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Bell Work R Find the 6 trig functions for <R. 22 cm S 14 cm sin R = csc R = cos R = sec R = tan R = cot R = T 5-3 Trigonometric Functions on the Unit Circle More Special Triangles…Find the missing sides. 1 ? 1 ? 45° 60° ? ? Our Goal today is to learn to use the Unit Circle to evaluate values of angles The Unit Circle: A circle with a radius of 1 that is placed on the xy coordinate plane with center at the origin. More Unit Circle • Find the sine, cosine, and tangent of a 30° angle using the unit circle. Sin 30° = Cos 30° = Tan 30° = Always draw the triangle to the x axis!!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 60° angle using the unit circle. Sin 60° = Cos 60° = Tan 60° = Always draw the triangle to the x axis!!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 45° angle using the unit circle. Sin 45° = Cos 45° = Tan 45° = Always draw the triangle to the x axis!!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 210° angle using the unit circle. Sin 210° = Cos 210° = Tan 210° = Always draw the triangle to the x axis!!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 150° angle using the unit circle. Sin 150° = Cos 150° = Tan 150° = Always draw the triangle to the x axis!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 600° angle using the unit circle. Sin 600° = Cos 600° = Tan 600° = Always draw the triangle to the x axis!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of a 225° angle using the unit circle. Sin 225° = Cos 225° = Tan 225° = Always draw the triangle to the x axis!! Always write the ordered pair, and the answer is in that point!!!! More Unit Circle • Find the sine, cosine, and tangent of θ using the unit circle. Sin θ = y/1 = y Cos θ = x/1 = x Tan θ = y/x These are always the ratios for an angle on the unit circle….but remember that the radius must be 1!!! Signs in Each Quadrant. • • • • All Student Take Calculus (-, +) (+, +) (-, -) (+, -) Find each value using the Unit Circle. • Cos 210° • Sin 300° • Cos 135 ° • Tan 480° Build the Unit Circle What about Reciprocal Functions? Csc θ = 1/y (reciprocal of Sin θ) Sec θ = 1/x (reciprocal of Cos θ) Cot θ = x/y (reciprocal of Tan θ) Find each value using the Unit Circle. • Sec -135° • Csc 660° • Cot 240 ° • Sec -225° What about Quadrantal Angles? • • • • • • Sin 90° Cos 90° Tan 90 ° Csc 90 ° Sec 90 ° Cot 90 ° Find each value using the Unit Circle. • csc 270° • Sin -225° • Cot 495 ° • Sec -240° Day 2 Values not on the unit circle. Finding trig values when it is NOT a unit circle. (Radius is not one) Trig Ratios Sin θ = y Csc θ = r Cos θ = x r y Tan θ = x r y r Sec θ = x x Cot θ = y Find the values of the six trig functions for angle θ in standard position if a point with coordinates (5, -12) lies on its terminal side. Sin θ = Csc θ = Cos θ = = Sec θ Tan θ = Cot θ = Always draw the triangle to the x axis!!! Find the values of the six trig functions for angle θ in standard position if a point with coordinates (-3, -4) lies on its terminal side. Sinθ= Cosθ= Tanθ= Always draw the triangle to the x axis!!! Csc θ= Sec θ = Cot θ= Find the values of the six trig functions for angle θ in standard position if a point with coordinates (-4, 2) lies on its terminal side. Sinθ= Cosθ= Tanθ= Always draw the triangle to the x axis!!! Csc θ= Sec θ = Cot θ= Suppose θ is an angle in standard position whose terminal side lies in Quadrant III. If sin θ = -4/5, find the values of the remaining five trig functions. Sinθ= Cscθ= Cosθ= Tanθ= Secθ= Cotθ = Suppose θ is an angle in standard position whose terminal side lies in Quadrant IV. If sec θ = √3, find the values of the remaining five trig functions. Sinθ= Csc θ= Cosθ= Secθ= Tanθ= Cotθ=