trigonometry

COMPLEMENTARY ANGLES *The sum of the measures Of two positive Angles is 90 º Supplementary angles *the sum of the measures Of two positive Angles is 180 º

Example of how to find the measure of a complementary angle: 6m + 3m = 90 add 6m and 3m 9m = 90 divide by 9 m = 10 The two angles have measures of 6(10) = 60º and 3(10) = 30º (6m) º (3m) º

Example of how to find the measure of a supplementary angle: 4k + 6k = 180 add 4k and 6k 10k = 180 divide by 10 k = 18 The two angles have measures of 4(18) = 72º and 6(18) = 108º (4k) º (6k) º

Vertical angles Angles that have equal measures parallel lines transversal

5 4 Alternate interior angles (also 3 and 6) Alternate exterior angles (also 2 and 7) 1 8 Interior angles on same side of transversal (also 3 and 5) 6 4 Corresponding angles (also 1 and 5, 3 and 7, 4 and 8) VERTICALANGLESVERTICALANGLES 2 6

Example of finding angle measures (3x + 2) º (5x – 40) º These are alternate exterior angles, set angles equal to each other. 3x + 2 = 5x – = 2x subtract 3x; add = x divide by 2 One angle has measure: 3x + 2 = 3(21) + 2 = 65º The other has measure: 5x – 40 = 5(21) – 40 = 65º

Trigonometric functions - (x,y) is a point other than the origin sin = y/r cos = x/r tan = y/x csc = r/y sec = r/x cot = x/y y x O r x y P(x,y)

Soh cah toa Sine = opposite/hypotenuse Cosine = adjacent/hypotenuse Tangent = opposite/adjacent

Right triangle 45º Reference angle h=15 o=x a Sin 45=opposite/hypotenuse sin 45 = x/15 x = = 8.06

Sin(45) = 1/√2 Cos(45) = 1/√2 Tan(45) = 1 Sin(30) = 1/2 Cos(30) = √3/2 Tan(30) = 1/√3 Sin(60) = √3/2 Cos(60) = 1/2 Tan(60) = √3/1 = √3 Trigonometric function values for special angles

Unit circle (cos,sin)

Converting between degrees and radians Converting degrees to radians: Multiply a degree measure by  /180 radian and simplify to covert to radians. Ex: 45º = 45º (  /180 radians) =  /4 radian Converting radians to degrees: Multiply a radian measure by 180º/  And simplify to convert to degrees. Ex: 9  /4 = 9  /4(180º/  ) = 405º

Signs and Ranges of Function Values Quad I Quad II Quad III Quad IV Quad I II III IV sin + + _ _ cos _ _ _ __ _ ___ _ tancotseccscAll positivesin/csc tan/cotcos/sec

Recap Learned about Complimentary and Supplementary Angles Learned how to find measure of angles Learned about Vertical Angles Learned how to find angle measures Learned about Trigonometric functions Learned Soh Cah Toa and how to use Show the unit circle Learned to convert between radians and degrees