Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Mrs. Grieser Name: Date: Block: Trig Functions in a Circle Circle with radius r, centered around origin (x 2 + y 2 = r 2 ) Drop perpendicular to make a triangle, label the point where the radius touches the circle (x, y) Find the six trig functions for angle θ: o sin θ csc θ o cos θ sec θ o tan θ cot θ Example: Let (-4, 3) be a point on the terminal side of an angle θ in standard position. Find the six trig function values of θ. Find the length of the radius Find the six trig functions for angle θ: o sin θ csc θ o cos θ sec θ o tan θ cot θ You Try Use the given point on the terminal side of an angle θ in standard position to evaluate the six trig functions of θ. a) (-12, 5) b) (6, -8) sin θ csc θ cos θ sec θ tan θ cot θ sin θ csc θ cos θ sec θ tan θ cot θ
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 2 The Unit Circle Circle with radius 1, centered around origin (x 2 + y 2 = 1) Drop a perpendicular from the radius to the x-axis to form a right triangle: Let θ be the angle formed by the x-axis and radius Find the six trigonometric functions relative to θ: sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = Of particular interest o x = cos θ o y = sin θ o tan θ = y sinθ is the of the radius/hypotenuse x cosθ o cos 2 θ + sin 2 θ = 1 by the Evaluate the angles of a unit circle based on given coordinates on the circle o Given point 3 1, 2 2 on the unit circle What are the 6 trigonometric values of the angle θ? sin θ = csc θ = cos θ = tan θ = sec θ = cot θ = What is θ?
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 3 o Given point 2 2, 2 2 on the unit circle What are the 6 trigonometric values of the angle θ? sin θ = csc θ = cos θ = tan θ = sec θ = cot θ = What is θ? 3 What about angles on the axes (quadrantal angles)?... 0,,, 2 2 θ = 0: coordinate (1, 0) θ = 2 : coordinate (0, 1) sin θ = csc θ = sin θ = csc θ = cos θ = sec θ = cos θ = sec θ = tan θ = cot θ = tan θ = cot θ = θ = : coordinate (-1, 0) 3 θ = : coordinate (0, -1) 2 sin θ = csc θ = sin θ = csc θ = cos θ = sec θ = cos θ = sec θ = tan θ = cot θ = tan θ = cot θ =
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 4 Important values to know on the unit circle
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 5 Reference Angles Associated with every angle in standard position is a reference angle Acute angle formed by the terminal ray and the x-axis Always positive Always between 0 o and 90 o Finding: o "Unwind" by adding or subtracting 360 o (2π) until angle between 0 and 90 o o Sketch the angle to find the quadrant o Easy in quadrant I - the reference angle is the angle o Other quadrants Remember the bow tie o Create reference triangles by dropping perpendiculars to the x-axis Examples: a) What is the reference angle for 120 o? 11 b) What is the reference angle for? 6 Quadrant Think of bow tie Quadrant Think of bow tie
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 6 You Try: Find the reference angles a) 210 o b) -260 o c) 7 d) 9 15 4 Evaluating Trigonometric Functions Reference angles help us find the value of trigonometric functions for any angle Find the reference angle; determine the trig value for that angle Determine the sign based on the quadrant the angle lies in Quadrant I signs: o x and y both positive o sin θ = y sign: o cos θ = x sign: o tan θ = x y sign: o ALL trig functions are positive in quadrant I Quadrant II signs: o x is, y is o sin θ = y sign: o cos θ = -x sign: o tan θ = y - x sign: o and are positive in quadrant II; all the rest are negative
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 7 Quadrant III signs: o x is, y is o sin θ = -y sign: o cos θ = -x sign: o tan θ = - y - x sign: o and are positive in quadrant III; all the rest are negative Quadrant IV signs: o x is, y is o sin θ = -y sign: o cos θ = x sign: - y o tan θ = x sign: o and are positive in quadrant IV; all the rest are negative Summary: o Quadrant I: ALL trig functions positive o Quadrant II: and positive o Quadrant III: and positive o Quadrant IV: and positive Classic Memory Aid: o ALL Students Take Calculus (all) (sin) (tan) (cos)
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Page 8 Evaluating trig functions WITHOUT a calculator o Find reference angle o Determine the sign based on quadrant o Use known trig value for the reference angle 5 o Example: Find cos 4 Reference angle: Quadrant / Sign: Value: Examples: Evaluate the trig functions WITHOUT a calculator a) sin (120 o ) b) cos 2 c) tan (-240 o ) b) csc 3 17 6 You try: Evaluate the trig functions WITHOUT a calculator a) cos (-225 o ) b) cot 10 3 c) sec (300 o ) b) tan 5 6