Unit 1: Fundamentals of trigonometry - Subjective Questions

MTH110 — Remedial Mathematics • Practice Questions with Detailed Answers

20 questions

Define trigonometric ratios with reference to a right-angled triangle. List all six ratios and their reciprocal relationships.

Find the values of all six trigonometric ratios for the angle $30^\circ$ and explain how they are derived.

State and prove the three fundamental trigonometric (Pythagorean) identities.

Explain the relation between radian measure and degree measure. Convert $150^\circ$ to radians and $\dfrac{5\pi}{6}$ radians to degrees.

Describe how radian measure connects angles with real numbers, and explain why this connection makes trigonometric functions functions of real variables.

State the domain and range of the six trigonometric functions.

The domain is the set of allowed input values and the range is the set of possible output values.

Function	Domain	Range
$\sin x$	$\mathbb{R}$	$[-1, 1]$
$\cos x$	$\mathbb{R}$	$[-1, 1]$
$\tan x$	$\mathbb{R} - \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$	$\mathbb{R}$
$\csc x$	$\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$	$\mathbb{R} - (-1, 1)$
$\sec x$	$\mathbb{R} - \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$	$\mathbb{R} - (-1, 1)$
$\cot x$	$\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$	$\mathbb{R}$

Notes:

$\sin x$ and $\cos x$ are defined for all reals because they come directly from coordinates on the unit circle.
$\tan x$ and $\sec x$ are undefined where $\cos x = 0$ .
$\cot x$ and $\csc x$ are undefined where $\sin x = 0$ .
$\csc x$ and $\sec x$ never take values between $-1$ and $1$ (exclusive).

Derive the formula for $\cos(A - B)$ and use it to find $\cos 15^\circ$ .

Derive the formula for $\sin(A + B)$ and hence find the value of $\sin 75^\circ$ .

Distinguish between trigonometric ratios and trigonometric functions.

Trigonometric ratios and trigonometric functions are related but conceptually different.

Basis	Trigonometric Ratios	Trigonometric Functions
Definition	Ratios of sides of a right-angled triangle	Functions defined for all real numbers (angles)
Domain	Acute angles ( $0^\circ$ to $90^\circ$ )	All real numbers (with restrictions for some)
Basis	Geometry of triangles	Unit circle / real analysis
Nature	Fixed numerical values for a given angle	Mapping $f: \mathbb{R} \to \mathbb{R}$
Example	$\sin 30^\circ = \frac{1}{2}$	$\sin x$ where $x \in \mathbb{R}$

Explanation:

Trigonometric ratios came first, historically defined only for acute angles inside a right triangle.
Trigonometric functions generalize these ratios using the unit circle so they apply to any angle, including negative and reflex angles.
This generalization allows periodicity, graphing, and use in calculus, which ratios alone cannot support.

Prove that $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta \csc\theta$ .

Explain the concept of the unit circle and how trigonometric functions are defined using it.

Prove that $\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$ using the sum formulas for sine and cosine.

Describe the signs of the trigonometric functions in the four quadrants and state the rule used to remember them.

The sign of a trigonometric function depends on the signs of the coordinates $(x, y)$ of the point on the unit circle in each quadrant.

Quadrant	Angle Range	$\sin$	$\cos$	$\tan$
I	$0$ to $90^\circ$	$+$	$+$	$+$
II	$90^\circ$ to $180^\circ$	$+$	$-$	$-$
III	$180^\circ$ to $270^\circ$	$-$	$-$	$+$
IV	$270^\circ$ to $360^\circ$	$-$	$+$	$-$

Memory Rule — "All Silver Tea Cups" (ASTC):

A (Quadrant I): All functions positive.
S (Quadrant II): Sin (and csc) positive; rest negative.
T (Quadrant III): Tan (and cot) positive; rest negative.
C (Quadrant IV): Cos (and sec) positive; rest negative.

The reciprocal functions ( $\csc, \sec, \cot$ ) share the same sign as their corresponding primary functions ( $\sin, \cos, \tan$ ).

If $\sin\theta = \dfrac{3}{5}$ and $\theta$ lies in the first quadrant, find the values of all other trigonometric ratios.

Prove that $\cos(A + B)\cos(A - B) = \cos^2 A - \sin^2 B$ .

Explain the periodicity of trigonometric functions. State the periods of all six functions.

Prove the identity $\dfrac{1 + \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 - \cos\theta}$ .

Compare the graphs of $\sin x$ and $\cos x$ . Describe their key features and the relationship between them.

Prove that $\sin(A + B) + \sin(A - B) = 2\sin A \cos B$ and use it to express $\sin 3x + \sin x$ as a product.

