Chapter 8 Introduction to Trigonometry (Concepts)

Basic Terms Related to Trigonometry

Trigonometry is a fascinating branch of mathematics that establishes and studies the relationships between the lengths of the sides and the measures of the angles of triangles. The word itself provides a clue about its focus: 'tri' (three), 'gon' (sides), and 'metron' (measure), combining to mean 'measuring the sides and angles of triangles'.

While trigonometry can be applied to any triangle, the fundamental concepts and ratios are first defined and most easily understood in the context of right-angled triangles. This chapter serves as an introduction to these basic ideas.

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Right-Angled Triangle and its Sides

A right-angled triangle is a triangle that contains one interior angle measuring exactly $90^\circ$. The other two angles must be acute angles (angles less than $90^\circ$), because the sum of all three angles in any triangle is $180^\circ$.

In a right-angled triangle, the sides are given specific names based on their position relative to the right angle and, importantly, relative to one of the two acute angles we choose to focus on.

Consider a right triangle $\triangle \text{ABC}$, where the angle at B is the right angle ($\angle \text{B} = 90^\circ$). Let's choose one of the acute angles, say $\angle \text{C}$, and denote its measure by the Greek letter $\theta$ (theta).

• Hypotenuse: This is always the side that is directly opposite the right angle. It is the longest side in any right-angled triangle. In $\triangle \text{ABC}$, side AC is the hypotenuse.
• Side Opposite to the angle (Perpendicular): This is the side that is directly across from or facing the acute angle $\theta$ we are currently considering. Relative to $\angle \text{C}$ (our $\theta$), the side AB is the side opposite to angle $\theta$.
• Side Adjacent to the angle (Base): This is the side that is next to the acute angle $\theta$ we are considering, but it is not the hypotenuse. Relative to $\angle \text{C}$ (our $\theta$), the side BC is the side adjacent to angle $\theta$.

It is crucial to remember that the terms 'opposite' and 'adjacent' sides are always defined with respect to a specific acute angle in the triangle. If we were to consider the other acute angle, $\angle \text{A}$ (let's call it $\phi$), the names of the opposite and adjacent sides would swap:

Relative to $\angle \text{A}$ (our $\phi$):

• Hypotenuse: AC (remains the same, as it's opposite the right angle).
• Side Opposite to $\angle \text{A}$ (or $\phi$): BC.
• Side Adjacent to $\angle \text{A}$ (or $\phi$), not the hypotenuse: AB.

Correctly identifying these sides relative to the angle of interest is the first essential step in applying trigonometry to right triangles.

Trigonometric Ratios (Concept Introduction)

For a given acute angle in a right-angled triangle, the ratio of the lengths of two sides of the triangle is constant, regardless of the size of the triangle. This is due to the property of similar triangles – if two right triangles have one acute angle equal, they are similar, and thus the ratios of corresponding sides are equal.

These constant ratios of the sides are called trigonometric ratios. There are six primary trigonometric ratios, each defined as the ratio of a specific pair of sides relative to an acute angle. These ratios are the foundation of trigonometry and will be introduced and defined in the next section.

Trigonometric Ratios

In trigonometry, the trigonometric ratios are the ratios of the lengths of the sides of a right-angled triangle with respect to its acute angles. These ratios are constant for a given angle, regardless of the size of the triangle, because similar triangles (which have equal angles) have proportional sides. There are six main trigonometric ratios.

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Definition of Trigonometric Ratios

Consider a right-angled triangle $\triangle \text{ABC}$, right-angled at B ($\angle \text{B} = 90^\circ$). Let $\theta$ be the measure of one of the acute angles, say $\angle \text{C}$.

Relative to the acute angle $\theta$ at C, the sides are identified as:

• Hypotenuse (H): AC
• Side Opposite to $\theta$ (P - Perpendicular): AB
• Side Adjacent to $\theta$ (B - Base): BC

The six trigonometric ratios for the angle $\theta$ are defined as follows:

1. Sine of $\theta$ (short form: $\sin \theta$): The ratio of the length of the side opposite to the angle $\theta$ to the length of the hypotenuse.

2. Cosine of $\theta$ (short form: $\cos \theta$): The ratio of the length of the side adjacent to the angle $\theta$ to the length of the hypotenuse.

3. Tangent of $\theta$ (short form: $\tan \theta$): The ratio of the length of the side opposite to the angle $\theta$ to the length of the side adjacent to the angle $\theta$.

