Unit Circle and Reference Angles 1 3 1 3 − , 2 2 (0,1) , 2 2 2π π π 2 2 2 2 2 − 3 3 2 , 2 2 , 2 cosine 3π 120○ 90 ○ 60○ π value 4 4 135○ 45○ Numerator is 3 1 3 1 − always one 2 , 2 Numerator is π 2 , 2 5π less than denominator always π 30 ○ 6 6 150○ times π sine value 0 0 (− 1,0) π 180 ○ Numerator is Numerator is 360○ 2π (1,0) 7π ○ always one always one less 210 3 1 more than than twice 11π − 6 330○ 3 1 2 ,− 2 denominator denominator 6 225 ○ times π times π 315○ 2 ,− 2 5π 240○ 300○ 4 7π 2 2 270○ − ,− 4 2 2 2 2 ,− 4π 3π 5π 2 2 3 2 3 1 3 1 3 − ,− ,− 2 2 (0,−1) 2 2 Value of Denominator Reference Angle ? Some hints when 6 30○ dealing with radians. ? 4 45○ ? 3 60○ A. Reference Angles: π Example: Draw 60° in Quadrant II. 3 1. (a) We are in the Quadrant II with a reference angle of 60 degrees. The angle in radians has a denominator of 3. Thus, the angle in Quadrant II is 120 degrees or 2π/3. (b)Formulas for Reference Angles: Note: Let θ be the reference angle given. Quadrant II Quadrant I 180 - θ θ Quadrant II Quadrant II θ-180 360- θ In this example, we in are in Quadrant II. Thus, the angle is 180 degrees-60 degrees = 120 degrees. Solution: 120º (2π/3) 60º Note: Knowing reference angles speeds up things considerably. Positive and Negative Quadrants S A Quadrant II Quadrant I (+) All trigonometric sine and cosecant functions are positive (-) cosine secant tangent cotangent T C Quadrant III Quadrant IV (+) (+) tangent and cotangent cosine and secant (-) (-) sine cosecant sine cosecant cosine secant tangent cotangent All trigonometric functions are positive in Quadrant I Sine and cosecant are positive in Quadrant II Tangent and cotangent are positive in Quadrant III Cosine and secant are positive in Quadrant IV *Note: This information is used in conjunction with reference angles. B. Using Reference Angles To Solve Trigonometric Equations. 2 Example: Solve : sin 2θ = − . 2 To solve this equation, take the inverse sine (arcsin) of both sides. 2 sin -1 (sin 2θ ) = sin −1 − 2 Note: 2θ means you have to make two revolutions around the unit circle. nθ determines the number of revolutions. Steps: 1. Determine which quadrants your desired angles lie in. In this example, sine is negative in Quadrants III and IV. S A T C 2. Find corresponding angles in those quadrants. In this example, we need an angle in Quadrant III and an angle in Quadrant IV which has a sine of 2 − . The reference angle is 45 degrees which means the angles have a denominator 2 of four. First time around: --Quadrant III (angle in radians is one more than denominator) 5π 4 --Quadrant IV (angle in radians is 1 less and twice denominator) 7π 4 Second time around (just add 2π to the previous angles): --Quadrant III (angle in radians is one more than denominator) 5π 13π + 2π = 4 4 --Quadrant IV (angle in radians is 1 less and twice denominator) 7π 15π + 2π = 4 4 5π 7π 13π 15π Solution: , , , 4 4 4 4

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