Prove that $\sin^6\theta + \cos^6\theta = 1 - 3\sin^2\theta\cos^2\theta$ .

Question 1

Define trigonometric ratios with reference to a right-angled triangle. List all six ratios and their reciprocal relationships.

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Answer 1

Trigonometric ratios are ratios of the sides of a right-angled triangle with respect to one of its acute angles.

Consider a right-angled triangle with an acute angle $\theta$ , where:

Hypotenuse = side opposite the right angle
Opposite = side opposite to $\theta$
Adjacent = side adjacent to $\theta$

The six ratios are:

$\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$
$\csc\theta = \dfrac{\text{Hypotenuse}}{\text{Opposite}}$
$\sec\theta = \dfrac{\text{Hypotenuse}}{\text{Adjacent}}$
$\cot\theta = \dfrac{\text{Adjacent}}{\text{Opposite}}$

Reciprocal relationships:

$\csc\theta = \dfrac{1}{\sin\theta}$
$\sec\theta = \dfrac{1}{\cos\theta}$
$\cot\theta = \dfrac{1}{\tan\theta}$

Also, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ and $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$ .

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Question 5

Describe how radian measure connects angles with real numbers, and explain why this connection makes trigonometric functions functions of real variables.

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Answer 5

Radian measure and real numbers:

When we measure angles in radians, every angle corresponds to a unique real number, and every real number corresponds to a unique angle. This one-to-one correspondence arises because the radian measure of an angle is defined as the ratio:
$\theta = \frac{\text{arc length}}{\text{radius}} = \frac{l}{r}$

Since this is a ratio of two lengths, it is a dimensionless real number.

Key points:

A positive real number represents an angle measured counter-clockwise.
A negative real number represents an angle measured clockwise.
Real numbers beyond $2\pi$ represent angles obtained after one or more complete rotations.

Why trigonometric functions become functions of real variables:

Because each real number $x$ maps to a unique angle of $x$ radians, we can define $\sin x$ , $\cos x$ , etc., where $x$ is any real number.
This allows us to treat trigonometric functions as functions $f: \mathbb{R} \to \mathbb{R}$ , enabling their use in calculus, graphing, and analysis.
It removes the dependence on degrees and unifies trigonometry with the study of real-valued functions.

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Question 6

State the domain and range of the six trigonometric functions.

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Answer 6

The domain is the set of allowed input values and the range is the set of possible output values.

Function	Domain	Range
$\sin x$	$\mathbb{R}$	$[-1, 1]$
$\cos x$	$\mathbb{R}$	$[-1, 1]$
$\tan x$	$\mathbb{R} - \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$	$\mathbb{R}$
$\csc x$	$\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$	$\mathbb{R} - (-1, 1)$
$\sec x$	$\mathbb{R} - \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$	$\mathbb{R} - (-1, 1)$
$\cot x$	$\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$	$\mathbb{R}$

Notes:

$\sin x$ and $\cos x$ are defined for all reals because they come directly from coordinates on the unit circle.
$\tan x$ and $\sec x$ are undefined where $\cos x = 0$ .
$\cot x$ and $\csc x$ are undefined where $\sin x = 0$ .
$\csc x$ and $\sec x$ never take values between $-1$ and $1$ (exclusive).

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Question 9

Distinguish between trigonometric ratios and trigonometric functions.

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Answer 9

Trigonometric ratios and trigonometric functions are related but conceptually different.

Basis	Trigonometric Ratios	Trigonometric Functions
Definition	Ratios of sides of a right-angled triangle	Functions defined for all real numbers (angles)
Domain	Acute angles ( $0^\circ$ to $90^\circ$ )	All real numbers (with restrictions for some)
Basis	Geometry of triangles	Unit circle / real analysis
Nature	Fixed numerical values for a given angle	Mapping $f: \mathbb{R} \to \mathbb{R}$
Example	$\sin 30^\circ = \frac{1}{2}$	$\sin x$ where $x \in \mathbb{R}$

Explanation:

Trigonometric ratios came first, historically defined only for acute angles inside a right triangle.
Trigonometric functions generalize these ratios using the unit circle so they apply to any angle, including negative and reflex angles.
This generalization allows periodicity, graphing, and use in calculus, which ratios alone cannot support.

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Question 11

Explain the concept of the unit circle and how trigonometric functions are defined using it.

