Trigonometry Reciprocal functions II KUS objectives BAT prove

Trigonometry: Reciprocal functions II KUS objectives BAT prove identities using reciprocal functions BAT know and use new trig identities

You need to be able to simplify expressions, prove identities and solve equations involving secθ, cosecθ and cotθ This is similar to work covered earlier, but there are now more possibilities You must practice as much as possible in order to get a ‘feel’ for what to do and when… WB 10 a Remember how we can rewrite cotθ from earlier? Group up as a single fraction Numerator and denominator are equal

WB 10 b Rewrite the part in brackets Multiply each fraction by the opposite’s denominator Group up since the denominators are now the same Multiply the part on top by the part outside the bracket Cancel the common factor to the top and bottom

WB 10 c Putting them together Replace numerator and denominator Left side Numerator Rewrite both Group up Denominator Rewrite both Multiply by the opposite’s denominator Group up From C 2 sin 2θ+ cos 2θ = 1 This is just a division Change to a multiplication Group up Simplify

Notes: New Trig Identities Write down two Trig Identities that you know: (1) (2) Try this: Divide (2) through by (3) This is a new Trig identity (you may be asked to derive it yourself) Can you make another identity dividing by cos 2 ? (4)

WB 11 Left hand side Factorise into a double bracket Replace cosec 2θ The second bracket = 1 Rewrite Group up into 1 fraction Rearrange the bottom (as in C 2) 1

WB 11 b Right hand side Multiply out the bracket Replace sec 2θ Rewrite the second term Replace the fraction The 1 s cancel out… Rewrite both terms based on the inequalities This requires a lot of practice and will be slow to begin with. The more questions you do, the faster you will get!

WB 12 A general strategy is to replace terms until they are all of the same type (eg cosθ, cotθ etc…) 4/ 5 -1 90 180 270 y = Tanθ 360 Replace cosec 2θ Multiply out the bracket Group terms on the left side Factorise Solve Invert so we can use the tan graph Use a calculator for the first answer Be sure to check for others in the given range or or

WB 13 Not possible

a) sec 2 x+ tanx = 3 b) 2 cot 2 x + cosecx + 1 = 0

a) 2 tan 2 x – 7 secx + 5 = 0 c) cosec 2 x = 3 cotx + 5 b) 2 sec 2 x = 9 tanx + 7

a) sec 2 x+ tan 2 x = 6 b) cot 2 x = cosecx c) tanx + cotx = 2 d) 3 tanx cotx = 5 secx

KUS objectives BAT prove identities using reciprocal functions BAT know and use new trig identities self-assess using: R / A / G ‘I am now able to ____. To improve I need to be able to ____’