Mathematics for Engineers Week 1 Francis Hunt Francis.Hunt@southwales.ac.uk

Trigonometry: why? just means triangle-measuring why do we want it? example uses: what's height of the building? (NB: diagram not to scale.) what's the position of ship? 5° 10° 500m A B 3km 30° 40° ship LAND SEA C D

Trigonometry: why? v  further uses: mechanics, resolving vectors projectile fired at velocity v, angle  to the horizontal horizontal velocity is v cos , vertical velocity v sin  we can show the range of the projectile is (v2 sin 2)/g (don't worry if you don't understand the words, just understand that what we cover in trigonometry is essential for subjects like mechanics) v 

Trigonometry: why? more advanced uses: any oscillations where the restoring force is proportional to the displacement can be written in the form sin(t). (Don't worry if you don't know what this means, just understand that what we cover in trigonometry is used extensively in any subject concerning waves and oscillations) solving differential equations, the boundary conditions can be broken down into sums of sine and cosines ("Fourier series") (again don't worry if you don't understand the words, just understand that what we cover in trigonometry is useful later on)

This week's learning outcomes After this week, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles

Degree, Minutes, Seconds (DMS) A whole rotation divides up into 360 degrees (written 360°) If not a whole number, we can write it as a decimal fraction: e.g. 56.7914° In surveying instead of .7914 on the end they specify how many sixtieths of a degree this is. These are called minutes. .7914/1 = 47.484/60 so a little more than 47 minutes (written 47') For even more precision we give the . 484 minutes left over in terms of sixtieths of a minute, called seconds. .484/60 = 29.04/3600 so a little more than 29 seconds (written 29'') so 56.7914° = 56°47'29'' (you may have a button on your calculator that converts for you) Degrees, minutes, seconds are not needed further in this course. Work through the details on pages 1-1 and 1-2 outside the lecture for examples of how to manipulate DMS quantities

This week's learning outcomes After this week, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles find all solutions 0-360° of trig equations like sin 3θ = 0.7

Radians A radian is another (more natural?) way to measure angles it is denoted with a c instead of a °, 5 radians could be written 5c Why? :it makes speed calculations simple: a wheel with 0.5m radius rotates at 10 radians a second. What speed is the lorry moving? Answer 10 × 0.5 = 5m/s a wheel with 0.5m radius rotates at 10 degrees a second. What speed is the lorry moving? Answer 10 × 0.5 × 2/360 = 0.087 m/s essential for calculus (next term) and complex numbers (this term) A B

Converting between radians and degrees The radian measure of an angle is the arc length divided by the radius r. Since the arc length of the circle is 2πr, the radian angle is a circle = 2πr/r = 2π. Converting degrees to radians: 180 degrees is  radians, so 1 degree is /180 radians so multiply the number of degrees by /180 to get the radians Converting radians to degrees:  radians is 180 degrees, so 1 radian is 180/ degrees so multiply the number of radians by 180/ to get the degrees Look at examples 4 and 5 on p1-3 after the lecture to see how this works

This week's learning outcomes From today's sessions, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles

Trigonometric Ratios hyp opp  adj sin  = cos  = tan  = remember these with the word SOHCAHTOA or “Some old hag cracked all her teeth on apples” In a right-angled triangle: knowing one side and  we can find the other sides knowing two sides we can find 

Example question Train climbs a 5° incline, how far along the track does it need to go to climb 100m? draw a diagram set up an equation: sin 5° = 100/y solve the equation: y = 100/ sin 5° = 1147 m Work through example 2 on p1-4 after lecture for more examples 100m y 5°

This week's learning outcomes From this week's sessions, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles find all solutions 0-360° of trig equations like sin 3θ = 0.7

a2 + b2 = c2 c b a Example from the audience? http://en.wikipedia.org/wiki/Image: Kapitolinischer_Pythagoras_adjusted.jpg

Sin, cos, tan of 30° and 60° sin 30° = 1/2 sin 60° = √3/2 tan 60° = √3 Using Pythagoras we can see what the trigonometric ratios of 30° and 60° are, without using a calculator. sin 30° = 1/2 sin 60° = √3/2 tan 60° = √3 tan 30° = 1/√3 = cos 60° = cos 30° 2 60° 30° 2 √3 60° 60° 1 1 2 memorise these

