Trigonometry

2 Unit 4:Mathematics Aims Introduce Pythagoras therom. Look at Trigonometry Objectives Investigate the pythagoras therom. Calculate trigonometric functions.

Mae'r onglau triongl yn adio i 180  180 

–Using our knowledge of the internal angles of a triangle we can find an unknown angle in a triangle given the other two angles. –Using Pythagoras’ theorem we can find the length of a side given the lengths of the other two sides in a right angled triangle. –Where we have a combination of lengths of sides and angles in a right angled triangle, we can use trigonometry to calculate the lengths of another side or an unknown angle.

About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

Pythagorus realized that if you have a right triangle, 3 4 5

and you square the lengths of the two sides that make up the right angle, 3 4 5

and add them together, 3 4 5

you get the same number you would get by squaring the other side

And it is true for any right triangle

Find the length of a diagonal of the rectangle: 15" 8" ?

Find the length of a diagonal of the rectangle: 15" 8" ? b = 8 a = 15 c

b = 8 a = 15 c

Practice using The Pythagorean Theorem to solve these right triangles:

5 12 c 10 b b 15 Prawf pye 1 & 2 Test pye 1 & 2

Right angled triangle –In a right angled triangle the side opposite the right angle (the longest side) is known as the hypotenuse. –The side opposite a given angle (other than the right angle) is known as the opposite. –The side next to the given angle is known as the adjacent. hypotenuse opposite adjacent

–Label each side on the diagrams below. –1. –2. –3. –4. hypotenuse opposite adjacent Prawf trig 1 Test trig 1

Sine (sin) ratio –In a right-angled triangle hypotenuse opposite x

–Calculate the length b in the diagram – – 23  sin 52 =  b = 18.1 (to 3 significant figures) 52  23 b Prawf 3 Test 3

Calculate the angle x in the diagram – 7 3 x Sin X = 3 = Sin -1 ( ) =25.377°

–Calculate the length p in the diagram 9.3 p 17.3 

Cosine (cos) ratio –In a right-angled triangle hypotenuse x adjacent

–Calculate the length c in the diagram – – – – 27  15 c

–Calculate the angle y in the diagram – y Cos y = adj (6.1) = hyp (9.7) Cos -1 ( ) = 51° 2”

Calculate the length q in the diagram 34  q 9.3

Tangent (tan) ratio –In a right-angled triangle opposite x adjacent

Calculate the length d in the diagram – – 37  40 d Tan 37 = opp (d) = adj (40) Opp (d) =tan 37 x 40 =

–Calculate the angle z in the diagram – z Tan z = 2.6 = Z = tan-1 (0.6341) = – 32 =.381 x60 = ° 22min

–Calculate the length marked r in the diagram 62  79.2 r Prawf trig 2 Test trig 2