Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MAT1A01: Trigonometry (Appendix D) Dr Craig 6 February 2013 Information Make sure that you have a MAT1A01 Learning Guide. Extra copies are available at the department (C-Ring). My office: C-Ring 533A (Stats Dept corridor) Consulting hours: Monday 12pm-1pm, Wednesday 12pm-1pm Email: [email protected] Tel: 011 559 3024 Or, just drop in and see if I am there. If I am free, I will help. Radian measure In this course, angles will be measured in radians. Radians measure the ratio between the arc length and the radius. When a =arc length, we have: a θ= and a = r θ. r Note: these formulas are only valid when θ is measured in radians. Example: 90◦ = π/2 radians. How do we get this? Consider a circle of radius 1. The circumference of the circle is given by 2πr = 2π × 1 = 2π. So, the arc length of 90◦ is 2π/4 = π/2. ∴ 90◦ = (π/2)/1 = π/2 radians. Formulas to convert between degrees and radians: 1 radian = 180 π 1◦ = Example: convert (a) 36◦ to radians, (b) −7π/2 to degrees. π radians. 180 30◦ = π 6 45◦ = π 4 60◦ = π 3 90◦ = π 2 120◦ = 2π 3 135◦ = 3π 4 150◦ = 5π 6 180◦ = 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 cos θ = adj hyp tan θ = opp adj csc θ = hyp opp sec θ = hyp 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 ) θ r 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 tan θ > 0 T C cos θ > 0 Special angles π 6 √ π/4 √ 2 2 1 π/3 π/4 1 1 Example: calculate all of the trig ratios for θ = 4π/3. 3 Trig identities Trig identities are useful relationships between trig functions. Some of the basic identities are: 1 1 csc θ = sec θ = sin θ cos θ cot θ = 1 tan θ cot θ = cos θ sin θ tan θ = sin θ cos θ More trig identities sin2 θ + cos2 θ = 1 More trig identities sin2 θ + cos2 θ = 1 Divide both sides by cos2 θ to get 1 sin2 θ cos2 θ + = cos2 θ cos2 θ cos2 θ ∴ tan2 θ + 1 = sec2 θ Now divide both sides of the original by sin2 θ: sin2 θ cos2 θ 1 + = sin2 θ sin2 θ sin2 θ ∴ 1 + cot2 θ = csc2 θ 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 )? 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) = cos2 x − sin2 x cos(2x) = 2 cos2 x − 1 cos(2x) = 1 − 2 sin2 x Example: Find all of the values of x in the interval [0, 2π] such that sin x = sin 2x. Graphs of trig functions Graph of f (x) = sin(x). Note: −1 6 sin x 6 1. Question: why is sin(π/2) = 1? Graph of f (x) = cos(x). Again, −1 6 cos x 6 1. Note: cos 0 = 1. Also sin(x + 2π) = sin x and cos(x + 2π) = cos x. Graph of f (x) = tan(x). Note the vertical asymptotes at π/2 and −π/2. 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.