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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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• unit circle: center at origin, radius is 1
• Radian measure of 𝜃 is 𝑠.
• π radians = π rad = 180°= a straight angle.
•
MATH 1013 Lecture Notes 3
Topics covered in tutorial 03:
1. Trigonometric function
2. Inverse trigonometric function
3. Trigonometric formula
1. Trigonometric function
What you need to know:
• Radian measure
• Sine, Cosine and Tangent values
• Trigonometric table
• Sine and Cosine function
Radian measure:
Example 3.1 Complete the following table:
Degree measure
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360° Radian measure
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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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𝑡𝑎𝑛 (𝜋
4) = 1
𝑠𝑖𝑛 (𝜋
4) = 𝑐𝑜𝑠 (
𝜋
4) =
√2
2
𝑡𝑎𝑛 (𝜋
6) = 𝑐𝑜𝑡 (
𝜋
3) =
√3
3
𝑐𝑜𝑡 (𝜋
6) = 𝑡𝑎𝑛 (
𝜋
3) = √3
𝑐𝑜𝑠 (𝜋
6) = 𝑠𝑖𝑛 (
𝜋
3) =
√3
2
𝒔𝒊𝒏𝜽 = 𝒚
𝒄𝒐𝒔𝜽 = 𝒙
𝒕𝒂𝒏𝜽 =𝒚
𝒙
Definition:
𝒔𝒊𝒏𝜽
𝒄𝒐𝒔𝜽
𝒕𝒂𝒏𝜽
Special sine and cosine values:
Example 3.2 Complete the following trigonometric table: +x 1
st quadrant +y 2nd quadrant -x 3
rd quadrant -y 4th quadrant +x
Degree measure
𝟎°
𝟑𝟎°
𝟒𝟓°
𝟔𝟎°
𝟗𝟎°
𝟏𝟐𝟎°
𝟏𝟑𝟓°
𝟏𝟓𝟎°
𝟏𝟖𝟎°
𝟐𝟏𝟎°
𝟐𝟐𝟓°
𝟐𝟒𝟎°
𝟐𝟕𝟎°
𝟑𝟎𝟎°
𝟑𝟏𝟓°
𝟑𝟑𝟎°
𝟑𝟔𝟎° Radian measure
0 𝝅
𝟔
𝝅
𝟒
𝝅
𝟑
𝝅
𝟐
𝟐𝝅
𝟑
𝟑𝝅
𝟒
𝟓𝝅
𝟔
𝝅 𝟕𝝅
𝟔
𝟓𝝅
𝟒
𝟒𝝅
𝟑
𝟑𝝅
𝟐
𝟓𝝅
𝟑
𝟕𝝅
𝟒
𝟏𝟏𝝅
𝟔
𝟐𝝅
𝑠𝑖𝑛𝜃 0 𝟏𝟐
√𝟐
𝟐
√𝟑
𝟐
1 √𝟑
𝟐
√𝟐
𝟐
𝟏
𝟐
0 −
𝟏
𝟐 −
√𝟐
𝟐 −
√𝟑
𝟐 -1
−√𝟑
𝟐 −
√𝟐
𝟐 −
𝟏
𝟐
0
𝑐𝑜𝑠𝜃 1 √𝟑𝟐
√𝟐
𝟐
𝟏
𝟐
0 −
𝟏
𝟐 −
√𝟐
𝟐 −
√𝟑
𝟐 -1
−√𝟑
𝟐 −
√𝟐
𝟐 −
𝟏
𝟐
0 𝟏
𝟐 √𝟐
𝟐
√𝟑
𝟐
1
𝑡𝑎𝑛𝜃 0 √𝟑𝟑
1
√𝟑 −√𝟑 −𝟏 −
√𝟑
𝟑 0 √𝟑
𝟑
1
√𝟑
−√𝟑
−𝟏 −
√𝟑
𝟑 0
r = 1
