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SINE, COSINE & TANGENT

Here is a mnemonic phrase from Giles Marlow of Woking, Surrey for the trigonometrical ratios sine (sin), cosine (cos) and tangent (tan) of any unknown angle Ø within a right-angled triangle:

" SOH - CAH - TOA! "

Pronounced "...soaker toe-er..." where:
Sin Ø = Opposite/Hypotenuse
Cos Ø = Adjacent/Hypotenuse
Tan Ø = Opposite/Adjacent sides
Alternatives

Trigonometry is the branch of mathematics dealing with the measurement of sides and angles of triangles. A right-angled triangle has three sides (the two at 90° to each other usually shown as horizontal and vertical sides, with the remaining third side being the hypotenuse). The sine of an angle is the ratio between the side opposite the angle concerned and the hypotenuse, while the cosine of the same angle is the ratio between the other remaining side (ie. the one adjacent to the angle) and the hypotenuse, and the tangent of the same angle is the ratio between the opposite and adjacent sides.

Natural tables are used to convert sine, cosine and tangent values into actual degrees and vice-versa. The ratio formulas can be transposed (into Opp=Hyp*Sin, A=H*C and O=A*T) so that one can always find (1) an angle given any two sides and (2) a side given an angle and one other side. Otherwise Pythagoras' Theorem is used to find a side given any other two sides.


One alternative mnemonic for the ratios is:

" Oh Heck - Another Hour Of Algebra! "
Or O/H (= Sin), A/H (= Cos), O/A (= Tan)

Mark Alcock has never forgotten the variation his old maths teacher gave him for remembering the rules of tan, cos and sine this way...

" To Oil A Car Always Have Some Oil Handy "


For a non-right-angled triangle, different ratio formulas apply, leading to another established mnemonic:

" plus... All Stations To Coventry... "
(ie. All+, Sin+, Tan+, Cos+)

To understand its significance, consider the different trigonometrical ratios aplying to non-right-angled triangles:

The sine rule for: (1) a side when one side and two angles are known, or (2) an angle knowing one angle and two sides:

side a/sin A (ie. angle A opposite side a)
= side b/sin B = side c/sin C

The cosine rule for: (1) a third side when two sides and the included angle A are known, or (2) an angle A when all three sides are known:

side a²=b²+c²-(2bc * cos A)
from which cos A=(b²+c²-a²)/2bc

Also in a non-right-angled triangle one angle may be obtuse (ie. greater than 90%), whereupon one must deduct it from 180° and make its cosine value negative:

sin Ø=sin(180°-Ø) and
cos Ø= -[minus]cos(180°-Ø)

The cosine of a obtuse angle is negative because the angle lies in the second quadrant of an imaginary circle. A quadrant is a quarter of a circle, and measuring angles in an anti-clockwise direction between a radial startpoint X (equivalent to 3 on a clock face) and another radial point P on the circle, values are positive for all the functions of an angle Ø in the 1st quadrant (P lying between 3-12, or Ø up to 90°) but only the sines in the 2nd quadrant (12-9, or 90-180°), only the cosines in the 3rd quadrant (9-6) and only the tangents in the 4th quadrant (6-3). All other functions in each quadrant are negative. This is summarised in the mnemonic sentence:

" plus... All Stations To Coventry... "
(ie. All+, Sin+, Tan+, Cos+)


Having now come so far, it seems appropriate to end with the formula that summarises the overall relationship between sine, cosine and tangent values. Easily transposeable, it is best remembered "mnemonically" by recalling its parts in straight S,C,T order:

- SOCKET -
sin over cosine equals tangent!
ie.( sin Ø / cos Ø ) = tan Ø

 

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