A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 6: Trigonometric functions · Lesson 1

Secant, cosecant and cotangent and their graphs

117 marks

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Quick questions to start.

A1
Needed today
Write down the exact values of $sin \frac{π}{3}$ , $cos \frac{3 π}{4}$ and $tan \frac{5 π}{6}$ .
[3 marks]
Answer
A2
Needed today
Write down the period, in radians, of $y = cos 2 x$ and of $y = tan \frac{x}{2}$ .
[2 marks]
Answer
A3
Common slip
Sam writes $sec 6 0^{\circ} = \frac{1}{sin 6 0 ^{\circ}}$ . Give the correct definition of $sec x$ and find the exact value of $sec 6 0^{\circ}$ .
[2 marks]
Answer
A4
Last chapter
Describe the two transformations that map $y = cos x$ onto $y = 3 cos x - 1$ , and state the range of $y = 3 cos x - 1$ .
[3 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the exact values of $sec 13 5^{\circ}$ , $cosec (- \frac{π}{3})$ and $cot \frac{4 π}{3}$ .

Hint Find sin, cos or tan of the angle first, with its sign. Then take the reciprocal.

B1
Find the exact values of $sec (- \frac{5 π}{6})$ , $cosec 21 0^{\circ}$ and $cot \frac{11 π}{6}$ .
[4 marks]
Answer

Example 2

Sketch $y = cosec x$ for $- 2 π \leq x \leq 2 π$ . Give the equations of the asymptotes, the coordinates of the turning points and the range.

Hint Draw y = sin x lightly first. Where sin x = 0 there is an asymptote; where sin x = ±1 there is a turning point.

B2
Sketch $y = sec x$ for $- \frac{5 π}{4} \leq x \leq \frac{5 π}{4}$ . Give the equations of the asymptotes, the coordinates of the turning points and the range.
[5 marks]
Answer

Example 3

Sketch $y = 1 + 2 sec x$ for $- π \leq x \leq π$ . Give the asymptotes, the turning points and the range.

Hint Stretch y = sec x by 2 vertically, then move it up 1. The asymptotes do not move.

B3
Sketch $y = 4 - 3 cosec 2 x$ for $0 < x < π$ . Give the equation of the asymptote, the coordinates of the turning points, saying which is a maximum and which a minimum, and the range.
[6 marks]
Answer

Example 4

Explain why $sec x = \frac{1}{2}$ has no solutions. For which values of $k$ does $cosec x = k$ have exactly two solutions with $0 < x < 2 π$ ?

Hint Look at the range of each graph and draw a horizontal line y = k across it.

B4
Find the values of $k$ for which the equation $2 sec x - 1 = k$ has exactly one solution with $- \frac{π}{2} < x < \frac{3 π}{2}$ , $x \neq = \frac{π}{2}$ .
[5 marks]
Answer

Example 5

Show that $cot x = - tan (x - \frac{π}{2})$ . Hence describe how the graph of $y = cot x$ can be obtained from the graph of $y = tan x$ .

Hint Write tan as sin over cos, and use sin(x − π/2) = −cos x and cos(x − π/2) = sin x.

B5
Show that $sec x = cosec (x + \frac{π}{2})$ . Hence describe the single transformation that maps $y = cosec x$ onto $y = sec x$ .
[4 marks]
Answer

Practice

A-Level

C1
Jo works out $cosec \frac{5 π}{3} - sec \frac{5 π}{3}$ and gets $2 + \frac{2 3}{3}$ . Explain the mistake Jo makes and find the correct exact value.
[4 marks]
Answer

C2
The function f is defined by $f (x) = 5 - 2 cosec x$ , $0 < x < 2 π$ , $x \neq = π$ . A student says that the range of f is $3 \leq f (x) \leq 7$ . Explain why the student is wrong and find the range of f.
[5 marks]
Answer
C3
For the curve $y = cosec (\frac{x}{2} + \frac{π}{6})$ , $- 2 π \leq x \leq 2 π$ , find the equations of the asymptotes and the coordinates of the turning points, stating which is a maximum and which is a minimum.
[6 marks]
Answer
C4
The curve $y = a + b sec x$ , $- \frac{π}{2} < x < \frac{3 π}{2}$ , $x \neq = \frac{π}{2}$ , where $a$ and $b$ are constants and $b > 0$ , has a minimum point at $(0, 7)$ and a maximum point at $(π, - 1)$ . Find $a$ and $b$ , and state the range of the curve.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The function f is defined by $f (x) = 3 sec 2 x + 1$ , $0 \leq x \leq π$ , $x \neq = \frac{π}{4}$ , $x \neq = \frac{3 π}{4}$ .
(a)
Show that the graph of $y = f (x)$ has asymptotes at $x = \frac{π}{4}$ and $x = \frac{3 π}{4}$ , and no others.
[2 marks]
Answer
(b)
Find the coordinates of the points on the graph where the gradient is zero.
[3 marks]
Answer
(c)
State the range of f.
[2 marks]
Answer
(d)
Hence find the set of values of $k$ for which the equation $f (x) = k$ has exactly two solutions.
[3 marks]
Answer
Total for D1: 10 marks