The other three ratios are the reciprocals of the first three:

4. Cosecant of $\theta$ (short form: $\text{cosec } \theta$): The reciprocal of $\sin \theta$. It is the ratio of the hypotenuse to the side opposite to $\theta$. Cosec $\theta$ is defined when $\sin \theta \neq 0$ (i.e., when $\theta \neq 0^\circ$ and $\theta \neq 180^\circ$, etc.).

5. Secant of $\theta$ (short form: $\text{sec } \theta$): The reciprocal of $\cos \theta$. It is the ratio of the hypotenuse to the side adjacent to $\theta$. Sec $\theta$ is defined when $\cos \theta \neq 0$ (i.e., when $\theta \neq 90^\circ, 270^\circ$, etc.).

6. Cotangent of $\theta$ (short form: $\cot \theta$): The reciprocal of $\tan \theta$. It is the ratio of the side adjacent to $\theta$ to the side opposite to $\theta$. Cot $\theta$ is defined when $\tan \theta \neq 0$ (i.e., when $\theta \neq 0^\circ, 180^\circ$, etc.).

From the definitions, we can also derive relationships between the ratios:

Mnemonic to Remember Ratios:

For a right-angled triangle, the fundamental trigonometric ratios relate the angles to the ratio of the lengths of its sides. The sides are defined relative to an angle $\theta$:

• Perpendicular (P): The side opposite to the angle $\theta$.
• Base (B): The side adjacent to the angle $\theta$.
• Hypotenuse (H): The longest side, opposite the right angle.

A popular mnemonic used in India to remember the definitions of Sine ($\sin$), Cosine ($\cos$), and Tangent ($\tan$) is the phrase:

"Pandit Badri Prasad, Har Har Bole, Sonā Chāndi Tole"

Now, write this mnemonic as follows:-

The mnemonic "Pandit Badri Prasad, Har Har Bole, Sonā Chāndi Tole" is a clever way to remember the trigonometric ratios for sine, cosine, and tangent. Each part of the phrase has a specific meaning corresponding to the sides of a right-angled triangle (Perpendicular, Base, Hypotenuse) and the ratios themselves.

Let's break down the meaning of each word in this mnemonic structure:

$\begin{matrix} \textbf{Sonā} & \textbf{Chāndi} & \textbf{Tole} \\ \downarrow & \downarrow & \downarrow \\ \frac{\textbf{P}\text{andit}}{\textbf{H}\text{ar}} & \frac{\textbf{B}\text{adri}}{\textbf{H}\text{ar}} & \frac{\textbf{P}\text{rasad}}{\textbf{B}\text{ole}} \end{matrix}$

Word	First Letter	Represents	Role in the Ratio
Pandit	P	Perpendicular	Numerator for Sine and Tangent
Badri	B	Base	Numerator for Cosine
Prasad	P	Perpendicular	(This reinforces 'P' for the Tangent ratio's numerator)
Har	H	Hypotenuse	Denominator for Sine and Cosine
Har	H	Hypotenuse	(Reinforces 'H' for Cosine's denominator)
Bole	B	Base	Denominator for Tangent
Sonā	S	Sine ($\sin$)	Links $\frac{P}{H}$ to the Sine function
Chāndi	C	Cosine ($\cos$)	Links $\frac{B}{H}$ to the Cosine function
Tole	T	Tangent ($\tan$)	Links $\frac{P}{B}$ to the Tangent function

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Putting It All Together

The mnemonic is structured as three pairs of words, where each pair forms a fraction. The final three words tell you which trigonometric function corresponds to which fraction.

1. Sonā $\implies$ Sine: The first fraction is formed by the first words of "Pandit Badri Prasad" and "Har Har Bole".

   $\sin \theta = \frac{\textbf{P}\text{andit}}{\textbf{H}\text{ar}} \implies \frac{\textbf{P}\text{erpendicular}}{\textbf{H}\text{ypotenuse}}$

2. Chāndi $\implies$ Cosine: The second fraction is formed by the second words of the phrases.

   $\cos \theta = \frac{\textbf{B}\text{adri}}{\textbf{H}\text{ar}} \implies \frac{\textbf{B}\text{ase}}{\textbf{H}\text{ypotenuse}}$

3. Tole $\implies$ Tangent: The third fraction is formed by the third words of the phrases.

   $\tan \theta = \frac{\textbf{P}\text{rasad}}{\textbf{B}\text{ole}} \implies \frac{\textbf{P}\text{erpendicular}}{\textbf{B}\text{ase}}$

This phrase can be broken down to form the ratios, where the first letter of each word corresponds to P, B, H, and the last three words refer to $\sin$, $\cos$, $\tan$.

