This is a summary of the subject Calculus I, this summary is written with respect to the examination. This means that the examples and the solutions in this summary are written in such a way that the questions of the exam are easier to answer.
1
Area of triangle formed by 𝑎, 𝑏 & 𝑐 = 2 |(𝑏 − 𝑎) × (𝑐 − 𝑎)|
Area of parallelogram spanned by vectors 𝑎, 𝑏 & 𝑐 = |(𝑏 − 𝑎) × (𝑐 − 𝑎)|
Example:
If 𝐴 = (−2, 3, 1), 𝐵 = (1, 0, −2) & 𝐶 = (2, −3, 1). What is the area of triangle ABC?
1
Area of triangle formed by 𝐴, 𝐵 & 𝐶 = 2 |(𝐵 − 𝐴) × (𝐶 − 𝐴)|
𝐵 − 𝐴 = (3, −3, −3) & 𝐶 − 𝐴 = (4, −6, 0)
1
Area of triangle formed by 𝐴, 𝐵 & 𝐶 = 2 |(−18, −12, −6)| = 3|(3, 2, 1)| = 3√14
(Inverse) Trigonometric Functions
Keep in mind the following equations, these will also be provided on
the formula sheet:
sin(2𝑎) = 2 sin(𝑎) cos(𝑎)
cos(2𝑎) = 2 cos2 (𝑎) − 1 = 1 − 2 sin2 (𝑎) = cos2 (𝑎) − sin2 (𝑎)
If we get a difficult expression to rewrite with inverse trigonometric
functions and normal trigonometric functions the easiest thing to do
is to make a small sketch and just take an angle.
Example:
What is the expression cos(2 arctan(𝑥)) equal to?
State 𝑦 = arctan(𝑥), thus 𝑥 = tan(𝑦)
This gives: cos(2 arctan(𝑥)) = cos 2(arctan(𝑥)) − sin2 (arctan(𝑥))
2
If we look at the triangle on the right, this is the situation described in the √𝑥 + 1
question. 𝑥
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