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Precalculus Trigonometry Angles Remember: 360◦ = 2π radians. Therefore, 2π . 360◦ 360◦ • To convert radians to degrees, multiply by . 2π • To convert degrees to radians, multiply by THE UNIT CIRCLE The unit circle is the circle of radius 1 centered at the origin, (0, 0). Each angle θ corresponds to a point on the circumference of this circle. The x-coordinate of this point is cos θ, and the y-coordinate of this point is sin θ. (sin θ) y − 21 √ √ 0 4 2 , 2 π 3 π 4 = θ 3π 4 1 2 π 150 ◦ = θ= 5π 6 √ = θ= 7π 6 4 5◦ = 31 4π 3 3π 2 = 240 ◦ θ= θ = 270◦ = = 7π 4 θ = 22 ◦ 5 −1 √ 2 2 , − 2 2 5π 3 √ 1 3 , − 2 2 √ √ 1 3 , − 2 2 −1 √ √ 0 4 , − 2 2 −1 − 12 √ = √ − √ 3 1 , − 2 2 ◦ 11 π 6 √ 300 √ 2 2 , − 2 2 √ − θ= = = √ 3 1 , − 2 2 330 ◦ θ √ x (cos θ) θ = 360◦ = 2π 5π 210 √ 4 0 , 2 2 θ = 0◦ = 0 θ = 180◦ = π θ= − 6 s=1 ◦ 30 radiu √ 4 0 , 2 2 (0, 0) − 21 = ◦ √ − √ 3 1 2 , 2 √ = ◦ 5 2π 3 13 θ= = = = 1 2 √ 2 2 , 2 2 √ θ= ◦ θ √ √ − 23 , 21 1 √ 1 3 , 2 2 √ π 2 √ √ 2 2 , 2 2 120 − θ= √ 1 3 , 2 2 √ − θ = 90◦ = 1 60 ◦ 1 1 2 45 ◦ = −1 − 12 1 1 2 √ √ √ √ √ The coordinates in the picture above are written to show a pattern: 20 , 21 , 22 , 23 , 24 . This pattern appears in both the x-coordinates and the y-coordinates (but the pattern goes one direction for the x-coordinates and the opposite direction for the y-coordinates). Some of these numbers are usually √ √ √ simplified: 20 = 0, 21 = 21 , and 24 = 1. Right-triangle trigonometry (for 0 < θ < π 2) hyp opp θ adj sin θ = opp hyp cos θ = adj hyp tan θ = opp adj csc θ = hyp opp sec θ = hyp adj cot θ = adj opp Pythagorean theorem: In a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 . c a b Trigonometric identities tan θ = sin θ cos θ cot θ = 1 cos θ = tan θ sin θ sec θ = 1 cos θ csc θ = 1 sin θ sin2 θ + cos2 θ = 1, for every value of θ cos(−θ) = cos θ sin(−θ) = − sin θ ) for every value of θ There are many more trigonometric identities, but the ones above are the most important. See Chapter 15 of the textbook, “Trigonometric Identities,” for more identities, including the angle addition formulas.