Phrase Part 1	Phrase Part 2	Resulting Ratio	Trigonometric Function (S-C-T)
Pandit	Har	$\frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{P}{H}$	Sonā $\implies \sin \theta$
Badri	Har	$\frac{\text{Base}}{\text{Hypotenuse}} = \frac{B}{H}$	Chāndi $\implies \cos \theta$
Prasad	Bole	$\frac{\text{Perpendicular}}{\text{Base}} = \frac{P}{B}$	Tole $\implies \tan \theta$

This gives us the primary trigonometric ratios:

• $\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{P}{H}$
• $\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{B}{H}$
• $\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{P}{B}$

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Reciprocal Trigonometric Ratios

The other three trigonometric ratios—Cosecant ($\text{cosec}$), Secant ($\text{sec}$), and Cotangent ($\cot$)—are the reciprocals of the primary ratios. Their relationship is as follows:

Reciprocal Ratio	Formula	Relationship
Cosecant ($\text{cosec} \; \theta$)	$\frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{H}{P}$	$\text{cosec} \; \theta = \frac{1}{\sin \theta}$
Secant ($\text{sec} \; \theta$)	$\frac{\text{Hypotenuse}}{\text{Base}} = \frac{H}{B}$	$\text{sec} \; \theta = \frac{1}{\cos \theta}$
Cotangent ($\cot \theta$)	$\frac{\text{Base}}{\text{Perpendicular}} = \frac{B}{P}$	$\cot \theta = \frac{1}{\tan \theta}$

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Invariance of Ratios

It is important to note that the values of these trigonometric ratios for a given angle (e.g., $45^\circ$) do not change with the size of the triangle. This is because any two right-angled triangles that share an identical acute angle are similar triangles. Due to the properties of similar triangles, the ratios of their corresponding sides are always equal.

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Trigonometric Ratios of Complementary Angles

In geometry, two angles are called complementary if the sum of their measures is exactly $90^\circ$. In any right-angled triangle, the two acute angles are always complementary because the sum of all angles is $180^\circ$, and one angle is $90^\circ$, leaving $180^\circ - 90^\circ = 90^\circ$ for the sum of the other two.

There is a specific relationship between the trigonometric ratios of an acute angle and the trigonometric ratios of its complementary angle. Understanding these relationships simplifies many trigonometric problems.

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Relationship Between Trigonometric Ratios of Complementary Angles

Consider a right-angled triangle $\triangle \text{ABC}$, right-angled at B ($\angle \text{B} = 90^\circ$). The two acute angles are $\angle \text{A}$ and $\angle \text{C}$. Since $\angle \text{A} + \angle \text{B} + \angle \text{C} = 180^\circ$ and $\angle \text{B} = 90^\circ$, we have $\angle \text{A} + \angle \text{C} = 90^\circ$. Thus, $\angle \text{A}$ and $\angle \text{C}$ are complementary angles.

Let $\theta$ be the measure of angle C ($\angle \text{C} = \theta$). Then the measure of angle A is $\angle \text{A} = 90^\circ - \theta$.

Let the sides relative to angle C ($\theta$) be:

• Opposite side to $\theta$ (P) = AB
• Adjacent side to $\theta$ (B) = BC
• Hypotenuse (H) = AC

The trigonometric ratios for angle $\theta$ are:

Now consider the angle $\angle \text{A} = 90^\circ - \theta$. Relative to angle A, the sides are:

• Side Opposite to $(90^\circ - \theta)$: BC
• Side Adjacent to $(90^\circ - \theta)$: AB
• Hypotenuse: AC

The trigonometric ratios for angle $90^\circ - \theta$ are:

Comparing the ratios for $\theta$ and $90^\circ - \theta$, we can see the following relationships:

