Trigonometry Inverse functions KUS objectives BAT use and understand applications and graphs of the inverse Trig functions Copy and complete, using surds where appropriate… 0° 30° 45° 60° 90° Sinθ 0.5 1/√2 or √2/2 √3/2 1 Cosθ 1 √3/2 1/√2 or √2/2 0.5 Tanθ 1/√3 or √3/3 1 √3 Undefined

π/6 π/4 π/3 π/2 Sinθ 0.5 1/√2 or √2/2 √3/2 1 Cosθ 1 √3/2 1/√2 or √2/2 The same values apply in radians as well… π/6 π/4 π/3 π/2 Sinθ 0.5 1/√2 or √2/2 √3/2 1 Cosθ 1 √3/2 1/√2 or √2/2 0.5 Tanθ 1/√3 or √3/3 1 √3 Undefined

𝒂𝒓𝒄𝒔𝒊𝒏 𝒙, 𝒂𝒓𝒄𝒄𝒐𝒔 𝒙, 𝒂𝒓𝒄𝒕𝒂𝒏 𝒙 𝒂𝒓𝒄𝒔𝒊𝒏 𝒙, 𝒂𝒓𝒄𝒄𝒐𝒔 𝒙, 𝒂𝒓𝒄𝒕𝒂𝒏 𝒙 These are the inverse functions of sin, cos and tan respectively However, an inverse function can only be drawn for a one-to-one function (when reflected in y = x, a many-to-one function would become one-to many, hence not a function) y = x y = arcsinx π/2 1 y = sinx -π/2 -1 1 π/2 -1 Remember that from a function to its inverse, the domain and range swap round (as do all co-ordinates) -π/2 y = sinx y = arcsinx Domain: -π/2 ≤ x ≤ π/2 Domain: -1 ≤ x ≤ 1 Range: -1 ≤ sinx ≤ 1 Range: -π/2 ≤ arcsinx ≤ π/2

We can’t use –π/2 ≤ x ≤ π/2 as the domain for cos, since it is many-to-one… y = arccosx (when reflected in y = x, a many-to-one function would become one-to many, hence not a function) y = x π/2 Remember that from a function to its inverse, the domain and range swap round (as do all co-ordinates) 1 -1 1 π/2 π -1 y = cosx y = cosx y = arccosx Domain: 0 ≤ x ≤ π Domain: -1 ≤ x ≤ 1 Range: -1 ≤ cosx ≤ 1 Range: 0 ≤ arccosx ≤ π

Domain: -π/2 < x < π/2 Domain: x ε R y = tanx π/2 y = arctanx -π/2 π/2 -π/2 Subtle differences… The domain for tanx cannot equal π/2 or –π/2 The range can be any real number! y = tanx y = arctanx Domain: -π/2 < x < π/2 Domain: x ε R Range: x ε R Range: -π/2 < arctanx < π/2

π y = arccosx π/2 -1 1 π/2 y = arcsinx π/2 y = arctanx -1 1 -π/2 -π/2

Work out in degrees, the value of arcsin − 2 2 Practice question 1a Work out in degrees, the value of arcsin − 2 2 Work out in radians, the value of cos [arcsin −1 ] a) Arcsin just means inverse sin… 1 y = sinx √2/2 Ignore the negative for now, and remember the values from earlier… -π/4 -π/2 π/4 π/2 -√2/2 Sin(-θ) = -Sinθ (or imagine the Sine graph…) -1

Work out in degrees, the value of arcsin − 2 2 Practice question 1b Work out in degrees, the value of arcsin − 2 2 Work out in radians, the value of cos [arcsin −1 ] b) Arcsin just means inverse sin… y = sinx 1 1 y = cosx Think about what value you need for x to get Sin x = –1 -π/2 π/2 -π/2 π/2 Cos(-θ) = Cos(θ) -1 -1 Remember it, or read from the graph…

Work out, in radians, the value of arcsin (0.5) Practice question2 Work out, in radians, the value of arcsin (0.5) Work out, in radians, the value of arctan ( 3 ) b) a) Arctan just means inverse tan… Arctan just means inverse sin… Remember the exact values from earlier… Remember the exact values from earlier…

Practice question 3 a) Write down the value of arccos(-1) arcos √3 2 arcsin(-½) arccos −1 √2 In both degrees and radians b) Solve these where possible arcsinx = 𝜋 4 arcsinx = 𝜋 6 arcsinx =𝜋 180 30 −𝝅 𝟔 𝟑𝝅 𝟒 𝟏 √𝟐 ½ no solution

self-assess using: R / A / G ‘I am now able to ____ . KUS objectives BAT use and understand applications and graphs of the inverse Trig functions self-assess using: R / A / G ‘I am now able to ____ . To improve I need to be able to ____’