Trigonometric functions Mathematics content

Main ideas: circular measure of angles. angle, arc, sector trig functions and their graphs periodicity and amplitude approximation to sin x, cos x, tan x when x is small differentiating trig functions primitives of trig functions extension to functions of the form a sin (bx+c) etc

Radian Measure A radian is defined as the angle made by taking the radius of a circle and ‘wrapping’ it around the circumference radian demonstration radian measure

So: There are π radians in a half circle And also 180° in a half circle So π radians = 180° So 1 radian = 180°/π = 57.2958° (approximately) and also 1° = 0.01745 radians (approximately) The circular measure of angles involves the measurement of angles in radians.

Express 90° in radians 180° = π radians Find the equivalent of π radians in degrees 3 π radians = 180° π radians = 180° 3 3 = 60° 1° = π radians 180 so 90°= 90 x π radians = π radians 2 Evaluate cos 1.145 using your calculator: (use RAD mode!) cos 1.145 = 0.413046135 = 0.413 (to 3 dec pl)

NOTE: Exact value triangles make a reappearance in this topic!

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Length of Circular Arc In a circle of radius r units, an angle measuring θ radians stands on an arc of length l units. As l increases, so does θ, until the arc becomes the circumference (2πr) and the angle becomes a complete revolution (2π radians). l = θ 2πr 2π and l = rθ

Area of a Sector The area of the shaded sector below is found as a fraction of the entire circle: Area of sector = θ πr2 2π A = ½ r2θ

e.g. Find the length of the arc in a circle of radius 6 cm, subtending an angle of 0.86 radians at the centre; and find the area of the corresponding sector A = ½ r2θ = ½ x 62 x 0.86 = 15.48 cm2 l = rθ = 6 x 0.86 = 5.16

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Groves

Cambridge

Fitzpatrick EXTENSION ONLY

Fitzpatrick EXTENSION ONLY

Fitzpatrick EXTENSION ONLY

Small Angles What happens when we use small angles, like 0.02 radians? sin x = 0.019999, tan x = 0.0200 cos x = 0.9998 Thus sin x ≈ x ≈ tan x for small x and cos x ≈ 1 and it follows that: lim x→0 sin x = 1 x

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Graphs of Trigonometric Functions We have already drawn graphs of trig functions...now we can draw them using radians: y = sin x y = cos x y = tan x

And the reciprocal ratios:

Period and Amplitude Trig curves repeat after a certain distance. This distance is called the period. The amplitude of the curve is half the distance between the maximum and minimum heights of the curve.

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Cambridge

Differentiating Trigonometric Functions: sin x Consider the gradient of the curve y = sin x The gradient function is actually y = cos x

EXTENSION ONLY

Function of a Function Rule (Chain Rule)

Derivative of cos x We can similarly graph the gradient of cos x The gradient function is y = -sin x

Derivative of tan x The gradient function of y = tan x is harder to sketch: The gradient function is actually y = sec2x

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Cambridge

Cambridge

Primitives of Trigonometric Functions Working backwards it is quite straightforward to find the primitives of trigonometric functions:

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Cambridge

Cambridge

Integration of sin2x and cos2x EXTENSION ONLY

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Make sure you: create a summary of the topic can work with angles, sectors and arcs can differentiate and integrate trig functions and apply this to solving problems