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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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Example 3.3 Complete the following table:
𝑠𝑖𝑛(2π + 𝜃) = 𝑠𝑖𝑛(𝜃) 𝑐𝑜𝑠(2π + 𝜃) = 𝑐𝑜𝑠(𝜃) (All +) 𝑡𝑎𝑛(2π + 𝜃) =
𝑡𝑎𝑛(𝜃)
𝑠𝑖𝑛(π − 𝜃) = 𝒔𝒊𝒏(𝜽) 𝑐𝑜𝑠(π − 𝜃) = − 𝑐𝑜𝑠(𝜃) 𝑡𝑎𝑛(π − 𝜃) =
−𝑡𝑎𝑛(𝜃)
𝑠𝑖𝑛(π + 𝜃) = − 𝑠𝑖𝑛(𝜃) 𝑐𝑜𝑠(π + 𝜃) = − 𝑐𝑜𝑠(𝜃) 𝑡𝑎𝑛(π + 𝜃) =
𝒕𝒂𝒏(𝜽)
𝑠𝑖𝑛(−𝜃) = − 𝑠𝑖𝑛(𝜃) 𝑐𝑜𝑠(−𝜃) = 𝒄𝒐𝒔(𝜽) 𝑡𝑎𝑛(−𝜃) = −𝑡𝑎𝑛(𝜃)
𝑠𝑖𝑛 (𝜋
2− 𝜃) = 𝑐𝑜𝑠(𝜃)
𝑐𝑜𝑠 (𝜋
2− 𝜃) = 𝑠𝑖𝑛(𝜃) (All +)
𝑡𝑎𝑛 (𝜋
2− 𝜃) = 𝑐𝑜𝑡(𝜃)
𝑠𝑖𝑛 (𝜋
2+ 𝜃) = 𝒄𝒐𝒔(𝜽)
𝑐𝑜𝑠 (𝜋
2+ 𝜃) = − 𝑠𝑖𝑛(𝜃)
𝑡𝑎𝑛 (𝜋
2+ 𝜃) = − 𝑐𝑜𝑡(𝜃)
Example 3.4 Find values of 𝑐𝑜𝑠𝜃 and 𝑡𝑎𝑛𝜃, given that 𝑠𝑖𝑛θ =
−4
5, 𝑤ℎ𝑒𝑟𝑒 π ≤ θ ≤
3𝜋
2.
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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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𝒚 = 𝒔𝒊𝒏 𝒙 𝒂𝒏𝒅 𝒚 = 𝐜𝐨𝐬 𝒙:
function 𝑦 = 𝑠𝑖𝑛 𝑥 𝑦 = cos 𝑥 𝑦 = tan 𝑥
graph
domain 𝑥 ∈ ℝ 𝑥 ∈ ℝ x ∈ {𝑥: 𝑥 ≠
π
2+ 𝑘π, k ∈ ℤ}
range 𝑦 ∈ [−1, 1] 𝑦 ∈ [−1, 1] 𝑦 ∈ ℝ
period 2π 2π π
2. Inverse trigonometric function.
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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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Example 3.5 Simplify sin (2𝑐𝑜𝑠−1𝑥)
Example 3.6 Find the domain and the range of 𝑔(𝑥) = 𝑠𝑖𝑛−1(3𝑥 +
1).
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MATH1012 Calculus IA (2018 Fall) Lecture Notes 3 (Phyllis
LIANG)
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Reference Page
Angle-Sum and -Difference Identities
𝒔𝒊𝒏(𝒙 ± 𝒚) = 𝒔𝒊𝒏(𝒙) ± 𝒔𝒊𝒏(𝒚)
𝒄𝒐𝒔(𝒙 ± 𝒚) = 𝒄𝒐𝒔(𝒙) ∓𝒄𝒐𝒔(𝒚)
𝒕𝒂𝒏(𝒙 ± 𝒚) =𝒕𝒂𝒏 (𝒙) ± 𝒕𝒂𝒏 (𝒚)
𝟏 ∓ 𝒕𝒂𝒏 (𝒙)𝒕𝒂𝒏 (𝒚)
