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Chris Fox's Engineering Section


Trigonometry

"The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides"

Pythagoras' Rule

Trigonometry is concerned with calculating the sides and angles of triangles. However, as will be seen, it can also be applied to any situation involving circular motion.

Trigonometry is a long-established branch of mathematics in my country; Angles came to England in the 5th century.

The degree (°), is commonly used as a unit of plane angle; there are 360 in a circle. An angle of 90° is termed a right angle; an angle measuring between 0 and 90° is an acute angle, and one measuring between 90 and 180° is an obtuse angle. The SI unit of plane angle is the radian (rad; c). 1c is the angle formed at the centre of a circle by two radii that cut off an arc equal in length to the radius; there are therefore 2πc in a circle. Because the circumference of a circle is given by C=2πr and the area by A=πr², if radians are used the arc length l=rθ and the sector area A=0.5r²θ.

Consider a right-angled triangle (One in which two sides join at a right angle). One of the angles other than the right angle is denoted by θ. The side connecting this angle to the right angle is termed the adjacent side; the side opposite the angle θ is termed the opposite side; and the side opposite the right angle is termed the hypotenuse.

The three basic functions of trigonometry are the sine, cosine and tangent. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. Substitution will show that tan θ = sin θ/cos θ.

Three other functions are the cosecant (cosec), secant (sec) and cotangent (cot). These are the reciprocals of the sine, cosine and tangent functions respectively.

Now consider a circle of radius r, whose centre lies at the origin of a graph. The angle θ denotes the anticlockwise turn from the x-axis to a particular radius. The co-ordinates of the other end of the radius are (r cos θ, r sin θ). This gives rise to the polar co-ordinate system, based on r and θ. Because x=r cos θ and y=r sin θ, r²=x²+y² (Pythagoras' Rule) and θ=tan-1 (y/x). Tan-1 denotes the inverse tangent, so that θ=tan -1 (tan θ).

If the radius is the hypotenuse of a right-angled triangle of which θ is an angle, using Pythagoras' Rule and cancelling shows that sin²θ+cos²θ=1 (a convenient way of writing (sin θ)² ... ). Dividing throughout by cos²θ gives tan²θ+1=sec²θ, and dividing by sin²θ gives 1+cot²θ=cosec²θ.

Consider a triangle with an angle A in the bottom left corner and a right angle in the bottom right corner, rotated anticlockwise through an angle B. Because the angles of a triangle total 180°, and because the angles on a straight line total 180°, the opposite side forms an angle B with the vertical. The B angles can each be considered part of another triangle, whose hypotenuse is also one side of the first. If the hypotenuse of the first triangle has length 1, the lengths of the opposite and adjacent sides are, respectively, sin A and cos A. Considering the co-ordinates of the top corner of the first triangle, and given that opposite sides of a rectangle have equal length:

sin (A+B) = sin A cos B + cos A sin B

cos (A+B) = cos A cos B - sin A sin B

These are the Addition Formulae.

The Addition Formulae can be used to expand tan (A+B). Because tan (A+B) = sin (A+B)/cos (A+B), replacing these terms with the Addition Formulae then dividing both the numerator and the denominator by cos A cos B gives:

tan (A+B) = (tan A + tan B)/(1 - tan A tan B).

Consider a triangle laid on one side. An imaginary line runs vertically from the top corner, dividing it into two right-angled triangles. The length of the imaginary line is equal to the sine of each of the bottom angles multiplied by the length of the adjoining hypotenuse. Doing this for each pair of sides, then combining the information into a single equation, we find that the sine of an angle is directly proportional to the length of the opposite side. This is the Sine Rule; because no constraints have been placed on either the triangles or angles used, it applies to all triangles.

Consider the same triangle and imaginary line. The bottom left angle is labelled A, and the side opposite is labelled a (lower case equivalent); the other two sides are labelled b and c. Applying Pythagoras' Rule to the right hand triangle, expanding and simplifying shows that a² = b² + c² - 2bc cos A. This is the Cosine Rule.


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