• $\sin (90^\circ - \theta) = \frac{\text{BC}}{\text{AC}}$ and $\cos \theta = \frac{\text{BC}}{\text{AC}}$. So, $\sin (90^\circ - \theta) = \cos \theta$.
• $\cos (90^\circ - \theta) = \frac{\text{AB}}{\text{AC}}$ and $\sin \theta = \frac{\text{AB}}{\text{AC}}$. So, $\cos (90^\circ - \theta) = \sin \theta$.
• $\tan (90^\circ - \theta) = \frac{\text{BC}}{\text{AB}}$ and $\cot \theta = \frac{\text{BC}}{\text{AB}}$. So, $\tan (90^\circ - \theta) = \cot \theta$.
• $\cot (90^\circ - \theta) = \frac{\text{AB}}{\text{BC}}$ and $\tan \theta = \frac{\text{AB}}{\text{BC}}$. So, $\cot (90^\circ - \theta) = \tan \theta$.
• $\text{sec } (90^\circ - \theta) = \frac{\text{AC}}{\text{AB}}$ and $\text{cosec } \theta = \frac{\text{AC}}{\text{AB}}$. So, $\text{sec } (90^\circ - \theta) = \text{cosec } \theta$.
• $\text{cosec } (90^\circ - \theta) = \frac{\text{AC}}{\text{BC}}$ and $\text{sec } \theta = \frac{\text{AC}}{\text{BC}}$. So, $\text{cosec } (90^\circ - \theta) = \text{sec } \theta$.

Formulas for Trigonometric Ratios of Complementary Angles:

For any acute angle $\theta$ (where $0^\circ < \theta < 90^\circ$):

• $\sin (90^\circ - \theta) = \cos \theta$
• $\cos (90^\circ - \theta) = \sin \theta$
• $\tan (90^\circ - \theta) = \cot \theta$
• $\cot (90^\circ - \theta) = \tan \theta$
• $\text{sec } (90^\circ - \theta) = \text{cosec } \theta$
• $\text{cosec } (90^\circ - \theta) = \text{sec } \theta$

These identities are valid for acute angles. Some definitions might require extension for boundary values $\theta = 0^\circ$ or $\theta = 90^\circ$ where certain ratios are undefined.

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Answer:

Trigonometric Ratios of Some Specific Angles

While trigonometric ratios are defined for any acute angle in a right-angled triangle, there are certain angles for which we can find the exact numerical values of these ratios using basic geometry. The specific angles commonly studied in Class 10 are $0^\circ, 30^\circ, 45^\circ, 60^\circ$, and $90^\circ$.

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Trigonometric Ratios for $45^\circ$

To find the trigonometric ratios for $45^\circ$, we consider a right-angled triangle where one acute angle is $45^\circ$. Since the sum of acute angles in a right triangle is $90^\circ$, the other acute angle must also be $90^\circ - 45^\circ = 45^\circ$. A right-angled triangle with two equal angles is an isosceles right-angled triangle.

Consider an isosceles right triangle $\triangle \text{ABC}$, right-angled at B, such that $\angle \text{A} = \angle \text{C} = 45^\circ$. Since the sides opposite to equal angles are equal, we have AB = BC. Let AB = BC = $a$, where $a$ is a positive real number.

By the Pythagoras theorem, the length of the hypotenuse AC is:

Now, calculate the trigonometric ratios for angle $45^\circ$ using $\angle \text{C} = 45^\circ$ (or $\angle \text{A} = 45^\circ$). Relative to $\angle \text{C}$:

• Opposite side (P) = AB = a
• Adjacent side (B) = BC = a
• Hypotenuse (H) = AC = $\sqrt{2}a$

The reciprocal ratios are:

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Trigonometric Ratios for $30^\circ$ and $60^\circ$

To find the trigonometric ratios for $30^\circ$ and $60^\circ$, we consider an equilateral triangle. All angles in an equilateral triangle are $60^\circ$. Let $\triangle \text{ABC}$ be an equilateral triangle with side length $2a$. We choose a side length $2a$ for convenience to avoid fractions later.

Draw an altitude AD from vertex A to side BC. In an equilateral triangle, the altitude is also a median and an angle bisector. So, AD is perpendicular to BC, D is the midpoint of BC, and AD bisects $\angle \text{BAC}$.

In $\triangle \text{ABC}$:

• AB = BC = AC = $2a$.
• $\angle \text{A} = \angle \text{B} = \angle \text{C} = 60^\circ$.

In $\triangle \text{ADB}$:

• $\angle \text{ADB} = 90^\circ$ (Altitude is perpendicular).
• $\angle \text{B} = 60^\circ$.
• $\angle \text{BAD} = \frac{1}{2} \angle \text{BAC} = \frac{1}{2}(60^\circ) = 30^\circ$.

So, $\triangle \text{ADB}$ is a right triangle with angles $30^\circ, 60^\circ, 90^\circ$.

