Match cards in pairs then try to fill in table 2 1 These can be useful! 0⁰= 0 30⁰ = π 6 45⁰ = 60⁰ = 90⁰= 180⁰= sin cos tan

Trigonometry Lesson 1 Aims: • To know what the curves of sin-1x , cos-1x and tan-1x look like and to be able to state their ranges and domains. • To know the graphical relationship between a trig curve and it’s inverse. • To know what the principle values are. • To know the relationship between the graphs and notation of cosine, sine and tan, with secant, cosecant and cotangent . • To know the domain & range of the 3 new trig functions • To be able to calculate exact values of these reciprocal trig functions, with use of a calculator.

The inverse of the sine function Suppose we wish to find θ such that sin θ = x This is written as θ = sin–1 x In this context, sin–1 x means the i______________ of sin x. This is not to be mixed up with (sin x)–1 = which is the r___________ and we will meet later. Do you think this sine graph has an inverse? ____ because ____________________ What can we do to allow the function to have an inverse? y y = sin x Stress that while we do write sin2x to mean (sin x)2, sin–1x does not mean (sin x)–1. x

The inverse of the sine function So, if we restrict the domain of f(x) = sin x to ≤ x ≤ we have a one-to-one function: There is only one value of sin–1 x in this range, called the p__________________value. The graph of y = sin–1 x is the reflection of y = sin x in the line y = x y 1 –1 y = sin–1 x 1 –1 x y 1 –1 y = sin–1 x y = sin x (Remember the scale used on the x- and y-axes must be the same.) The domain of sin–1 x is the same as the range of sin x : ≤ x ≤ The range of sin–1 x is the same as the restricted domain of sin x : ≤ sin–1 x ≤

The inverse of cosine and tangent We can restrict the domains of cos x and tan x in the same way as we did for sin x so that if f(x) = cos x for ≤ x ≤ f –1(x) = cos–1 x then for ≤ x ≤ . < x < And if f(x) = tan x for Remind students that tan x is undefined for x = –π/2 and x = π/2 and so these values are not included when restricting the domain. f –1(x) = tan–1 x then for The graphs cos–1 x and tan–1 x can be obtained by reflecting the graphs of cos x and tan x in the line y = x.

The graph of y = cos–1 x x y 1 –1 y = cos–1 x y 1 –1 y = cos–1 x 1 x y 1 –1 y = cos–1 x y 1 –1 y = cos–1 x 1 x –1 y = cosx The domain of cos–1 x is the same as the range of cos x : ≤ x ≤ The range of cos–1 x is the same as the restricted domain of cos x : ≤ cos–1 x ≤

The graph of y = tan–1 x x y y = tan x y = tan–1 x y y = tanx The domain of tan–1 x is the same as the range of tan x : x The range of tan–1 x is the same as the restricted domain of tan x : < tan–1 x <

Find the principle values of these in degrees & radians In exact form for qu 1 to 3. To 3 d.p. in qu 4 - 5 Hint: Put your calc in degrees mode and convert to radians after. Solve qu 6 - 8 5. cos-1 0.9 = Do exercise A on the worksheet – 10 minutes

The reciprocal trigonometric functions The reciprocal trigonometric functions are cosecant, secant and cotangent. They are related to the three main trigonometric ratios as follows: cosec x = sec x = cot x = This is short for cosecant. This is short for secant. This is short for cotangent. Notice that the first letter of sin, cos and tan happens to be the same as the third letter of the corresponding reciprocal functions cosec, sec and cot.

The graph of sec x What is the wave length (period) of cos x? __________ Drag the point through the graph of cos x and as the graph of sec x is traced out make the following observations: When y = cos x = 1, y = sec x = 1. As y = cos x decreases from 1 towards 0, y = sec x increases from 1 towards +∞. When y = cos x = 0, y = sec x is undefined. As y = cos x decreases from 0 towards –1, y = sec x increases from –∞ towards –1. When y = cos x = –1, y = sec x = –1. State the range and domain of sec x

The graph of cosec x What is the wave length (period) of sin x? __________ Drag the point through the graph of sin x and as the graph of cosec x is traced out make the following observations: When y = sin x = 0, y = cosec x is undefined. As y = sin x increases from 0 towards 1, y = cosec x decreases from +∞ towards 1. When y = sin x = 1, y = cosec x = 1. As y = sin x decreases from 1 towards 0, y = cosec x increases from 1 towards +∞. When y = sin x = –1, y = cosec x = –1. State the range and domain of cosec x

The graph of cot x What is the wave length (period) of tan x? __________ Drag the point through the graph of tan x and as the graph of cot x is traced out make the following observations: When y = tan x is 0, y = cot x is undefined. As y = tan x increases from 0 towards +∞, y = cot x decreases from +∞ towards 0. When y = tan x is undefined, y = cot x = 0. As y = tan x increases from –∞ towards 0, y = cot x decreases from 0 towards –∞. State the range and domain of cot x

Problems involving reciprocal trig functions Use a calculator to find, to 2 d.p., the value of: a) sec 85° b) cosec 220° c) cot –70° Find the exact value of: a) cosec b) cot c) sec – You can put your calculator in radians mode and recognise those special decimals! a) a) b) b) Ensure that students have their calculators set to degrees before working through these problems. Note that only the cot of an angle can have values between –1 and 1 (the sec or cosec of an angle is either greater than 1 or less than –1). c) c)

Problems involving reciprocal trig functions Extension try Simplify sec x cot x Do exercise B from worksheet