MAT01A1 Appendix D: Trigonometry Dr Craig 12 February 2019
Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
Important information Course code: MAT01A1 NOT: MAT1A1E, MAT1A3E, MATE0A1, MAEB0A1, MAA00A1, MAT00A1, MAFT0A1 Learning Guide: available on Blackboard. Please check Blackboard twice a week. Student email: check this email account twice per week or set up forwarding to an address that you check frequently.
Important information Lecture times: Tuesday 08h50 10h25 Wednesdays 17h10 18h45 Lecture venues: C-LES 102, C-LES 103 Tutorials: Tuesday afternoons 13h50 15h25: D-LES 104 or B-LES 102 OR 15h30 17h05: C-LES 203
Other announcements No tuts for MAT01A1 on Wednesdays. If you see this on your timetable, it is an error. (To move your Chem. prac., email Mr Kgatshe ckgatshe@uj.ac.za) CSC02A2 students. Email Dr Craig regarding tutorial clash. Maths Learning Centre in C-Ring 512: 10h30 15h25 Mondays 08h00 15h25 Tuesday to Thursday 08h00 12h05 Fridays
Lecturers Consultation Hours Monday: 14h40 15h25 Dr Craig (C-508) Tuesday: 11h20 13h45 Dr Robinson (C-514) Wednesday: 15h30 17h05 Dr Robinson (C-514) Thursday: 11h20 12h55 Dr Craig (C-508) Friday: 11h20 12h55 Dr Craig (C-508)
Warm up Let a > 0. Then 4. f(x) = a if and only if f(x) = a or f(x) = a. 5. f(x) < a if and only if a < f(x) < a. 6. f(x) > a if and only if f(x) > a or f(x) < a. Now solve: 1 < x + 1 < 4
The solution is the values of x that satisfy: 1 < x + 1 AND x + 1 < 4 Therefore x ( 5, 2) (0, 3). This algebraic solution agrees with the sketch:
The triangle inequality If a, b R, then a + b a + b. How do we prove this? First observe that a a and a a. Similarly, b b and b b.
Applications: Example: If x 4 < 0.1 and y 7 < 0.2, use the Triangle Inequality to estimate (x + y) 11. Exercise: show that if x + 3 < 1 2, then 4x + 13 < 3.
How big is a degree? If you didn t have a protractor, how would you draw an angle of size 30? You could do it using the 1, 2, 3 special triangle. But what about 32? Would it be possible to draw this angle with only a ruler, compass, and a piece of string?
Radian measure From now on in your mathematical life, angles will be measured in radians. Radians measure the ratio between the arc length and the radius. When a =arc length, we have: θ = a r and a = rθ. Note: these formulas are only valid when θ is measured in radians.
Example: 90 = π/2 radians. How do we get this? Consider any circle. Let the radius of the circle be r. The circumference of the circle is equal to 2πr. The arc length of 90 is a quarter of the total circumference, so a = 2πr 4 = πr 2. 90 = πr 2 1 r = π 2 radians.
Using the fact that 360 will be 2π radians, we can use the following formulas to convert between degrees and radians: 1 radian = 180 π Example: convert (a) 72 to radians, (b) 5π/2 to degrees. 1 = π 180 radians.
More examples: (a) If the radius of a circle is 5cm, what angle is subtended by an arc of 6cm? (b) If a circle has radius 3cm, what is the length of an arc subtended by a central angle of 3π/8?
30 = π 6 45 = π 4 60 = π 3 90 = π 2 120 = 2π 3 135 = 3π 4 180 = π 270 = 3π 2 360 = 2π
Trigonometric functions Trig functions take as input an angle (measured in radians) and output the ratio between two distances. hypotenuse opposite θ adjacent sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp
More generally, angles can be measured in a coordinate system: θ 0 θ < 0 A positive and negative angle drawn in the standard position.
Trig functions in a coordinate system P (x, y) r θ sin θ = y r cos θ = x r tan θ = y x csc θ = r y sec θ = r x cot θ = x y
Signs of trig functions sin θ 0 S A All 0 T tan θ 0 cos θ 0 C
Special angles 2 π/4 1 2 π 6 3 π/4 1 π/3 Exercise: calculate all of the trig ratios for θ = 5π/3. 1
Graphs of trig functions: f(x) = sin x Note: 1 sin x 1. Question: why is sin(3π/2) = 1? Think of how 3π/2 is sketched in xy-plane.
Graph of f(x) = cos(x). Again, cos x 1. Note: cos 0 = 1. Also sin(x + 2π) = sin x and cos(x + 2π) = cos x.
Now that we are using radians we should label our axes using radian intervals like π/2 or maybe π/4, depending what is most suitable.
Graph of f(x) = tan(x). Note the vertical asymptotes at π/2 and π/2.
Why does tan(x) get very big as x approaches π 2 from below?
Graph of f(x) = csc(x). Note: csc(x) 1, x / { z.π z Z }.
Graph of f(x) = csc(x) and g(x) = sin(x). Note: csc(x) 1, x / { z.π z Z }.
Graph of f(x) = sec(x).
Graph of f(x) = cot(x). Note: cot(x) = 0 wherever tan(x) has an asymptote.
Graph of f(x) = cot(x) and g(x) = tan(x). Note: cot(x) = 0 wherever tan(x) has an asymptote.
Trig identities Trig identities are useful relationships between trig functions. Some of the basic identities are: csc θ = 1 sin θ sec θ = 1 cos θ tan θ = sin θ cos θ cot θ = 1 tan θ = cos θ sin θ
More trig identities sin 2 θ + cos 2 θ = 1 Divide both sides by cos 2 θ to get sin 2 θ cos 2 θ + cos2 θ cos 2 θ = 1 cos 2 θ tan 2 θ + 1 = sec 2 θ Or, divide both sides of the original by sin 2 θ: sin 2 θ sin 2 θ + cos2 θ sin 2 θ = 1 sin 2 θ 1 + cot 2 θ = csc 2 θ
Example sin 2 θ + cos 2 θ = 1 Prove the following trig identities: cot 2 θ + sec 2 θ = tan 2 θ + csc 2 θ tan 2 α sin 2 α = tan 2 α sin 2 α
Addition formulas sin(x + y) = sin x. cos y + cos x. sin y cos(x + y) = cos x. cos y sin x. sin y How can we use sin(x + y) = sin x. cos y + cos x. sin y to get a formula for sin(x y)? Result: sin(x y) = sin x. cos y cos x. sin y
We can also use the addition formulas to get the double-angle formulas: sin(2x) = 2 sin x. cos x cos(2x) = cos 2 x sin 2 x cos(2x) = 2 cos 2 x 1 cos(2x) = 1 2 sin 2 x Example: Find all of the values of x in the interval [0, 2π] such that sin x = sin 2x.