We know AB = $2a$. Since D is the midpoint of BC, BD = $\frac{1}{2}$ BC $= \frac{1}{2}(2a) = a$.

By the Pythagoras theorem in $\triangle \text{ADB}$, $AD^2 + BD^2 = AB^2$.

Now we have the side lengths of $\triangle \text{ADB}$: AB = $2a$, BD = $a$, AD = $\sqrt{3}a$. We can find the trigonometric ratios for $30^\circ$ and $60^\circ$.

For angle $30^\circ$ ($\angle BAD$):

Relative to angle BAD = $30^\circ$:

• Opposite side (P) = BD = a
• Adjacent side (B) = AD = $\sqrt{3}a$
• Hypotenuse (H) = AB = $2a$

Reciprocal ratios:

For angle $60^\circ$ ($\angle B$):

Relative to angle B = $60^\circ$:

• Opposite side (P) = AD = $\sqrt{3}a$
• Adjacent side (B) = BD = a
• Hypotenuse (H) = AB = $2a$

Reciprocal ratios:

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Trigonometric Ratios for $0^\circ$ and $90^\circ$

The angles $0^\circ$ and $90^\circ$ are special cases. They cannot strictly be angles in a triangle (which must have positive angles summing to $180^\circ$). Their trigonometric ratios are defined by considering them as limiting cases of acute angles in a right triangle or by using the concept of the unit circle (which is usually covered in higher classes). For Class 10, the values are usually stated.

Imagine a point P(x, y) on a circle of radius r centered at the origin. The angle $\theta$ is measured counterclockwise from the positive x-axis to the line segment OP. The coordinates of P relate to the trigonometric ratios as $\cos \theta = x/r$ and $\sin \theta = y/r$.

• For $0^\circ$: Consider the point (r, 0) on the positive x-axis. Here $\theta = 0^\circ$, $x=r, y=0$.
  • $\sin 0^\circ = \frac{y}{r} = \frac{0}{r} = 0$
  • $\cos 0^\circ = \frac{x}{r} = \frac{r}{r} = 1$
  • $\tan 0^\circ = \frac{y}{x} = \frac{0}{r} = 0$
  • $\cot 0^\circ = \frac{x}{y} = \frac{r}{0}$, which is Undefined as division by zero is not possible.
  • $\sec 0^\circ = \frac{r}{x} = \frac{r}{r} = 1$
  • $\text{cosec } 0^\circ = \frac{r}{y} = \frac{r}{0}$, which is Undefined.
• For $90^\circ$: Consider the point (0, r) on the positive y-axis. Here $\theta = 90^\circ$, $x=0, y=r$.
  • $\sin 90^\circ = \frac{y}{r} = \frac{r}{r} = 1$
  • $\cos 90^\circ = \frac{x}{r} = \frac{0}{r} = 0$
  • $\tan 90^\circ = \frac{y}{x} = \frac{r}{0}$, which is Undefined.
  • $\cot 90^\circ = \frac{x}{y} = \frac{0}{r} = 0$
  • $\sec 90^\circ = \frac{r}{x} = \frac{r}{0}$, which is Undefined.
  • $\text{cosec } 90^\circ = \frac{r}{y} = \frac{r}{r} = 1$

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Table of Trigonometric Ratios for Specific Angles

Here is a summary table of the trigonometric ratios for the specific angles $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$. It's highly recommended to memorise these values.

Angle ($\theta$)	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin \theta$	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	1
$\cos \theta$	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0
$\tan \theta$	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	Undefined
$\text{cosec } \theta$	Undefined	2	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	1
$\text{sec } \theta$	1	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	2	Undefined
$\cot \theta$	Undefined	$\sqrt{3}$	1	$\frac{1}{\sqrt{3}}$	0

Notice the symmetry: The $\sin$ values go from 0 to 1, while the $\cos$ values go from 1 to 0. Also, $\sin \theta = \cos (90^\circ - \theta)$ and $\cos \theta = \sin (90^\circ - \theta)$, consistent with the complementary angle identities.

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Answer:

Trigonometric Identities

An identity is an equation that is true for all valid values of the variables involved. In trigonometry, a trigonometric identity is an equation involving trigonometric ratios of an angle that holds true for all values of the angle for which the trigonometric ratios appearing in the equation are defined.

Trigonometric identities are fundamental in simplifying trigonometric expressions, solving trigonometric equations, and proving other relationships between trigonometric ratios.

