Author: misamaliraza94
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Example 3:
Example 3: A force of 11 pounds and a force of 6 pounds act on an object at an angle of 41° with respect to one another. What is the magnitude of the resultant force, and what angle does the resultant force form with the 11‐pound force (Figure 7)? Figure 7 …
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Example 2 :
Example 2: A plane flies at 300 miles per hour. There is a wind blowing out of the southeast at 86 miles per hour with a bearing of 320°. At what bearing must the plane head in order to have a true bearing (relative to the ground) of 14°? What will be the plane’s groundspeed (Figure…
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Vector Operations
In the physical world, some quantities, such as mass, length, age, and value, can be represented by only magnitude. Other quantities, such as speed and force, also involve direction. You can use vectors to represent those quantities that involve both magnitude and direction. One common use of vectors involves finding the actual speed and direction…
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Examples
Example 4: Find a unit vector v with the same direction as the vector u given that u = ⟨7, − 1⟩. Two special unit vectors, i = ⟨1, 0⟩ and j = ⟨0, 1⟩, can be used to express any vector v = ⟨a, b⟩. Example 5: Write u = ⟨5, 3⟩ in terms of the i and j unit vectors (Figure 5 ). Figure 5 Drawing for Example 5.…
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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…
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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…
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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
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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…
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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…