1. Trigonometry

2. Reciprocal Trigonometry Functions
Connections to the Study Design:
AOS 1 – Functions and graphs
Graphs of the reciprocal circular functions cosecant, secant and cotangent, and simple transformations of these

3. Key Notation: 𝑃 : represents a point/coordinate 𝑃′: represents
Key Vocabulary: Functions Unit Circle Derive Relationship Magnitude Radians Compound- Angle Quadrant CAST Equivalent Reciprocal Abbreviate Cosecant Secant Cotangent Inverse Corresponding Rationalise Denominator Simplify Resulting
Key Notation: 𝑃 : represents a point/coordinate 𝑃′: represents

4. Trig Recap
sin 𝜃= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 = 𝑏 𝑐

5. Unit Circle - CAST

6. Exact Values in First Quadrant
Angle (degrees) 0°
30°
45°
60°
90°
180°
270°
360°
Angle (radians)
𝜋 6
𝜋 4
𝜋 3
𝜋 2 𝜋
3𝜋 2 2𝜋
sin(θ)
1 2
2 2
3 2 1 -1
cos(θ)
tan(θ)
3 3 3
Undefined

7. First Quadrant
Angle:
Degrees: 0° <𝜃<90°
Radians: 0 < 𝜃 < 𝜋 2
𝑥>0 𝑎𝑛𝑑 𝑦 >0
sin 𝜃 >0 𝑎𝑛𝑑 cos 𝜃 >0 => tan 𝜃 >0
Note: sin 30°+ sin 60° ≠ sin 90° In general: sin 𝐴+𝐵 ≠ sin 𝐴 +sin⁡(𝐵) cos 𝐴+𝐵 ≠ cos 𝐴 + cos 𝐵 tan 𝐴+𝐵 ≠ tan 𝐴 + tan 𝐵

8. Second Quadrant Angle: 180 − θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π −θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Degrees: 90° <𝜃<180°
Radians: 𝜋 2 < 𝜃 < 𝜋
𝑥<0 𝑎𝑛𝑑 𝑦 >0
sin 𝜃 >0 𝑎𝑛𝑑 cos 𝜃 <0 => tan 𝜃 >0
Point 𝑃(𝑎,𝑏) reflected through the y-axis becomes 𝑃 ′ −𝑎,𝑏
Point P makes angle θ, then P’ makes angle:
180 − θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π −θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
sin 180° − 𝜃 = sin 𝜃
cos 180° − 𝜃 =−cos⁡(𝜃)
tan 180° − 𝜃 =−tan⁡(𝜃)
sin 𝜋 − 𝜃 = sin 𝜃
cos 𝜋− 𝜃 =−cos⁡(𝜃)
tan 𝜋− 𝜃 =−tan⁡(𝜃)

9. Third Quadrant Angle: 180+ θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π+θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Degrees: 180° <𝜃<270°
Radians: π< 𝜃 < 3𝜋 2
𝑥<0 𝑎𝑛𝑑 𝑦 <0
sin 𝜃 <0 𝑎𝑛𝑑 cos 𝜃 <0 => tan 𝜃 >0
Point 𝑃(𝑎,𝑏) reflected through the y-axis becomes 𝑃 ′ −𝑎,−𝑏
Point P makes angle θ, then P’ makes angle:
180+ θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π+θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
sin 180°+ 𝜃 = −sin 𝜃
cos 180°+ 𝜃 =−cos⁡(𝜃)
tan 180°+ 𝜃 =tan⁡(𝜃)
sin 𝜋+ 𝜃 = −sin 𝜃
cos 𝜋+ 𝜃 =−cos⁡(𝜃)
tan 𝜋+ 𝜃 =tan⁡(𝜃)