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Basic Trigonometric Identities (Pythagorean Identities)

These are the most important trigonometric identities and are derived directly from the Pythagoras theorem applied to a right-angled triangle.

Consider a right-angled triangle $\triangle \text{ABC}$, right-angled at B ($\angle \text{B} = 90^\circ$). Let $\theta$ be the measure of the acute angle at C ($\angle \text{C} = \theta$). The sides are Opposite (AB), Adjacent (BC), and Hypotenuse (AC).

By the Pythagoras theorem in $\triangle \text{ABC}$, we have:

Identity 1: $\sin^2 \theta + \cos^2 \theta = 1$

To derive this identity, divide each term of the Pythagorean equation (1) by $\text{AC}^2$ (the square of the hypotenuse). Since AC is the hypotenuse of a right triangle, AC $\neq 0$, so we can divide by $\text{AC}^2$.

Rewrite the terms as squares of ratios:

From the definitions of trigonometric ratios for angle $\theta = \angle \text{C}$, we know that $\sin \theta = \frac{\text{AB}}{\text{AC}}$ and $\cos \theta = \frac{\text{BC}}{\text{AC}}$. Substitute these ratios into the equation:

This is conventionally written as:

This identity is true for all angles $\theta$ such that $0^\circ \leq \theta \leq 90^\circ$.

Identity 2: $1 + \tan^2 \theta = \sec^2 \theta$

To derive this identity, divide each term of the Pythagorean equation (1) by $\text{BC}^2$ (the square of the adjacent side relative to $\theta$). We assume $\text{BC} \neq 0$, which is true for $0^\circ \leq \theta < 90^\circ$.

Rewrite the terms as squares of ratios:

From the definitions of trigonometric ratios, we know that $\tan \theta = \frac{\text{AB}}{\text{BC}}$ and $\text{sec } \theta = \frac{\text{AC}}{\text{BC}}$. Substitute these ratios:

This is conventionally written as:

This identity is true for all angles $\theta$ such that $0^\circ \leq \theta < 90^\circ$ (since $\tan 90^\circ$ and $\text{sec } 90^\circ$ are undefined).

Identity 3: $1 + \cot^2 \theta = \text{cosec}^2 \theta$

To derive this identity, divide each term of the Pythagorean equation (1) by $\text{AB}^2$ (the square of the opposite side relative to $\theta$). We assume $\text{AB} \neq 0$, which is true for $0^\circ < \theta \leq 90^\circ$.

Rewrite the terms as squares of ratios:

From the definitions of trigonometric ratios, we know that $\cot \theta = \frac{\text{BC}}{\text{AB}}$ and $\text{cosec } \theta = \frac{\text{AC}}{\text{AB}}$. Substitute these ratios:

This is conventionally written as:

This identity is true for all angles $\theta$ such that $0^\circ < \theta \leq 90^\circ$ (since $\cot 0^\circ$ and $\text{cosec } 0^\circ$ are undefined).

The three Pythagorean Identities are:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $1 + \tan^2 \theta = \sec^2 \theta$
• $1 + \cot^2 \theta = \text{cosec}^2 \theta$

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Other Useful Identities

Besides the Pythagorean identities, there are other basic identities derived directly from the definitions of the trigonometric ratios (Section I2):

• Reciprocal Identities:
  • $\sin \theta = \frac{1}{\text{cosec } \theta}$ (and $\text{cosec } \theta = \frac{1}{\sin \theta}$)
  • $\cos \theta = \frac{1}{\text{sec } \theta}$ (and $\text{sec } \theta = \frac{1}{\cos \theta}$)
  • $\tan \theta = \frac{1}{\cot \theta}$ (and $\cot \theta = \frac{1}{\tan \theta}$)
• Quotient Identities:
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$

The Pythagorean identities can also be rearranged to derive alternative forms, such as:

• From $\sin^2 \theta + \cos^2 \theta = 1$: $\sin^2 \theta = 1 - \cos^2 \theta$ and $\cos^2 \theta = 1 - \sin^2 \theta$.
• From $1 + \tan^2 \theta = \sec^2 \theta$: $\sec^2 \theta - \tan^2 \theta = 1$.
• From $1 + \cot^2 \theta = \text{cosec}^2 \theta$: $\text{cosec}^2 \theta - \cot^2 \theta = 1$.

These identities are invaluable tools for simplifying trigonometric expressions and proving more complex trigonometric relationships.

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