Reciprocal Trigonometric functions

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… WB10a Simplify 𝑠𝑖𝑛𝜃 𝑐𝑜𝑡𝜃 𝑠𝑒𝑐 𝜃 Remember how we can rewrite cotθ from earlier? Group up as a single fraction Numerator and denominator are equal

Simplify sin 𝜃 cos 𝜃 sec 𝜃 +𝑐𝑜𝑠𝑒𝑐 𝜃 WB10b Simplify sin 𝜃 cos 𝜃 sec 𝜃 +𝑐𝑜𝑠𝑒𝑐 𝜃 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

WB10c Show that cot 𝜃 𝑐𝑜𝑠𝑒𝑐 𝜃 𝑠𝑒𝑐 2 𝜃+ 𝑐𝑜𝑠𝑒𝑐 2 𝜃 ≡ 𝑐𝑜𝑠 3 𝜃 Putting them together Replace numerator and denominator Left side Numerator Denominator This is just a division Rewrite both Rewrite both Change to a multiplication Multiply by the opposite’s denominator Group up Group up Group up From C2  sin2θ+ cos2θ = 1 Simplify

NEW Trigonometric Identities

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 cos2? (4)

a) Prove that 𝑐𝑜𝑠𝑒𝑐 4 𝜃− 𝑐𝑜𝑡 4 𝜃≡ 1+𝑐𝑜𝑠 2 𝜃 1−𝑐𝑜𝑠 2 𝜃 WB 11 a) Prove that 𝑐𝑜𝑠𝑒𝑐 4 𝜃− 𝑐𝑜𝑡 4 𝜃≡ 1+𝑐𝑜𝑠 2 𝜃 1−𝑐𝑜𝑠 2 𝜃 Left hand side Factorise into a double bracket Replace cosec2θ The second bracket = 1 1 Rewrite Group up into 1 fraction Rearrange the bottom (as in C2)

b) Prove that 𝑠𝑒𝑐 2 𝜃− 𝑐𝑜𝑠 2 𝜃≡ 𝑠𝑖𝑛 2 𝜃 1+ 𝑠𝑒𝑐 2 𝜃 WB11b b) Prove that 𝑠𝑒𝑐 2 𝜃− 𝑐𝑜𝑠 2 𝜃≡ 𝑠𝑖𝑛 2 𝜃 1+ 𝑠𝑒𝑐 2 𝜃 Right hand side Multiply out the bracket Replace sec2θ Rewrite the second term Replace the fraction 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! The 1s cancel out…

Trigonometric Identities and equations

Solve the equation 4 𝑐𝑜𝑠𝑒𝑐 2 𝜃−9= cot 𝜃 in the interval 0≤𝜃≤360 WB12 Solve the equation 4 𝑐𝑜𝑠𝑒𝑐 2 𝜃−9= cot 𝜃 in the interval 0≤𝜃≤360 A general strategy is to replace terms until they are all of the same type (eg cosθ, cotθ etc…) Replace cosec2θ Multiply out the bracket 4/5 y = Tanθ 90 180 270 360 -1 Group terms on the left side Factorise Solve or Invert so we can use the tan graph or Use a calculator for the first answer  Be sure to check for others in the given range

Solve these equations: a) 4𝑐𝑜𝑠 2 𝜃+5 sin 𝜃=3 in the interval −𝜋≤𝜃≤𝜋 WB13 Solve these equations: a) 4𝑐𝑜𝑠 2 𝜃+5 sin 𝜃=3 in the interval −𝜋≤𝜃≤𝜋 Not possible b) 2𝑡𝑎𝑛 2 𝜃+ sec 𝜃=1 in the interval −𝜋≤𝜃≤𝜋

WB14I Solve these equations: −𝝅≤𝒙≤𝝅 sec2x+ tanx = 3 b) 2cot2x + cosecx + 1 = 0 Ask students if they want the answers visible or not for checking

2tan2x – 7secx + 5 = 0 2sec2x = 9 tanx + 7 cosec2x = 3 cotx + 5 WB15I Solve these equations: 0≤𝑥≤2𝜋 a) 2tan2x – 7secx + 5 = 0 b) 2sec2x = 9 tanx + 7 c) cosec2x = 3 cotx + 5 Ask students if they want the answers visible or not for checking

sec2x+ tan2x = 6 cot2x = cosecx tanx + cotx = 2 3tanx cotx = 5 secx WB16I Solve these equations: -180≤𝑥≤180 a) sec2x+ tan2x = 6 b) cot2x = cosecx c) tanx + cotx = 2 d) 3tanx cotx = 5 secx Ask students if they want the answers visible or not for checking

self-assess using: R / A / G ‘I am now able to ____ . 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 ____’