10. Fourth Quadrant Angle: 180+ θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π+θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Degrees: 270° <𝜃<360°
Radians: 3𝜋 2 < 𝜃 <𝜋
𝑥>0 𝑎𝑛𝑑 𝑦 <0
sin 𝜃 <0 𝑎𝑛𝑑 cos 𝜃 >0 => tan 𝜃 >0
Point 𝑃(𝑎,𝑏) reflected through the y-axis becomes 𝑃 ′ −𝑎,−𝑏
Point P makes angle θ, then P’ makes angle:
180+ θ 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑟 π+θ 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
sin 360°− 𝜃 = −sin 𝜃
cos 360°−𝜃 =cos⁡(𝜃)
tan 360°−𝜃 =−tan⁡(𝜃)
sin 2𝜋−𝜃 = −sin 𝜃
cos 2𝜋−𝜃 =cos⁡(𝜃)
tan 2𝜋−𝜃 =−tan⁡(𝜃)

11. Summary
sin 𝜃 =𝑠𝑖𝑛 𝜋−𝜃 =− sin 𝜋+𝜃 =− sin 2𝜋−𝜃 cos 𝜃 =− cos 𝜋−𝜃 =− cos 𝜋+𝜃 = cos 2𝜋−𝜃 tan 𝜃 =− tan 𝜋−𝜃 = tan 𝜋+𝜃 = −tan 2𝜋−𝜃

12. Negative Angles
Point 𝑃(𝑎,𝑏) reflected through the x-axis becomes 𝑃 ′ 𝑎,−𝑏
𝑃 makes angle θ with x-axis => 𝑃 ′ makes angle - θ
sin −𝜃 =− sin 𝜃 cos −𝜃 = cos 𝜃 tan −𝜃 =−tan⁡(𝜃)
A negative angle − 𝜋 2 <𝜃<0 is just the equivalent angle in the fourth quadrant.
For positive angles greater than 360° or 2π, we can just subtract multiples of 360° or 2π.
sin 360°+ 𝜃 = sin 𝜃
cos 360°+𝜃 =cos⁡(𝜃)
tan 360°+𝜃 =tan⁡(𝜃)
sin 2𝜋+𝜃 = sin 𝜃
cos 2𝜋+𝜃 =cos⁡(𝜃)
tan 2𝜋+𝜃 =tan⁡(𝜃)

13. Reciprocal Trigonometric Functions
Sine Function
Cosine Function
Tangent Function
Reciprocal Trigonometric Function
Cosecant Function: cosec(θ)= 1 sin⁡(𝜃) , 𝑤ℎ𝑒𝑟𝑒 sin⁡(𝜃)≠0
Secant Function: sec θ = 1 cos⁡(𝜃) , 𝑤ℎ𝑒𝑟𝑒 cos⁡(𝜃)≠0
Cotangent Function: cot θ = 1 tan⁡(𝜃) , 𝑤ℎ𝑒𝑟𝑒 sin 𝜃 ≠0
Note: these are not inverse trigonometric functions.
cosec θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 opposite = 𝑐 𝑏
sec θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 adjacent = 𝑐 𝑎
cot θ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 = 𝑎 𝑏

14. Exact Values
Angles multiples of 30° and 45°: Exact values for reciprocal trigonometric functions can be found from corresponding trigonometric values.

15. Example 1: Exact Values
Find the exact value of 𝑐𝑜𝑠𝑒𝑐( 5𝜋 4 )

16. Example 2: Using triangles to find values
If 𝑐𝑜𝑠𝑒𝑐 𝜃 = 7 4 and 𝜋 2 <𝜃<𝜋, find the exact value of cot⁡(𝜃).

17. Trigonometric Identities using Reciprocal Trigonometric Functions
Connections to the Study Design
AOS 1 Functions and graphs
Compound and double angle formulas for sine, cosine and tangent and the identities:
𝑠𝑒𝑐 2 𝑥 =1+ 𝑡𝑎𝑛 2 𝑥
𝑐𝑜𝑠𝑒𝑐 2 𝑥 =1+ 𝑐𝑜𝑡 2 (𝑥)

18. Key Vocabulary: Identity Equation Mathematical Convention Transform Simplifying Factorising Cancelling Common Factors Denominators Numerators Prove Fundamental Relations Quotient
Key Notation: holds for all values 𝑛∈𝑍 𝑄 𝑅 LHS and RHS