Sin, cos, tan of 45° 1 sin 45° = 1/√2 cos 45° = 1/√2 tan 45° = 1 √2 1 Cut a unit square along the diagonal 1 sin 45° = 1/√2 cos 45° = 1/√2 tan 45° = 1 √2 1 1 45° 1 memorise these

Three more names memorise these (3rd letter is the clue) cosecant of an angle is the reciprocal of sine of the angle secant of an angle is the reciprocal of cosine of the angle cotangent of an angle is the reciprocal of the tangent of the angle memorise these (3rd letter is the clue)

Names: why? Sine Hindu mathematician Aryabhata (~475-~550) called sine a chord-half, or jya-ardha, in his book, shortened to jya or jiva Arab mathematicians kept the word when translating, omitted the consonants, so it could be pronounced jiva, jiba or jaib jaib means bay, and in Latin this is sinus The English version of the Latin sinus is sine Cosine sine of the complementary angle

Names: why? R P  Q 1 Tangent In a unit circle, a line touching the circle like that through RQ is called a tangent. The length RQ cut off with an angle  has length tan  (if you do the calculation) Secant In a unit circle, a line cutting the circle like PR is called a secant. The length PR in a unit circle has length sec  (if you do the calculation) R P  Q 1

This week's learning outcomes From this week's sessions, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles

Extending the definition of sine in a unit circle the height PQ equals sin , for 0    90° if we give point P coordinates ( x, y ) we will use this as the new definition of sin  we now have a definition of sin  for any angle  P (x,y) 1 1  sin  = y O Q

Extending the definition of cosine in a unit circle the x coordinate of P equals cos , for 0    90° we will use this as the new definition of cos  we now have a definition of cos  for any angle  P (x,y) 1 1  cos  = x O Q

Extending the definition of tangent in a unit circle the tan  is y/x, for 0    90° we will use this as the new definition of tan  we now have a definition of tan  for any angle  how do these new definitions behave? : Geogebra demo P (x,y) 1 tan  =  O Q

Signs of sin, cos, tan in different quadrants + - sin + - cos + - tan A S where each is positive (A=all) T C

Key points the magnitude of a trigonometric ratio of an angle (e.g. sin 150°) is determined by the acute angle to the horizontal. (e.g. 30° here). the sign is given by the quadrant chart. (e.g. 150° is in quadrant 2, where sine is positive so sin 150° = + sin 30°)

Using the quadrant chart Find cos 240° without a calculator: Find angle to horizontal (60°) Take cos of this angle (0.5) Attach appropriate sign (-0.5) 240° Q A S T C O P

Periodicities Adding any multiple 360° to an angle does not change the sine or cosine we say sine and cosine have periodicity 360° or 2π radians tangent has periodicity 180° or π radians 360+α α P

This week's learning outcomes From this week's sessions, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles

Extra slides follow – not used this week

Solving Trig Equations Solve sin 3θ = 0.7 for all values 0 ≤ θ < 360° Take inverse sine of both sides 3θ = sin-1 0.7 = 44.4° (using calculator) Symmetry says there is another value for 3θ in quadrant 2 with same acute angle to horizontal, and same sign i.e. 3θ = 180 - 44.4 = 135.6° Periodicity says we can add any integer multiple of 360° to these and still have solutions: 3θ = 44.4 + 360n or 3θ = 135.6 + 360n so  3 gives θ = 14.8 + 120n or θ = 45.2 + 120n We have 6 solutions 0 ≤ θ < 360°: θ = 14.8° , 134.8 °, 254.8 °, 45.2 °, 165.2 °, 285.2 ° Check the answers satisfy sin 3θ = 0.7

Today's learning outcomes From today's sessions, you should be able to: convert between degrees, minutes, seconds and decimals define a radian and convert between degrees and radians define and use trigonometric ratios for angles up to 90° use Pythagoras' theorem define and use trigonometric ratios for arbitrary angles Now do Exercises 1.1 and 1.2 to check you can