A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 6: Trigonometric functions · Lesson 3

Inverse trigonometric functions

114 marks

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Quick questions to start.

A1
Needed today
$f (x) = x^{2} + 1$ , $x \geq 0$ . Find $f^{- 1} (x)$ and state its domain.
[3 marks]
Answer
A2
Needed today
Write down the exact values of $sin \frac{π}{3}$ , $cos \frac{3 π}{4}$ and $tan (- \frac{π}{6})$ .
[3 marks]
Answer
A3
Needed today
Solve $cos x = - \frac{1}{2}$ for $0 \leq x \leq π$ .
[2 marks]
Answer
A4
Today's idea
Kai says that $arcsin x$ means $\frac{1}{sin x}$ . Explain why Kai is wrong, and find $arcsin \frac{1}{2}$ .
[2 marks]

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 $arccos \frac{1}{2}$ , $arcsin (- \frac{3}{2})$ and $arctan (- 1)$ .

Hint arcsin and arctan give an answer between −π/2 and π/2. arccos gives an answer between 0 and π.

B1
Find the exact values of $arcsin (- \frac{1}{2})$ , $arccos (- \frac{2}{2})$ , $arctan 3$ and $arccos 0$ .
[4 marks]
Answer

Example 2

Find the exact values of $cos (arcsin \frac{3}{5})$ and $tan (arccos (- \frac{4}{5}))$ .

Hint Call the angle θ, draw a right-angled triangle for its size, then decide the signs from the range of the inverse function.

B2
Find the exact values of $sin (arccos \frac{8}{17})$ and $tan (arcsin (- \frac{5}{13}))$ .
[4 marks]
Answer

Example 3

Show that $cos (arcsin x) = 1 - x^{2}$ for $- 1 \leq x \leq 1$ .

Hint Let y = arcsin x. Use sin² y + cos² y = 1, then decide the sign of cos y from the range of arcsin.

B3
Let $y = arctan x$ . Show that $cos y = \frac{1}{1 + x ^{2}}$ , and hence that $sin y = \frac{x}{1 + x ^{2}}$ .
[4 marks]

Example 4

Find the exact values of $arcsin (sin \frac{5 π}{6})$ and $arccos (cos (- \frac{π}{3}))$ .

Hint The answer must lie in the range of the inverse function, so it is not always the angle you started with.

B4
Find the exact values of $arcsin (sin (- \frac{π}{3}))$ , $arccos (cos \frac{5 π}{4})$ and $arctan (tan \frac{3 π}{4})$ .
[3 marks]
Answer

Example 5

The curve C has equation $y = 2 arcsin (x - 1) + \frac{π}{2}$ . Find the domain and range of the function, and where C meets the coordinate axes.

Hint arcsin needs an input from −1 to 1, and gives an output from −π/2 to π/2.

B5
The curve C has equation $y = 3 arccos (2 x) - π$ . Find the domain and range of the function, and the coordinates of the points where C meets the coordinate axes.
[6 marks]
Answer

Practice

A-Level

C1
Sam says $arccos (- \frac{3}{2}) = - \frac{π}{6}$ . Explain why Sam is wrong and give the correct value.
[4 marks]
Answer

C2
Solve $3 arccos (2 x - 1) = π$ .
[4 marks]
Answer
C3
(a)
Solve $4 arctan x = π$ .
[2 marks]
Answer
(b)
Explain why $4 arctan x = 3 π$ has no solution.
[2 marks]
Total for C3: 4 marks
C4
Prove that $arcsin x + arccos x = \frac{π}{2}$ for $- 1 \leq x \leq 1$ .
[5 marks]
C5
Find the exact values of $arcsin (sin 2)$ and $arccos (cos 4)$ , where the angles are in radians.
[4 marks]
Answer
C6
The curve C has equation $y = 2 arctan x + \frac{π}{2}$ . State the range of the function, the equations of the asymptotes of C, and where C meets the y-axis.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The function f is defined by $f (x) = 2 arccos (x - 1)$ , $0 \leq x \leq 2$ .
(a)
State the range of f.
[2 marks]
Answer
(b)
Find $f^{- 1} (x)$ and state its domain.
[4 marks]
Answer
(c)
Hence solve $f (x) = \frac{4 π}{3}$ .
[3 marks]
Answer
Total for D1: 9 marks

D2
Ella writes: " $cos (arccos (- \frac{7}{25})) = - \frac{7}{25}$ is negative, so $sin (arccos (- \frac{7}{25})) = - \frac{24}{25}$ ."
Ella is wrong.
(a)
Explain Ella's mistake and give the correct value.
[2 marks]
(b)
Find the exact value of $tan (arccos (- \frac{7}{25}))$ .
[2 marks]
Answer
(c)
Show that $tan (arcsin x) = \frac{x}{1 - x ^{2}}$ for $- 1 < x < 1$ .
[3 marks]
(d)
Hence find the exact value of $tan (arcsin \frac{1}{3})$ , with a rational denominator.
[2 marks]
Answer
Total for D2: 9 marks
D3
Solve $arcsin x = arccos 2 x$ , giving your answer as an exact value with a rational denominator.
[7 marks]
Answer
D4
(a)
Show that $arctan x + arctan \frac{1}{x} = \frac{π}{2}$ for $x > 0$ .
[4 marks]
(b)
Hence solve $arctan x = 2 arctan \frac{1}{x}$ for $x > 0$ .
[3 marks]
Answer
(c)
State the value of $arctan x + arctan \frac{1}{x}$ when $x < 0$ , giving a reason.
[2 marks]
Answer
Total for D4: 9 marks
D5
The function h is defined by $h (x) = π - 3 arcsin \frac{x}{2}$ , $- 2 \leq x \leq 2$ .
(a)
Sketch the graph of $y = h (x)$ . Show the coordinates of the end points of the graph and of the point where it meets the y-axis.
[3 marks]
(b)
State the range of h.
[2 marks]
Answer
(c)
Find the exact x-coordinate of the point where the graph crosses the x-axis.
[2 marks]
Answer
(d)
Solve $h (x) = arcsin \frac{x}{2}$ .
[2 marks]
Answer
Total for D5: 9 marks
D6
This question is about the equation $arctan \frac{x}{2} = arccos \frac{1}{x}$ , where $∣ x ∣ \geq 1$ .
(a)
Let $θ = arccos \frac{1}{x}$ . Show that $tan^{2} θ = x^{2} - 1$ .
[2 marks]
(b)
By taking tan of both sides, solve the equation. Give your answer in exact form, and explain why any other value you find is not a solution.
[5 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
Solve $arcsin x = arctan 2 x$ . Give exact answers.
[8 marks]
Hint 1 · What to try
Let $θ = arcsin x = arctan 2 x$ . Then $sin θ = x$ and $tan θ = 2 x$ .
Hint 2 · The first line
So $\frac{sin θ}{cos θ} = 2 sin θ$ . Bring everything to one side and factorise; do not divide by $sin θ$ .
Hint 3 · The full method
$θ$ must lie between $- \frac{π}{2}$ and $\frac{π}{2}$ . Check each value of $x$ in the original equation.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can state the domain and range of arcsin, arccos and arctan.
I can find exact values such as arccos(−√2/2) and sin(arccos(8/17)).
I can sketch a curve made from an inverse trig function, with its domain, range and axis crossings.
I can prove and use results such as arcsin x + arccos x = π/2.

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