𝒂 𝒄𝒐𝒔𝜽 + 𝒃 𝒔𝒊𝒏𝜽
General Solution
cos 𝑘𝑥 = 𝑐
2𝜋𝑛 ± 𝛼
𝑥=
𝑘
sin 𝑘𝑥 = 𝑐
𝑛𝜋 + −1 𝑛 𝛼
𝑥=
𝑘
𝑊ℎ𝑒𝑟𝑒 𝛼 = sin−1 𝑐
𝑊ℎ𝑒𝑟𝑒 𝛼 = cos −1 𝑐
tan 𝑘𝑥 = 𝑐
𝑛𝜋 + 𝛼
𝑥=
𝑘
𝑊ℎ𝑒𝑟𝑒 𝛼 = tan−1 𝑐
𝑹𝒄𝒐𝒔(𝒙 − 𝜶)
𝑅 cos(𝑥 − 𝛼) = 𝑅(cos 𝑥 cos 𝛼 + sin 𝑥 sin 𝛼)
𝑅 cos(𝑥 − 𝛼) = 𝑅 cos 𝑥 cos 𝛼 + 𝑅 sin 𝑥 sin 𝛼
So, if we want to write an expression of the form a cos 𝑥 + 𝑏 sin 𝑥 in the form
𝑅 cos(𝑥 − 𝛼) we can do this by comparing
𝑎 cos 𝑥 + 𝑏 sin 𝑥 = 𝑅 cos 𝑥 cos 𝛼 + 𝑅 sin 𝑥 sin 𝛼
𝑹𝒄𝒐𝒔(𝒙 − 𝜶)
For both sides to be equal the terms in cos 𝑥 and sin 𝑥 must be equal. Therefore we
have to equate the coefficients.
𝑎 cos 𝑥 + 𝑏 sin 𝑥 = 𝑅 cos 𝑥 cos 𝛼 + 𝑅 sin 𝑥 sin 𝛼
𝑎 cos 𝑥 = 𝑅 cos 𝑥 cos 𝛼
𝑏 𝑠𝑖𝑛 𝑥 = 𝑅 𝑠𝑖𝑛 𝑥 𝑠𝑖𝑛 𝛼
𝑎 = 𝑅 cos 𝛼
𝑏 = 𝑅 𝑠𝑖𝑛 𝛼
𝑹𝒄𝒐𝒔(𝒙 − 𝜶)
We can now find R and α in terms of 𝑎 𝑎𝑛𝑑 𝑏
𝑎2 + 𝑏 2 = 𝑅2 cos 2 𝛼 + 𝑅2 sin2 𝛼
𝑎2 + 𝑏 2 = 𝑅2 (cos 2 𝛼 + sin2 𝛼)
𝑎2 + 𝑏 2 = 𝑅2
𝑅=
𝑎2 + 𝑏 2
𝑹𝒄𝒐𝒔(𝒙 − 𝜶)
We can now find R and α in terms of 𝑎 𝑎𝑛𝑑 𝑏
Recall
𝑎 = 𝑅 cos 𝛼
𝑏 = 𝑅 𝑠𝑖𝑛 𝛼
𝑏
𝑅 sin 𝛼
=
𝑎 𝑅 cos 𝛼
𝑏
tan 𝛼 =
𝑎
𝛼=
tan−1
𝑏
𝑎
𝑎 cos 𝑥 + 𝑏 sin 𝑥 can be written as
𝑅 cos(𝑥 − 𝛼) where:
𝑅=
𝛼=
𝑎2 + 𝑏 2
tan−1
𝑏
𝑎
𝒃
𝜶
𝒂
Examples
Express 2 cos 𝜃 + sin 𝜃 in the form 𝑟 cos(𝜃 − 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
2 = 𝑅 cos 𝛼
1 = 𝑅 sin 𝛼
𝑎=2
𝑏=1
𝑟=
𝑎2
+ 𝑏2
𝑟=
12 + 22
𝑟= 5
−1
𝑏
𝑎
tan−1
1
2
𝛼 = tan
𝛼=
𝛼 = 26.6°
∴ 2 cos 𝜃 + sin 𝜃 = 5 cos(𝜃 − 26.6°)
Examples
Express 3 cos 𝜃 + 4 sin 𝜃 in the form 𝑟 cos(𝜃 − 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
3 = 𝑅 cos 𝛼
4 = 𝑅 sin 𝛼
𝑎=3
𝑏=4
𝑟=
𝑎2
𝑟=
42 + 32
𝑟=5
+ 𝑏2
−1
𝑏
𝑎
tan−1
4
3
𝛼 = tan
𝛼=
𝛼 = 53.1°
∴ 3 cos 𝜃 + 4 sin 𝜃 = 5 cos(𝜃 − 53.1°)
Examples
