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1.2 Derivatives Basic Concepts – Tangent and Derivatives

Chord, Secant, and Tangent

  • A chord is a line segment joining two points on a curve, like the line AB in the previous example.
  • A secant is a line that passes through two points, which means that it can be longer than a chord for any given two points A and B.
Picture 1
  • Graphically, ‘a tangent though a curve at point P’ is defined as a line that passes through P and its gradient equals the curve’s instantaneous rate of change at P.
  • Suppose P has coordinates P(p,\:f(p)), then \text {Gradient of tangent at } P=\lim _{h \rightarrow 0} \frac{f(p+h)-f(p)}{h}

Note: Here, h \rightarrow 0 is used and not h \rightarrow 0^{+}.

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5.3 Solution of Trigonometric Equations

Basic Trigonometric Equations (Sine and Cosine)

  • The most basic equations in polynomials would be linear equations, such as ax+b=c (which gives you x=\frac{c-b}{a}). The trigonometric equivalent ones would be \sin t =a or \cos t =b, and you are supposed to solve for t.
  • Be careful, as solving trigonometric equations are not as simple. Refer to the examples below.
  • For each such equations, unless a restriction on x (or more commonly used in trigonometric, \theta), there will be infinite or no solutions for x. This is because of the periodic and symmetric properties that they have.
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1.5 Trigonometric Functions Background Knowledge

Degrees and Radians

  • All these while we are familiar with degrees. We know that a straight line is 180^{\circ}, and for a full circle it is 360^{\circ}.
  • There is another unit to represent degrees, it is known as radians.
  • The short form of radians can be written in several ways. Take 1 radian for example. The most formal way would be written as1^{c}. It can be written as 1\ \mathrm{rad} as before, or the most common way, 1 (without any units).
  • The term ‘radian’ actually originated when we wanted to find a neat expression for the angle when the corresponding arc length of a circle with 1 unit radius, for this angle, is also 1 unit. Using degrees, the number will not look nice, hence a new unit called radians are invented.
  • Radians are useful as it can be treated as numbers (without units).
  • Radians are defined as positive for angles moving anticlockwise. For the diagram at the right, that is a positive angle (1 rad). It is similar to how we define 90^{\circ} and -90^{\circ}.
  • The conversion from radian to degrees is:
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