TRIGONOMETRY(i)

 

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Right Angled Triangles: Trigonometry equations interconnect the angles and sides of right angled triangles.

The longest side is always called the hypotenuse and is always opposite the right angle. The other two sides of the triangle are named in relation to the subject angle. The first triangle in the diagram above shows the subject angle in blue. These other sides make up the right angle and are named as opposite to the subject angle, and adjacent to the subject angle. The second triangle above shows the names of the sides when the subject angle is changed.

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Formulae: where is the subject angle:

The ratios between the sides are functions of the subject angle.

sin = oppositehypotenuse.

cos = adjacenthypotenuse.

tan = oppositeadjacent.

These are useful where an angle and the length of one side is given, then all the other lengths can be found.

= sin oppositehypotenuse.

= cos adjacenthypotenuse.

= tan oppositeadjacent.

(The means the inverse, usually a shift key on the calculator). These are useful where the lenght of at least two sides are known, but no angles given (other than the right angle).

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One Angle & One Side Examples: (A calculator is required to follow these examples). The names of the sides are given after the subject angle is known. Here the subject angle is drawn in blue in the diagram below.

opp = opposite, adj = adjacent, hyp = hypotenuse

(i) One angle and the hypotenuse is given, find the other two sides:

Formula: sin 27 = opphyp. Rearrange: opp = sin 27x hyp. Therefore opp = 36.

Formula: cos 27 = adjhyp. Rearrange: adj = cos 27 x hyp. Therefore adj = 71.

(ii) One angle and the opposite side is given, find the other two sides:

Formula: sin 27 = opphyp. Rearrange: hyp = oppsin 27. Therefore hyp = 132.

Formula: tan 27 = oppadj. Rearrange: adj = opptan 27. Therefore adj = 118.

(iii) One angle and the adjacent side is given, find the other two sides:

Formula: cos 27 = adjhyp. Rearrange: hyp = adjcos 27. Therefore hyp = 79.

Formula: tan 27 = oppadj. Rearrange: opp = adj x tan 27. Therefore opp = 36.

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Exercise (a):

Exercise (b):

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Two Sides & No Angles Examples: (A calculator is required to follow these examples). The names of the sides are given with respect to the required angle. Here the required angle is drawn in blue in the diagram below.

opp = opposite, adj = adjacent, hyp = hypotenuse.

(iv) Two sides are given, find the angle 'a':

The known sides are opposite and hypotenuse, therefore use sin function. Formula: a = sin x opphyp. = 30.

(The other angle is therefore 60 (all angles sum to 180). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths are now the hypotenuse and the adjacent sides. Formula: angle = cos x adjhyp. = 60.)

(v) Two sides are given, find the angle 'b':

The known sides are the adjacent and the hypotenuse, therefore use the cos function. Formula: b = cos x adjhyp. = 389.

(The other angle is therefore 511(all angles sum to 180). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths are now the hypotenuse and the opposite sides. Formula: angle = sin x opphyp. = 511.)

(vi) Two sides are given, find the angle 'c':

The known sides are the opposite and the adjacent, therefore use the tan function. Formula: c = tan x oppadj. = 575.

(The other angle is therefore 325(all angles sum to 180). Or to calculate the other angle using trigonometry, look at the sides in relation to this angle. The known lengths still the opposite and the adjacent sides but reversed. Formula: angle = tan x oppadj. = 325.)

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Exercise (c):

Exercise (d):

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Applied Examples:

(vii)

A church tower stands 35 metres high, its shadow is measured at 50 metres. Find the angle of elevation of the sun ?

 

 

Answer: Draw the right angled triangle: Use the formula where two sides are known but no angle: From the angle in question the two sides known are the opposite and the adjacent. Therefore the formula is: = tan x 3550 = 35.

(viii) A plane is flying at 10,000 metres altitude directly above, 12 seconds later the angle of elevation is 75, find the speed of the plane.

Answer: Draw the right angled triangle: Use the formula where one side is known and one angle is known: The side that is needed is the side A B. Using rules from Geometry(i) the angle within the triangle next to the angle of elevation is 15 also the angle at B is 75. Using this angle at B the formula is tan 75 = 10000adj. Rearranging adj or A B = 10000tan 75 = 2680 metres. The speed is 2680 metres in 12 seconds or (x 300) 804000 metres in an hour ( 804 kmh).

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Obtuse Angles:

 

The trigonometric ratios of sin, cos and tan are defined as follows:

sin (180 - ) = opphyp.

-cos (180 - ) = adjhyp.

-tan (180 - ) = oppadj.

Example: if = 160; sin = sin 20; cos = - cos 20; tan = - tan 20.

Example: find the obtuse angle when sin = 06.

Answer: sin 06 = 369. Therefore: 180 - 369 = 1431.

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