D2
Ravi is finding the asymptotes of $y = 2 cosec (x - \frac{π}{4})$ , $0 \leq x \leq 2 π$ . He writes: "The graph of $y = 2 cosec x$ has asymptotes at $x = π$ and $x = 2 π$ . Moving it by $\frac{π}{4}$ gives asymptotes at $x = \frac{3 π}{4}$ and $x = \frac{7 π}{4}$ ."
Ravi is wrong.
(a)
Explain Ravi's mistake and give the correct asymptotes.
[3 marks]
(b)
Solve $2 cosec (x - \frac{π}{4}) = - 22$ for $0 \leq x \leq 2 π$ .
[4 marks]
Answer
(c)
State the range of the curve.
[2 marks]
Answer
Total for D2: 9 marks
D3
Find the coordinates of every point where the curve $y = sec x$ meets the curve $y = 2 cos x$ , for $0 \leq x \leq 2 π$ . Give your answers exactly.
[7 marks]
Answer
D4
(a)
Show that $cot x = tan (\frac{π}{2} - x)$ .
[2 marks]
(b)
Using part (a), describe two transformations that map $y = tan x$ onto $y = cot x$ .
[2 marks]
Answer
(c)
Hence solve $cot 2 x = tan x$ for $0 < x < π$ , giving your answers in terms of $π$ .
[5 marks]
Answer
Total for D4: 9 marks
D5
The figure shows part of the curve with equation $y = p + q sec x$ , $- \frac{π}{2} < x < \frac{3 π}{2}$ , $x \neq = \frac{π}{2}$ , where $p$ and $q$ are constants. The curve has a minimum point at $(0, 2)$ and a maximum point at $(π, - 6)$ .
(a)
Find the value of $p$ and the value of $q$ .
[3 marks]
Answer
(b)
State the range of the curve.
[2 marks]
Answer
(c)
Hence find the exact values of $x$ where the curve meets the line $y = - 10$ .
[4 marks]
Answer
(d)
Explain why the curve does not meet the $x$ -axis.
[2 marks]
Total for D5: 11 marks
D6
Given that $sec θ = - \frac{5}{4}$ and $π < θ < \frac{3 π}{2}$ ,
(a)
find the exact value of $cosec θ$ ,
[3 marks]
Answer
(b)
find the exact value of $cot θ$ ,
[2 marks]
Answer
(c)
hence find the exact value of $cosec θ - cot θ$ .
[2 marks]
Answer
Total for D6: 7 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The function f is defined by $f (x) = a + b cosec (x + c)$ , $0 \leq x \leq 2 π$ , where $a$ , $b$ and $c$ are constants, $b > 0$ and $0 < c < π$ . The graph of $y = f (x)$ has an asymptote at $x = \frac{2 π}{3}$ , and the range of f is $f (x) \leq - 1$ or $f (x) \geq 5$ . Find $a$ , $b$ and $c$ , and solve $f (x) = 2 + 23$ .
[10 marks]
Hint 1 · What to try
An asymptote is where $sin (x + c) = 0$ . Which multiple of $π$ can $\frac{2 π}{3} + c$ be, when $0 < c < π$ ?
Hint 2 · The first line
The two ends of the range come from $cosec = 1$ and $cosec = - 1$ : so $a + b = 5$ and $a - b = - 1$ . Check both values are reached on the domain.
Hint 3 · The full method
For the equation you need $sin (x + \frac{π}{3}) = \frac{3}{2}$ with $\frac{π}{3} \leq x + \frac{π}{3} \leq \frac{7 π}{3}$ . Look at both ends of the interval.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find exact values of sec, cosec and cot for angles in degrees or radians.
I can sketch y = sec x, y = cosec x and y = cot x and give their asymptotes and turning points.
I can sketch a transformed graph such as y = a + b sec(kx) and state its range.
I can use a graph to say how many solutions an equation has.

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