7. radian measure

7.1 introducing radian measure

• radian: distance travelled around the circle (degrees into radians : x pi, divide by 180)

7.2 inverse trig functions and solving trig equations

• f(x) = sin (x) ; f^-1(x) = arc sin x , domain [-1,1], range [-pi/2 , pi/2]

• f(x) = cos (x) ; f^-1(x) = arc cos x , domain [-1,1], range [0 , pi]

• f(x) = tan (x) ; f^-1(x) = arc tan x , domain is real, range (-pi/2 , pi/2)

7.3 modelling with trigonometric functions

• y = a sin bx and y = a cos bx ; amplitude a and period 2pi/b

• (reminder amplitude is from the midline to the peak, period is from each peak wave)

• max/min points after a translation, you can get the max/min points on the original equation and apply the translations to those coordinates.

• for accuracy on the amplitude, especially after a translation, use:

  • y_max - y_min / 2

• in y = a sin (bx + c) + d : find c or d using x₁ + x₂ / 2 or y₁ + y₂ / 2 using the max/min coordinates.

7.4 arcs and sectors

• (segment is smaller than a sector, segments are cut off by a chord)

• length of arc = pi x radius

• area of sector = ½ angle x radius²

7.5 triangles and circles

• questions asking for something like the area of the sector - the area of the triangle

• (cosine rule: a² = b² - c² - 2bc cos A)

• (sin rule: ½ a b sin C)

7.6 small angle approximations

• in formula book, practically just substitution.

• percentage error: |question - answer| / question x100

• ‘x is small enough to ignore terms in feta³ and higher’ type of phrasing, then it is the binomial expansion AFTER using small angle formula.