YEAR 9 TRIGONOMETRY Where you see the picture below copy the information on the slide into your bound reference. What is Trigonometry & why study Trigonometry? Trigonometry is the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Trigonometry is necessary in other branches of mathematics, including calculus, vectors and complex numbers. What do you already know about right angled triangles? The sum of the angles is 180o

Unknown sides are represented by letters in lower case. The side opposite the right angle is the longest side Angles are written in the inside corners of the triangle One angle is equal 90o to and the sum of the other two angles is 90o Isosceles (90o, 45o, 45o) and scalene triangles (two angles unequal) are the two types of right angled triangles

The little square in the corner of the triangle tells us this angle is 90o The points, or corners, of the triangle are labelled by letters in upper case. Unknown angles are represented by symbols such as the Greek letters: Theta , Beta , Alpha Side names of a right angled triangle In a right angled triangle the three sides are given special names.

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. In this case (Theta) is the angle you are working with. e ot p hy se u n

opposite The opposite (o) is the side opposite the angle . adjacent The adjacent (a) is the side adjacent (next to) the angle . Label the sides of these triangles

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. 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 hy te o p s nu e adjacent

opposite opposite = sin x hypotenuse 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). 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. opposite sin = o h sin x h = o

o = sin x h adjacent Example for: opposite = sin x hypotenuse 1. Put the given values (hypotenuse and angle) into the formula: sin = o h b sin 43o = b 16 43o 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 43o x 16

3. Calculate, and remember the units. b = 10.9m hypotenuse = opposite x 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). sin = o h sin x h = o h= o sin Example for: hypotenuse = opposite x sin 1. Put the given values (opposite and angle) into the formula: sin = o h sin 41o = 19

q 41o 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 41o 3. Calculate, and remember the units. q = 28.96m 19m 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 hy te o p s nu e adjacent

opposite Example for: adjacent = cos x hypotenuse 1. Put the given values (hypotenuse and angle) into the formula: cos = a h cos 43o = b 16 43o b 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 43o x 16 3. Calculate, and remember the units. b = 11.7m

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

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 hy te o p s nu e

adjacent opposite Example for: opposite = tan x adjacent 1. Put the given values (hypotenuse and angle) into the formula: tan = o a b tan 43o = b 16 43o 16 m 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 43o x 16 3. Calculate, and remember the units. b = 14.9m Example for: adjacent = opposite x tan 1. Put the given values (opposite and angle) into the formula: tan = o a tan 41o = 19 q 41o 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 41o 3. Calculate, and remember the units. q = 21.86m 19m q Remembering all the trigonometric ratios 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. Mixed and practical problems 55o H 1. Label the sides o, a, h for the 8.7cm A given angle. f 2. Use SOH CAH TOA to determine which ratio to use. 3. Put the values into the O 2. have a and h, so use cos formula.

4. Rearrange the formula to 3. cos = a make the unknown the h subject. cos 55o = f 5. Calculate, remembering 8.7 units. 4. f = cos 55o x 8.7 Note: When solving practical problems draw the diagram first. 5. 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. sin = o h cos = a h tan = o a = sin-1 x o h = cos-1 x a h

= tan-1 x o a Example for: Calculating unknown angles. have o and h, so use sin sin = o h sin = 11 16 = sin-1 11 16 = 43.43 H 16m O

11m A