1. Reciprocal
Trigonometric
functions

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

3. 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

4. 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

5. 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

6. NEW Trigonometric
Identities

7. 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)

8. 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)

9. 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…

10. Trigonometric
Identities and
equations

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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 ____’