “Students,” only sine and its reciprocal function, cosecant, are positive. In quadrant III, “Take,” only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Calculus” only cosine and its reciprocal function, secant, are positive. How To: Find the trigonometric value for any angle Measure the angle between the terminal side of the given angle and the horizontal axis. This is the reference angle. Apply the trig function to the reference angle. Apply the appropriate sign using the table above. Try another version of this question Give exact values using fractions and/or radicals, but not decimals. If 𝜃 = 𝜋 3 , thenTry another version of this question If 𝜃 =− 11⁢𝜋 3 , then find exact values for the following. If the trigonometric function is undefined for 𝜃 =− 11⁢𝜋 3 , enter DNE. sec⁡(𝜃) equals csc⁡(𝜃) equals tan⁡(𝜃) equals cot⁡(𝜃) equals sin⁡(𝜃) = cos⁡(𝜃) =