Introduction to Trigonometry Core 2 and 3

Trig basics + - y = sin θ y P x = cos θ x θ Radius OP is the hypotenuse of a right-angled triangle whose adjacent side is given by the x-coordinate of P and whose opposite side is given by the y-coordinate of P. A normal set of axes… Draw in a unit circle (a circle of radius 1) + - O y y = sin θ Now take the point (1,0) and rotate it anticlockwise about the origin, so that radius OP makes an angle θ with the positive x-axis (1,0) (0,1) θ P cos θ = adj / hyp and hyp = radius = 1 => x-coord. of P = adj = cos θ x = cos θ x sin θ = opp / hyp and hyp = radius = 1 => y-coord. of P = opp = sin θ Hence no matter how far round the circle P moves, x = cos θ and y = sin θ (Note: sin & cos are still in alphabetical order – or you can think of the vertical axis as a “sine-post”) One more thing: Knowing that tan θ = opp / adj, how could you express tan θ in terms of sin θ and cos θ?

Sketching trig graphs + - y = sin θ P x = cos θ θ Can you predict what the graph of y = sin θ will look like as θ changes? Think about what is happening at each stage: At the starting point (1,0), what is sin θ? What is sin θ when you get to (0,1)? What about when θ=45 degrees, i.e. when P is half way from (1,0) to (0,1)? At what angle will sin θ be ½? What will happen in the other quadrants? What will happen if you go round again? + - O y = sin θ (1,0) (0,1) θ P x = cos θ How about the graph of y = cos θ? What will that look like? For more of a challenge, can you predict how the graph of y = tan θ will look? Click here to see the sine curve Click here to see the cosine curve Click here to see the tangent curve

Trig mnemonics Think about the quadrants where the different trig ratios are positive (the “Curves compared” slide may help you with this)… 90° Between 0 and 90°, sin θ goes from 0 up to 1 cos θ goes from 1 down to 0 tan θ = sin θ / cos θ = +ve / +ve so sin, cos and tan θ are ALL positive. + - O y = sin θ x = cos θ S A Between 90 and 180°, sin θ goes from 1 down to 0 cos θ goes from 0 down to -1 tan θ = sin θ / cos θ = +ve / -ve so only sin θ is positive. 180° 0° 360° T C Carry on and fill in the other two quadrants. 270° The traditional mnemonic for this diagram is “All Stations To Coventry” - but you can invent your own if you prefer! Curves compared

Solving trig equations 1 Now let’s look at how this diagram helps us to identify different angles with the same value of sin θ, cos θ or tan θ. 90° + - O y = sin θ x = cos θ Imagine that you are asked to solve the equation sin θ = 0.4 for 0 ≤ θ ≤ 360°. 180°-θ The principal solution, θ (in this case 23.6°), is the acute angle that your calculator gives you for “sin-10.4”. However, the symmetry of the curve means that there will be another solution in the same period. Look at the sine curve and see if you can work out where it will be. Now use this diagram to find the second solution: The sine curve is symmetrical about 90°, so you reflect the line showing where θ is, in the 90° line, i.e. the vertical axis. S A θ θ 180° 0° 360° T C How far round have we come from the 0° line? So the second solution in the range specified is 180° - θ, giving 180 - 23.6 = 156.4°. What would the additional solutions be if the range were 0 ≤ θ ≤ 720°? 270° Answer: Add 360° to each of the previous solutions. Sine curve Cosine curve Tangent curve

Solving trig equations 2 Solving equations in tan θ is easier, because the tangent curve simply repeats itself every 180°. (Check that you can see this from the tangent graph.) So all you need to do is extend the line showing the principal angle, straight through the origin. Now try solving cos θ = 0.8 for 0 ≤ θ ≤ 360°. Use the same approach, but this time we have a different line of symmetry. 90° + - O y = sin θ x = cos θ The cosine curve is symmetrical about 0° and about 180°, so the horizontal axis is our reflection line. S A θ 180° 0° Try solving tan θ = 0.8 for 0 ≤ θ ≤ 360°. θ 360° 180°+θ T C θ How far round have we come from the 0° line this time? 360°-θ Solution: θ = 38.7°, 218.7° Solution: θ = 36.9°, 323.1° 270° What would the additional solutions be for the interval -360 ≤ θ ≤ 360°? Answer: Subtract 360° from each of the previous solutions. Sine curve Cosine curve Tangent curve

Solving trig equations 3 We already know that the first quadrant of the graph contains the principle value of θ. 90° + - O y = sin θ x = cos θ Now we also know how to find the second solution in a given period: With sin θ, the second solution is given by 180°- θ 180°- θ θ S A With cos θ, it’s 360°- θ 180° ±360° With tan θ, it’s 180°+ θ 0° 360° T C If solutions are required outside the interval 0 to 360° then we add 360° to / subtract 360° from each solution already found, until we have all the solutions in the interval specified. 180°+θ 360°-θ 270° Sine curve Cosine curve Tangent curve

Solving trig equations in radians The same approach can be used for problems where the angle is in radians. π/2 + - O y = sin θ x = cos θ π radians are equivalent to 180 degrees so: With sin θ, the second solution is given by π - θ π-θ θ With cos θ, it’s 2π - θ S A With tan θ, it’s π + θ π ±2π 2π T C If solutions are required outside the interval 0 to 2π then we add 2π to / subtract 2π from each solution already found, until we have all the solutions in the interval specified. π+θ 2π-θ 3π/2

Calculus with trig (Core 3) If y = sin x then dy/dx = cos x If y = cos x then dy/dx = -sin x + - O y = sin θ x = cos θ It’s essential that you remember when the sign needs to change, otherwise you could lose a lot of exam marks! The same axes we used for “ASTC” can help you here. Int. Diff. All you have to remember is to go clockwise for differentiation anticlockwise for integration. So differentiating, we have: sin x → cos x → -sin x → -cos x → sin x And integrating, we have: sin x → -cos x → -sin x → cos x → sin x Remember that calculus can only be used with trig functions if the angle is in radians!

Sine curve Cosine curve Tangent curve Back to “Sketching trig graphs” slide

Cosine curve Sine curve Tangent curve Back to “Sketching trig graphs” slide

Tangent curve Cosine curve Sine curve Back to “Sketching trig graphs” slide

Curves compared y = sin x y = cos x y = tan x Back to “Trig mnemonics” slide