19. Identities vs Equations
[ sin 𝜃 ] 2 = 𝑠𝑖𝑛 2 (𝜃) similary [ 𝑐𝑜𝑠 𝜃 ] 2 = 𝑐𝑜𝑠 2 (𝜃)
𝑠𝑖𝑛 2 𝜃 + 𝑐𝑜𝑠 2 𝜃 =1 – Identity not equation, since it holds true for all values of 𝜃
tan 𝜃 = sin⁡(𝜃) cos⁡(𝜃) holds for all values of 𝜃 for which tan(𝜃) is defined, that is for all values where cos(𝜃)≠0, or 𝜃≠(2𝑛+1) 𝜋 2 where 𝑛∈𝑍 or odd multiples of 𝜋 2

20. Identities and Relations
Trigonometric Identities Recap:
tan 𝜃 = sin⁡(𝜃) cos⁡(𝜃)
sec 𝜃 = 1 cos⁡(𝜃)
𝑐𝑜𝑠𝑒𝑐 𝜃 = 1 sin⁡(𝜃)
cot 𝜃 = 1 tan⁡(𝜃)
Fundamental Relations: 1+ 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑜𝑠𝑒𝑐 2 (𝜃) 𝑡𝑎𝑛 2 𝜃 +1= 𝑠𝑒𝑐 2 (𝜃)

21. Work Example 3: Identity Proof
Prove the identity tan 𝜃 + cot 𝜃 = sec 𝜃 𝑐𝑜𝑠𝑒𝑐(𝜃)

22. Work Example 4: Identity Proof
Prove the identity 1+ 𝑡𝑎𝑛 2 (𝜃) 1+ 𝑐𝑜𝑡 2 (𝜃) = 𝑡𝑎𝑛 2 (𝜃)

23. Compound-angle Formulas
Connections to the Study Design AOS 1 Compound and double angle formulas for sine, cosine and tangent and the identities: 𝑠𝑒𝑐 2 𝑥 =1+ 𝑡𝑎𝑛 2 𝑥 𝑐𝑜𝑠𝑒𝑐 2 𝑥 =1+ 𝑐𝑜𝑡 2 (𝑥)

24. Key Vocabulary: Compound Perpendicular Properties Supplementary Substituting Derive Ratio Corresponding Theorems Expanding Expressions Complementary Multiples Argument
Key Notation: ∠𝐴𝐵𝐶 sin 𝑛𝜋 2 ±𝜃 𝑓𝑜𝑟 𝑛∈𝑍 0<𝐴< 𝜋 2

25. Summary of the compound-angle formulas
sin 𝐴+𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 sin 𝐴−𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 cos 𝐴+𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 cos 𝐴−𝐵 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 tan 𝐴+𝐵 = tan 𝐴 +tan⁡(𝐵) 1− tan 𝐴 tan⁡(𝐵) tan 𝐴−𝐵 = tan 𝐴 −tan⁡(𝐵) 1+ tan 𝐴 tan⁡(𝐵)

26. Proof of the compound-angle formula: tangent

27. Worked Examples: Collaborative
Work Example 5: Using compound-angle formulas
Evaluate 𝐬𝐢𝐧 𝟐𝟐° 𝐜𝐨𝐬 𝟑𝟖° + 𝐜𝐨𝐬 𝟐𝟐° 𝐬𝐢𝐧 𝟑𝟖° | Jay and Vincent
Work Example 6: Expanding trigonometric expressions with phase shifts
Expand 𝟐 𝐜𝐨𝐬 𝜽+ 𝝅 𝟑 | Alex and Star
Work Example 7: Simplification of 𝑠𝑖𝑛 𝑛𝜋 2 ±𝜃 and cos 𝑛𝜋 2 ±𝜃
Use compound-angle formulas to simplify 𝐜𝐨𝐬 𝟑𝝅 𝟐 −𝜽 | Yanira and Ruby
Work Example 8: Exact values for multiples of 𝜋 12
Find the exact value of 𝐬𝐢𝐧 𝟏𝟑𝝅 𝟏𝟐 | Christian and Vish
Work Example 9: Using triangles to find values
If 𝐜𝐨𝐬 𝑨 = 𝟏𝟐 𝟏𝟑 and 𝐬𝐢𝐧 𝑩 = 𝟕 𝟐𝟓 , where 𝟎<𝑨< 𝝅 𝟐 and 𝝅 𝟐 <𝑩<𝝅, find the exact value of 𝐬𝐢𝐧⁡(𝑨−𝑩) | James and Emily

