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Trigonometry is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Trigonometry is the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics.

 The sum of the angles is 180 o One angle is equal 90 o to and the sum of the other two angles is 90 o The side opposite the right angle is the longest side Angles are written in the inside corners of the triangle Unknown angles are represented by symbols such as the Greek letters: Theta, Beta β, Alpha α The little square in the corner of the triangle tells us this angle is 90 o Isosceles (90 o, 45 o, 45 o ) and scalene triangles (two angles unequal) are the two types of right angled triangles Unknown sides are represented by letters in lower case. The points, or corners, of the triangle are labelled by letters in upper case.

In a right angled triangle the three sides are given special names. hypotenuse The hypotenuse (h) is always the longest side and is opposite the right angle.

The other two sides are labelled depending on the angle you are working with. hypotenuse opposite  adjacent In this case  (Theta) is the angle you are working with. The opposite (o) is the side opposite the angle. The adjacent (a) is the side adjacent (next to) the angle.

   

Unknown sides or angles can be found using Trigonometric ratios called sine, cosine and tangent. Each Trigonometric ratio can be used to calculate an unknown side length or to find out an unknown angle. First we will look into working out unknown side lengths, and we will begin by looking at Sine.

hypotenuse opposite  adjacent Sine is the ratio of the opposite and hypotenuse. The sine of angle . It is abbreviated to sin . sin  = opposite side length = o hypotenuse length h

opposite = sin  x hypotenuse hypotenuse opposite  adjacent To do this you multiply both sides by h to cancel out the divide by h. Note: h divide h = 1, and o x 1 = o. Then swap both sides of the equation. sin  = o h sin  x h = o o = sin  x h To work out the opposite side using sine you need to rearrange the formula to make the opposite (the unknown) the subject (on the left hand side of the equation).

16 m b 43 o 1.Put the given values (hypotenuse and angle) into the formula: sin  = o h sin 43 o = b 16 2.Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: o = sin  x h. b = sin 43 o x 16 3.Calculate, and remember the units. b = 10.9m Example for: opposite = sin  x hypotenuse

hypotenuse = opposite x sin  sin  = o h sin  x h = o h = o sin  To work out the hypotenuse using sine you need to rearrange the formula to make the hypotenuse (the unknown) the subject (on the left hand side of the equation).

q 19m 41 o 1.Put the given values (opposite and angle) into the formula: sin  = o h sin 41 o = 19 q 2.Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, h = o x sin . q = 19 sin 41 o 3.Calculate, and remember the units. q = 28.96m Example for: hypotenuse = opposite x sin 

hypotenuse opposite  adjacent Cosine is the ratio of the adjacent and hypotenuse. The cosine of angle  is abbreviated to cos . cos  = adjacent side length = a hypotenuse length h

16 m b 43 o 1.Put the given values (hypotenuse and angle) into the formula: cos  = a h cos 43 o = b 16 2.Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: a = cos  x h. b = cos 43 o x 16 3.Calculate, and remember the units. b = 11.7m Example for: adjacent = cos  x hypotenuse

q 19m 41 o 1.Put the given values (opposite and angle) into the formula: cos  = a h cos 41 o = 19 q 2.Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, h = a x cos . q = 19 cos 41 o 3.Calculate, and remember the units. q = 25.18m Example for: hypotenuse = adjacent x cos 

hypotenuse opposite  adjacent Tangent is the ratio of the opposite and adjacent. The tangent of angle  is abbreviated to tan . tan  = opposite side length = o adjacent length a

16 m b 43 o 1.Put the given values (hypotenuse and angle) into the formula: tan  = o a tan 43 o = b 16 2.Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: o = tan  x a. b = tan 43 o x 16 3.Calculate, and remember the units. b = 14.9m Example for: opposite = tan  x adjacent

q 19m 41 o 1.Put the given values (opposite and angle) into the formula: tan  = o a tan 41 o = 19 q 2.Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, a = o x tan . q = 19 tan 41 o 3.Calculate, and remember the units. q = 21.86m Example for: adjacent = opposite x tan 

The following mnemonic can be used to help you remember the trigonometric ratios. SOH – CAH – TOA The value of each of these ratios for any angle can be calculated by measuring two specific side lengths of the right-angled triangle containing that angle and dividing them. Remembering all the trigonometric ratios

Mixed and practical problems 1.Label the sides o, a, h for the given angle. 2.Use SOH – CAH –TOA to determine which ratio to use. 3.Put the values into the formula. 4.Rearrange the formula to make the unknown the subject. 5.Calculate, remembering units. Note:When solving practical problems – draw the diagram first. f 8.7cm 55 o H O A 2.have a and h, so use cos  3.cos  = a h cos 55 o = f f = cos 55 o x f = 5cm

Calculating an unknown angle To do this you need to divide by sin, cos or tan. This is called the inverse and is written and sin -1, cos -1 or tan -1. cos  = a h  = cos -1 x a h sin  = o h  = sin -1 x o h tan  = o a  = tan -1 x o a

Example for: Calculating unknown angles. 11m 16m  H A O have o and h, so use sin  sin  = o h sin  =  = sin  = 43.43