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Answer 11

The Unit Circle:

The unit circle is a circle of radius $1$ centred at the origin $(0,0)$ of the coordinate plane, described by the equation:
$x^2 + y^2 = 1$

Defining trigonometric functions:

For any real number $\theta$ , measure an angle of $\theta$ radians counter-clockwise from the positive $x$ -axis. This determines a point $P(x, y)$ on the unit circle. Then:

$\cos\theta = x$ (the x-coordinate)
$\sin\theta = y$ (the y-coordinate)
$\tan\theta = \dfrac{y}{x}$ (for $x \neq 0$ )
$\csc\theta = \dfrac{1}{y}$ , $\sec\theta = \dfrac{1}{x}$ , $\cot\theta = \dfrac{x}{y}$

Advantages of this definition:

Since $-1 \le x \le 1$ and $-1 \le y \le 1$ , it immediately shows that $\sin\theta$ and $\cos\theta$ lie in $[-1, 1]$ .
The signs of the functions in each quadrant follow from the signs of $x$ and $y$ .
$\sin^2\theta + \cos^2\theta = x^2 + y^2 = 1$ is automatic.
It extends the definitions to all angles, giving periodic functions with period $2\pi$ .

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Question 13

Describe the signs of the trigonometric functions in the four quadrants and state the rule used to remember them.

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Answer 13

The sign of a trigonometric function depends on the signs of the coordinates $(x, y)$ of the point on the unit circle in each quadrant.

Quadrant	Angle Range	$\sin$	$\cos$	$\tan$
I	$0$ to $90^\circ$	$+$	$+$	$+$
II	$90^\circ$ to $180^\circ$	$+$	$-$	$-$
III	$180^\circ$ to $270^\circ$	$-$	$-$	$+$
IV	$270^\circ$ to $360^\circ$	$-$	$+$	$-$

Memory Rule — "All Silver Tea Cups" (ASTC):

A (Quadrant I): All functions positive.
S (Quadrant II): Sin (and csc) positive; rest negative.
T (Quadrant III): Tan (and cot) positive; rest negative.
C (Quadrant IV): Cos (and sec) positive; rest negative.

The reciprocal functions ( $\csc, \sec, \cot$ ) share the same sign as their corresponding primary functions ( $\sin, \cos, \tan$ ).

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Question 16

Explain the periodicity of trigonometric functions. State the periods of all six functions.

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Answer 16

Periodicity:

A function $f$ is said to be periodic with period $T$ if $f(x + T) = f(x)$ for all $x$ in its domain, where $T$ is the smallest positive such number.

Trigonometric functions are periodic because they are defined via the unit circle. After completing one full rotation ( $2\pi$ radians), the point returns to the same position, so the values repeat.

Periods of the six functions:

$\sin x$ has period $2\pi$ : $\sin(x + 2\pi) = \sin x$
$\cos x$ has period $2\pi$ : $\cos(x + 2\pi) = \cos x$
$\csc x$ has period $2\pi$
$\sec x$ has period $2\pi$
$\tan x$ has period $\pi$ : $\tan(x + \pi) = \tan x$
$\cot x$ has period $\pi$

Why $\tan$ and $\cot$ have period $\pi$ :
Since $\tan x = \dfrac{\sin x}{\cos x}$ , adding $\pi$ changes the signs of both $\sin x$ and $\cos x$ , which cancel out, leaving the value unchanged after only $\pi$ instead of $2\pi$ .

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Question 18

Compare the graphs of $\sin x$ and $\cos x$ . Describe their key features and the relationship between them.

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Answer 18

Graph of $\sin x$ :

Domain: $\mathbb{R}$ ; Range: $[-1, 1]$
Passes through the origin $(0, 0)$
Period: $2\pi$
Maximum $= 1$ at $x = \frac{\pi}{2} + 2n\pi$ ; Minimum $= -1$ at $x = \frac{3\pi}{2} + 2n\pi$
Odd function: $\sin(-x) = -\sin x$ (symmetric about origin)

Graph of $\cos x$ :

Domain: $\mathbb{R}$ ; Range: $[-1, 1]$
Passes through $(0, 1)$
Period: $2\pi$
Maximum $= 1$ at $x = 2n\pi$ ; Minimum $= -1$ at $x = \pi + 2n\pi$
Even function: $\cos(-x) = \cos x$ (symmetric about y-axis)

Relationship between them:

Both are smooth, continuous, wave-like curves with the same shape, amplitude, and period.
The cosine graph is the sine graph shifted left by $\frac{\pi}{2}$ :
$\cos x = \sin\left(x + \frac{\pi}{2}\right)$
Equivalently, $\sin x = \cos\left(x - \frac{\pi}{2}\right)$ .
This phase shift is why they are called co-functions.

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Unit 1: Fundamentals of trigonometry - Subjective Questions

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