UPLI FT EDUCATI ON
- Applications of Trig Diff
- Diff of Inverse Functions
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YEAR 12 Ext 1 Math
Trigonometric Differentiation
Q1 If 𝑦 = sec 𝑥 tan 𝑥
a) Find the coordinates of any x-intercepts
b) Find 𝑦 ′ , hence, show that the curve has no stationary points.
c) Find 𝑦′′, hence, discuss the change of concavity of the curve for 0 ≤ 𝑥 ≤ 2𝜋
d) State the equation of the asymptotes, hence, sketch the curve for 0 ≤ 𝑥 ≤ 2𝜋
Q2 If 𝑦 = ln sin 𝑥
a) Solve sin 𝑥 > 0, hence state the domain.
b) Find 𝑦′, hence the co-ordinates of any stationary points.
c) Find 𝑦′′, hence, show that all these stationary points are maximum points.
d) Sketch the curve, for 0 ≤ 𝑥 ≤ 3𝜋
Q3 A damped oscillation is given by 𝑦 = 𝑒
1
4
− 𝑥
sin 3𝑡 for 𝑡 ≥ 0
a) Find the coordinates of any t-intercepts.
b) Find 𝑦′, hence the t-co-ordinates of the stationary points.
c) Discuss the behaviour of the curve as 𝑡 → +∞
d) Sketch the curve 𝑦 = 𝑒
1
4
− 𝑥
sin 3𝑥 for 0 ≤ 𝑥 ≤ 2𝜋
Q4. ABC is an isosceles triangle circumscribing about a circle of radius 𝑟 units.
a) If the base angle is 2𝜃, show that the area of ∆𝐴𝐵𝐶 =
2r2
tan 𝜃−tan3 𝜃
b) Hence, find the dimensions of the smallest isosceles triangle that can be circumscribed about a
circle of radius 𝑟 units.
Q5. 𝐴 is a point on the circumference of a circle of radius 𝑎 units. Using 𝐴 as the centre, an arc of radius
𝑟 units, 0 < 𝑟 < 2𝑎, is drawn to intercept the circle at two points 𝐵 and 𝐶.
a) If ∠𝐴𝐵𝐶 = 2𝜃, and the length of the arc 𝐵𝐶 (of the circle of centre A) is 𝑙, show that
𝑙 = 4𝑎𝜃 cos 𝜃
b) Hence, show that 𝑙 is maximum when 𝜃 = cot 𝜃
c) Approximate 𝜃 by graphical means.
Q6.2000
Q7. 2005
Q7. 2006
Derivatives of Inverse Trig. Functions
𝑦 = sin−1 𝑥
𝑦 = cos −1 𝑥
𝑦 = tan−1 𝑥
Proof of each:
𝑦′ =
1
1 − 𝑥2
1
′
𝑦 =−
1 − 𝑥2
1
′
𝑦 =
1 + 𝑥2
1. Differentiate the following:
𝑦 = sin−1 (4𝑥)
y = cos −1 (𝑒 𝑥 )
y = tan
−1
1
𝑥
y = tan−1 1 + 𝑥 2
2. Differentiate the following:
𝑦 = sin−1 (cos 𝑥)
𝑦 = tan(2 tan−1 𝑥)
𝑦 = tan−1 (cos 𝑥)
𝑦 = sin(2 sin−1 𝑥)
3. Differentiate the following:
𝑦 = 𝑥 2 cos−1 𝑥
ln(tan−1 𝑥)
sin−1 (ln 𝑥)
𝑦 = 𝑥𝑡𝑎𝑛−1 (𝑥)
4. Differentiate the following:
𝑥
−1
𝑦 = tan
1−𝑥
−1
𝑦 = sin
1−𝑥
1+𝑥
𝑦=
sin−1
𝑦 = tan
−1
𝑥
1−𝑥
1−𝑥
1+𝑥
Q5.
a)
b)
c)
d)
e)
𝑦 = 𝑥 2 sin−1 𝑥
Find the domain and range.
Find 𝑦 ′ , hence, determine the coordinates of any stationary points.
What happens at 𝑥 = 0, −1 𝑎𝑛𝑑 1
Find the coordinates of any points of inflection.
Sketch the curve.
Q6.
A picture 0.3m high is hung on a wall such that its bottom is 0.6m above the eye level of an observer
who stands at a distance of 𝑥 meters from the wall. Find 𝑥 so that the angle subtended by the picture is
a maximum.
Q7.
AB is a chord of length 2𝑥 in a circle of radius 𝑟.
a) Find the area of the largest triangle enclosed by the chord AB and the two radii that join to the
ends of the chord.
b) Prove that the area of the largest segment enclosed by the chord AB and the minor arc is equal to
half the area of the circle.
Challenge Problem 1
Challenge Problem 2
Challenge Problem 3