28. Double-angle formulas
Connections to the Study Design AOS 1 Compound and double angle formulas for sine, cosine and tangent and the identities: 𝑠𝑒𝑐 2 𝑥 =1+ 𝑡𝑎𝑛 2 𝑥 𝑐𝑜𝑠𝑒𝑐 2 𝑥 =1+ 𝑐𝑜𝑡 2 (𝑥)

29. Key Vocabulary: Equivalent Common Factor Null Factor Law Reciprocal
Key Notation: 𝐵=𝐴 𝑥∈ 0,2𝜋

30. Double Angle Formulas
sin 2𝐴 =2 sin 𝐴 cos 𝐴 cos 2𝐴 = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛 2 𝐴 cos 2𝐴 =1−2 𝑠𝑖𝑛 2 𝐴 cos 2𝐴 =2 𝑐𝑜𝑠 2 𝐴 −1

31. Half-angle formulas
1− cos 𝐴 =2 𝑠𝑖𝑛 2 𝐴 2 1+ cos 𝐴 =2 𝑐𝑜𝑠 2 ( 𝐴 2 )

32. Multiple-angle formulas
sin 3𝐴 =3 sin 𝐴 −4 𝑠𝑖𝑛 3 𝐴 cos 3𝐴 =4 𝑐𝑜𝑠 3 𝐴 −3 cos 𝐴 tan 3𝐴 = 3 tan 𝐴 − 𝑡𝑎𝑛 3 (𝐴) 1−3 𝑡𝑎𝑛 2 (𝐴) sin 4𝐴 =cos⁡(𝐴)(4 sin 𝐴 −8 𝑠𝑖𝑛 3 𝐴 ) cos 4𝐴 =8 𝑐𝑜𝑠 4 𝐴 −8 𝑐𝑜𝑠 2 𝐴 +1 tan 4𝐴 = 4tan⁡(𝐴)(1− 𝑡𝑎𝑛 2 𝐴 ) 1−6 𝑡𝑎𝑛 2 𝐴 + 𝑡𝑎𝑛 4 (𝐴)

33. Collaborative Work Examples
Worked Example 10: Using double-angle formulas in simplifying expressions
Find the exact value of sin 7𝜋 cos 7𝜋 12
Worked Example 11: Finding Trigonometry expressions involving double-angle formulas
If cos 𝐴 = 1 4 , determine the exact values of sin⁡(2𝐴), cos 2𝐴 and tan⁡(2𝐴)
Worked Example 12: Solving trigonometric equations involving double-angle formulas
Solve for x if sin 2𝑥 cos 𝑥 =0 for 𝑥∈[0,2𝜋]
Worked Example 13: Trigonometric identities using double—angle formula
Prove the identity cos 2𝐴 cos 𝐴 + sin 2𝐴 sin⁡(𝐴) sin 3𝐴 cos 𝐴 − cos 3𝐴 sin⁡(𝐴) = 1 2 𝑐𝑜𝑠𝑒𝑐(𝐴)
Worked Example 14: Half-angle formulas
Prove the identity 𝑐𝑜𝑠𝑒𝑐 𝐴 − cot 𝐴 = tan 𝐴 2
Worked Example 15: Multiple-angle formulas
Prove the identity cos 3𝐴 =4 𝑐𝑜𝑠 3 𝐴 −3cos⁡(𝐴)