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  • Coordinate System 2

    Figure 2               Drawing for Example 1. If the coordinates of point P are ( x, y), An algebraic vector is an ordered pair of real numbers. An algebraic vector that corresponds to standard geometric vector  is denoted as ⟨ a, b⟩ if terminal point P has coordinates of (a, b). The numbers a and b are called the components of vector ⟨ a, b⟩ (see Figure…

  • Rectangular Coordinate System

    The following discussion is limited to vectors in a two‐dimensional coordinate plane, although the concepts can be extended to higher dimensions. If vector  is shifted so that its initial point is at the origin of the rectangular coordinate plane, it is said to be in standard position. If vector  is equal to vector  and has its initial point at…

  • Pythagorean Identities

    sin2θ+cos2θ=1sin2⁡θ+cos2⁡θ=1 cot2θ+1=cosec2θcot2⁡θ+1=cosec2θ 1+tan2θ=sec2θ

  • Inverse Trigonometry Formulas

    sin-1 (–x) = – sin-1 x cos-1 (–x) = π – cos-1 x tan-1 (–x) = – tan-1 x cosec-1 (–x) = – cosec-1 x sec-1 (–x) = π – sec-1 x cot-1 (–x) = π – cot-1 x

  • Sum to Product Formula List

    We can prove these sum to product formulas using the product to sum formulas in trigonometry. The sum to product formula are expressed as follows: sin A + sin B = 2 sin [(A + B)/2] cos [(A – B)/2] sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2] cos A…

  • Half Angle Formula of Tan Derivation

    We know that tan (A/2) = [sin (A/2)] / [cos (A/2)] From the half angle formulas of sin and cos, tan (A/2) = [±√(1 – cos A)/2] / [±√(1 + cos A)/2] = ±√[(1 – cos A) / (1 + cos A)] This is one of the formulas of tan (A/2). Let us derive the other two…

  • Half Angle Formula of Cos Derivation

    Now, we will prove the half angle formula for the cosine function. Using one of the above formulas of cos A, cos A = 2 cos2(A/2) – 1 From this, 2 cos2(A/2) = 1 + cos A cos2 (A/2) = (1 + cos A) / 2 cos (A/2) = ±√[(1 + cos A) / 2]

  • Half Angle Formula of Sin Proof

    Now, we will prove the half angle formula for the sine function. Using one of the above formulas of cos A, we have cos A = 1 – 2 sin2 (A/2) From this, 2 sin2 (A/2) = 1 – cos A sin2 (A/2) = (1 – cos A) / 2 sin (A/2) = ±√[(1 – cos A) /…

  • Half Angle Formulas?

    In this section, we will see the half angle formulas of sin, cos, and tan. We know the values of the trigonometric functions (sin, cos , tan, cot, sec, cosec) for the angles like 0°, 30°, 45°, 60°, and 90° from the trigonometric table. But to know the exact values of sin 22.5°, tan 15°, etc,…

  • Triple Angle Identities

    Sin 3x = 3sin x – 4sin3x Cos 3x = 4cos3x-3cos x T a n   3 x = 3   t a n   x − t a n 3 x / 1 − 3 t a n 2 x

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