3.7 Inverse & Reciprocal Trig Functions

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 5c 6a 6b 7a 7b 8 9

Question 1a

Marks: 4

Show that

\frac{\cot x}{\sec x (1 - \sin x)} \equiv 1 + \frac{1}{\sin x}, x \neq \frac{n π}{2}, n \in ℤ

Key Concepts

Reciprocal Trig Functions
Trigonometry - Simple Identities

Question 1b

Marks: 4

(b)

Hence determine the set of values for

k

for which

\frac{\cot x}{\sec x (1 - \sin x)} = k

has no real solutions.

Key Concepts

Reciprocal Functions & Graphs

Question 2a

Marks: 4

(a)

Find the exact value for the expression

\cot^{2} (\frac{π}{6}) + cosec (\frac{2 π}{3})

Give your answer in the form

a + \frac{b \sqrt{c}}{d}

where

a, b, c, d \in ℤ

Key Concepts

Reciprocal Trig Functions
Double Angle Formulae

Question 2b

Marks: 3

(b)

Find the exact value for the expression

\frac{\sec (\frac{2 π}{3})}{\cot (\frac{π}{4})}

Key Concepts

Reciprocal Trig Functions
Double Angle Formulae

Question 3a

Marks: 2

(a)

Sketch the graph of

y = \sec x

for

- 2 π \leq x \leq π

Key Concepts

Reciprocal Trig Functions
Pythagorean Identities

Question 3b

Marks: 5

(b)

Given that

\sec 2 θ = \frac{8}{7}

and

- 2 π \leq θ \leq - \frac{3 π}{2}

, find the value of

\sin θ

Key Concepts

Quadratic Trigonometric Equations
Relationship Between Trigonometric Ratios

Question 4a

Marks: 4

(a)

Show that the equation

$5 \sin x - 9 \cos x = \frac{2 cosec x}{1 + \tan^{2} x}$

can be rewritten as

(a \sin x + \cos x) (\sin x - b \cos x) = 0

Key Concepts

Reciprocal Trig Functions

Question 4b

Marks: 3

(b)

Hence solve the equation

5 \sin x - 9 \cos x = \frac{2 \cos e c x}{1 + \tan^{2} x}, - π \leq x \leq π .

Key Concepts

Quadratic Trigonometric Equations
Using Trigonometric Graphs

Question 5a

Marks: 4

Consider the function $f (x) = \cos^{2} (2 \arcsin (x))$ , $0 \leq x \leq a$ where $a$ is a constant.

(a)

Given that

f

has an inverse, find an expression for

f^{- 1} (x)

Key Concepts

Inverse Trig Functions
Inverse Functions

Question 5b

Marks: 4

(b)

Show that

\cos^{2} (2 \arcsin (x)) = 4 x^{2} - 4 x^{2} + 1

for

- 1 \leq x \leq 1

Key Concepts

Double Angle Formulae
Inverse Trig Functions

Question 5c

Marks: 2

(c)

Given that

f

has an inverse, find the maximum value for

a

Key Concepts

Inverse Functions
Sketching Polynomial Graphs

Question 6a

Marks: 3

A function $f$ can be defined by $f (x) = \frac{1}{2} \tan (\arcsin (x)) + 5$ .

(a)

Sketch the graph of

f

indicating clearly any intercepts with the coordinate axes and asymptotes.

Key Concepts

Graphs of Trigonometric Functions

Question 6b

Marks: 5

(b)

Solve the inequality

\frac{1}{2} \tan (\arcsin (x)) + 5 > 2

Give your answer in the form

\frac{a \sqrt{b}}{c} < x < d

where

a, b, c, d \in ℤ

Key Concepts

Inverse Trig Functions

Question 7a

Marks: 2

Consider the equation $\frac{1 - \cos^{2} θ}{\cot θ} = - \frac{1}{\cos e c^{2} θ}$ .

(a)

Explain why the equation is undefined when

θ = \frac{1}{2} k π

for any

k \in ℤ

Key Concepts

Reciprocal Trig Functions

Question 7b

Marks: 6

(b)

Solve the equation

\frac{1 - \cos^{2} θ}{\cot θ} = - \frac{1}{\cos e c^{2} θ}, for - π \leq θ \leq 2 π

Key Concepts

Reciprocal Trig Functions
Trigonometry - Simple Identities

Question 8

Marks: 8

Show that $\sin^{2} (\frac{1}{2} arcta n (x)) = \frac{1 + x^{2} - \sqrt{1 + x^{2}}}{2 (1 + x^{2})}, x \in ℝ .$

Key Concepts

Inverse Trig Functions
Double Angle Formulae

Question 9

Marks: 6

Solve the equation $2 \sin x + \frac{\cot x}{\sec x} = 3 \cos x$ for the interval $- 90 ° < x < 90 °$ .

Key Concepts

Reciprocal Trig Functions
Quadratic Trigonometric Equations

DP IB Maths: AA HL

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3.7 Inverse & Reciprocal Trig Functions

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