Express 2 sin 𝜃 − cos 𝜃 in the form 𝑟 sin(𝜃 − 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
2 sin 𝜃 − cos 𝜃 = 𝑅 sin(𝜃 − 𝛼)
2 sin 𝜃 − cos 𝜃 = 𝑅(sin 𝜃 cos 𝛼 − cos 𝜃 sin 𝛼)
2 sin 𝜃 − cos 𝜃 = 𝑅 sin 𝜃 cos 𝛼 − 𝑅 cos 𝜃 sin 𝛼
2 sin 𝜃 = 𝑅 sin 𝜃 cos 𝛼
2 = 𝑅 cos 𝛼
𝑅2 sin2 𝛼 + 𝑅2 cos2 𝛼 = 12 + 22
𝑅2 (sin2 𝛼 + cos 2 𝛼) = 12 + 22
𝑟𝑒𝑐𝑎𝑙𝑙: sin2 𝑥 + cos 2 𝑥 = 1
𝑅2 = 5
− cos 𝜃 = −𝑅 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝛼
1 = 𝑅 sin 𝛼
∴ 2 sin 𝜃 − 𝑐𝑜𝑠 𝜃 = 5 𝑠𝑖𝑛(𝜃 − 26.6°)
𝑅= 5
1
𝛼 = tan
2
𝛼 = 26.6°
−1
Examples
Express 7 cos 𝑥 − 24 sin 𝑥 in the form 𝑟 cos(𝜃 + 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
Hence, solve the equation 7 cos 𝑥 − 24 sin 𝑥 = 4 for 0° < 𝑥 < 360°
7 = 𝑟 cos 𝛼
24 = 𝑟 sin 𝛼
𝑎=7
𝑏 = 24
𝑟=
72 + 242
𝑟 = 25
24
𝛼=
7
𝛼 = 73.7°
tan−1
∴ 7 cos 𝑥 − 24 sin 𝑥 = 25 cos(𝑥 + 73.7°)
Examples
Express 7 cos 𝑥 − 24 sin 𝑥 in the form 𝑟 cos(𝜃 + 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
Hence, solve the equation 7 cos 𝑥 − 24 sin 𝑥 = 4 for 0° < 𝑥 < 360°
25 cos 𝑥 + 73.7° = 4
4
cos 𝑥 + 73.7° =
25
4
−1
𝑥 + 73.7 = cos
25
𝑥 + 73.7 = 80.8°, 279.2°
𝑥 = 7.1°, 205.5°
Examples
Express 2 cos 𝑥 + sin 𝑥 in the form 𝑟 cos(𝜃 − 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
Hence, solve the equation 2 cos 𝑥 + sin 𝑥 = 1 for 0° < 𝑥 < 360°
2 = 𝑟 cos 𝛼
1 = 𝑟 sin 𝛼
𝑎=2
𝑏=1
𝑟=
12 + 22
𝑟= 5
1
𝛼=
2
𝛼 = 26.6°
tan−1
∴ 2 cos 𝑥 + sin 𝑥 = 5 cos(𝑥 − 26.6°)
Examples
Express 2 cos 𝑥 + sin 𝑥 in the form 𝑟 cos(𝜃 − 𝛼) 𝑤ℎ𝑒𝑟𝑒 𝑟 > 0, 0° < 𝛼 < 90°
Hence, solve the equation 2 cos 𝑥 + sin 𝑥 = 1 for 0° < 𝑥 < 360°
5 𝑐𝑜𝑠 𝑥 − 26.6° = 1
1
cos 𝑥 − 26.6° =
5
1
−1
𝑥 − 26.6 = cos
5
𝑥 − 26.6 = 63.4°, 296.6°
𝑥 = 90, 323.2°