CHAPTER 6: TRIGONOMETRY
Chapter objectives:
• Know the six trigonometric functions of angles of any magnitude (sine,
cosine, tangent, secant, cosecant, cotangent)
• Understand amplitude and periodicity and the relationship between
graphs of e.g. sin 𝑥 and sin 2𝑥
• Draw and use the graphs of
𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐
𝑦 = 𝑎 cos 𝑏𝑥 + 𝑐
𝑦 = 𝑎 tan 𝑏𝑥 + 𝑐
where 𝑎 and 𝑏 are positive integers and 𝑐 is an integer
• Use the relationships
sin 𝐴
= tan 𝐴
cos 𝐴
cos 𝐴
= cot 𝐴
sin 𝐴
sin0 𝐴 + cos 0 𝐴 ≡ 1
sec 0 𝐴 ≡ 1 + tan0 𝐴
cosec 0 𝐴 ≡ 1 + cot 0 𝐴
and solve simple trigonometric equations involving the six
trigonometric functions and the above relationships.
• Prove simple trigonometric identities
• Use addition formulae, sin(𝐴 ± 𝐵) , cos(𝐴 ± 𝐵), and application to
multiple angles.
• Use expression in the form of 𝑅 cos(𝜃 ± 𝛼) or 𝑅 sin(𝜃 ± 𝛼) to find
solutions to equations of the form 𝑎 cos 𝜃 + 𝑏 sin 𝜃 = 𝑐
103
,The six main trigonometric ratios
The trigonometric ratios are defined in terms of a right-angled triangle.
The angles of a triangle are named using capital letters while the side
opposite to the angle is named using the respective small letter.
The hypotenuse of a right-angled triangle is its longest side, which is also the
side opposite the right angle. In the above diagram, the hypotenuse is 𝑐.
The side opposite an angle is the side that does not form the angle. In the
diagram, the side opposite the angle 𝐴Ó is a.
The side adjacent to an angle is the side other than the hypotenuse which
forms the angle. The side adjacent to angle 𝐴Ó in figure 5.1 is 𝑏. There are six
trigonometric ratios for angles of any magnitude: tangent (tan), cotangent
(cot), sine (sin), cosecant (cosec), cosine (cos) and secant (sec).
opposite 𝑎
tan 𝐴 = =
adjacent 𝑏
adjacent 𝑏 1
cot 𝐴 = = =
opposite 𝑎 tan 𝐴
opposite 𝑎
sin 𝐴 = =
hypotenuse 𝑐
104
, hypotenuse 𝑐 1
cosec 𝐴 = = =
opposite 𝑎 sin 𝐴
adjacent 𝑏
cos 𝐴 = =
hypotenuse 𝑐
hypotenuse 𝑐 1
sec 𝐴 = = =
adjacent 𝑏 cos 𝐴
Trigonometric identities
The relationship between sine, cosine, tangent and cotangent
𝑎 𝑏
sin 𝐴 = and cos 𝐴 =
𝑐 𝑐
𝑎
sin 𝐴 x 𝑐 y
→ =
cos 𝐴 x𝑏y
𝑐
sin 𝐴 𝑎
→ = = tan 𝐴
cos 𝐴 𝑏
𝐬𝐢𝐧
∴ 𝐭𝐚𝐧 =
𝐜𝐨𝐬
1
cot 𝐴 =
tan 𝐴
𝐜𝐨𝐬
∴ 𝐜𝐨𝐭 =
𝐬𝐢𝐧
The identity 𝐜𝐨𝐬𝟐 𝑨 + 𝐬𝐢𝐧𝟐 𝑨 ≡ 𝟏
105
Chapter objectives:
• Know the six trigonometric functions of angles of any magnitude (sine,
cosine, tangent, secant, cosecant, cotangent)
• Understand amplitude and periodicity and the relationship between
graphs of e.g. sin 𝑥 and sin 2𝑥
• Draw and use the graphs of
𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐
𝑦 = 𝑎 cos 𝑏𝑥 + 𝑐
𝑦 = 𝑎 tan 𝑏𝑥 + 𝑐
where 𝑎 and 𝑏 are positive integers and 𝑐 is an integer
• Use the relationships
sin 𝐴
= tan 𝐴
cos 𝐴
cos 𝐴
= cot 𝐴
sin 𝐴
sin0 𝐴 + cos 0 𝐴 ≡ 1
sec 0 𝐴 ≡ 1 + tan0 𝐴
cosec 0 𝐴 ≡ 1 + cot 0 𝐴
and solve simple trigonometric equations involving the six
trigonometric functions and the above relationships.
• Prove simple trigonometric identities
• Use addition formulae, sin(𝐴 ± 𝐵) , cos(𝐴 ± 𝐵), and application to
multiple angles.
• Use expression in the form of 𝑅 cos(𝜃 ± 𝛼) or 𝑅 sin(𝜃 ± 𝛼) to find
solutions to equations of the form 𝑎 cos 𝜃 + 𝑏 sin 𝜃 = 𝑐
103
,The six main trigonometric ratios
The trigonometric ratios are defined in terms of a right-angled triangle.
The angles of a triangle are named using capital letters while the side
opposite to the angle is named using the respective small letter.
The hypotenuse of a right-angled triangle is its longest side, which is also the
side opposite the right angle. In the above diagram, the hypotenuse is 𝑐.
The side opposite an angle is the side that does not form the angle. In the
diagram, the side opposite the angle 𝐴Ó is a.
The side adjacent to an angle is the side other than the hypotenuse which
forms the angle. The side adjacent to angle 𝐴Ó in figure 5.1 is 𝑏. There are six
trigonometric ratios for angles of any magnitude: tangent (tan), cotangent
(cot), sine (sin), cosecant (cosec), cosine (cos) and secant (sec).
opposite 𝑎
tan 𝐴 = =
adjacent 𝑏
adjacent 𝑏 1
cot 𝐴 = = =
opposite 𝑎 tan 𝐴
opposite 𝑎
sin 𝐴 = =
hypotenuse 𝑐
104
, hypotenuse 𝑐 1
cosec 𝐴 = = =
opposite 𝑎 sin 𝐴
adjacent 𝑏
cos 𝐴 = =
hypotenuse 𝑐
hypotenuse 𝑐 1
sec 𝐴 = = =
adjacent 𝑏 cos 𝐴
Trigonometric identities
The relationship between sine, cosine, tangent and cotangent
𝑎 𝑏
sin 𝐴 = and cos 𝐴 =
𝑐 𝑐
𝑎
sin 𝐴 x 𝑐 y
→ =
cos 𝐴 x𝑏y
𝑐
sin 𝐴 𝑎
→ = = tan 𝐴
cos 𝐴 𝑏
𝐬𝐢𝐧
∴ 𝐭𝐚𝐧 =
𝐜𝐨𝐬
1
cot 𝐴 =
tan 𝐴
𝐜𝐨𝐬
∴ 𝐜𝐨𝐭 =
𝐬𝐢𝐧
The identity 𝐜𝐨𝐬𝟐 𝑨 + 𝐬𝐢𝐧𝟐 𝑨 ≡ 𝟏
105
