Chapter 8 — Introduction To Trigonometry

What is trigonometry? Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially right-angled triangles. It introduces trigonometric ratios (sine, cosine, tangent and their reciprocals) that connect an acute angle of a right triangle to the lengths of its sides.

Why learn it? (Motivation)

• Many practical problems ask for heights, distances or slopes which are difficult to measure directly. Trigonometry provides simple ratio-based methods to find these using an angle and one known length.
• It is fundamental in fields such as engineering, architecture, surveying, navigation, physics and computer graphics — anywhere angles and distances interact.
• Extending trigonometric ratios to all angles and using their graphs helps model periodic phenomena (waves, oscillations) and solve equations involving angles.

Basic idea (right-angled triangle)

Given a right triangle with acute angle θ, label the side opposite θ as the opposite side, the side adjacent to θ (but not the hypotenuse) as the adjacent side, and the longest side as the hypotenuse. Trigonometric ratios are defined as:

• sin θ = (length of opposite) / (length of hypotenuse)
• cos θ = (length of adjacent) / (length of hypotenuse)
• tan θ = (length of opposite) / (length of adjacent)

How this helps in real life: If you can measure an angle of elevation (or depression) and one length (for example, the distance from the base), you can compute unknown heights using these ratios. This avoids climbing, digging or direct measuring.

Extension & graphs: The same ratios can be extended to any angle using the unit circle (point on circle of radius 1 has coordinates (cos θ, sin θ)). Graphs of sin, cos and tan versus angle show periodic behaviour and help visualize how these ratios change with angle.

Remember: The initial introduction focuses on acute angles in right triangles and practical applications (heights and distances). Later topics extend definitions to all angles, add identities and study graphs.

Right triangle: A triangle with one angle equal to 90°. If triangle ABC is right-angled at B (∠B = 90°), the side opposite the right angle (AC) is called the hypotenuse. The other two sides (AB and BC) are the legs.

Angle notation and labeling: In trigonometry we commonly denote an acute angle by a Greek letter such as θ (theta). For triangle ABC with ∠B = 90° and ∠A = θ, the sides are named relative to θ as:

• Opposite side: the side opposite θ (here BC).
• Adjacent side: the side next to θ but not the hypotenuse (here AB).
• Hypotenuse: the side opposite the right angle (here AC).

Basic idea: Trigonometric ratios relate the angles of a right triangle to ratios of its sides. They depend only on the angle, not on the size of the triangle (similar triangles principle).

Complementary angles: In a right triangle the two acute angles are complementary: ∠A + ∠C = 90°. This gives relations such as sin(90° − θ) = cos θ and tan(90° − θ) = cot θ.

Degrees and radians: In Class 10 we usually use degrees (°). If needed, convert degrees to radians by multiplying degrees by π/180.

Use in problems: To find an unknown side, choose the appropriate ratio (sin, cos or tan) involving the known angle and known side, set up the equation, and solve. To find an angle from two known sides, take the inverse trigonometric function (e.g., θ = sin⁻¹(opposite/hypotenuse)).

Definition (right‑triangle approach): In a right‑angled triangle, for an acute angle θ we call the side opposite to θ the opposite, the side next to θ (but not the hypotenuse) the adjacent, and the longest side the hypotenuse. The six trigonometric ratios of θ are defined as ratios of these sides:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent
• cosec θ = hypotenuse / opposite
• sec θ = hypotenuse / adjacent
• cot θ = adjacent / opposite

Memorise with SOH‑CAH‑TOA for sin, cos, tan. The last three are reciprocals of the first three: cosec = 1/sin, sec = 1/cos, cot = 1/tan (where defined).

Important identities: From Pythagoras (if hypotenuse = h, opposite = o, adjacent = a) we get o² + a² = h². Dividing by h² yields the fundamental identity: sin²θ + cos²θ = 1. Using sin and cos, tan and sec satisfy 1 + tan²θ = sec²θ. Also tan θ = sin θ / cos θ.

Complementary angles: In a right triangle the two acute angles sum to 90°. Hence sin θ = cos(90° − θ) and cos θ = sin(90° − θ); similarly tan θ = cot(90° − θ).

Range and domain (in right‑triangle context): For acute angles (0° < θ < 90°) the ratios are positive. To work with angles outside this range we extend the definitions using the unit circle, which also introduces periodicity and sign changes for sine, cosine and tangent.

How to use: Given any two sides of a right triangle, choose θ, identify opposite/adjacent/hypotenuse, and apply the appropriate ratio to find the third side or the angle (use inverse trig functions for angles).

Idea in one line: In right triangles, the ratios of corresponding sides (sine, cosine, tangent, etc.) depend only on the acute angle, not on the size of the triangle. This follows from similarity of right triangles.

Reason / Proof (using similarity):

• Consider two right triangles △ABC and △PQR with right angles at B and Q respectively, and suppose ∠A = ∠P = θ (same acute angle).
• Because both triangles have the same angles (θ, 90° and the remaining angle), they are similar by AA (angle–angle) similarity.
• From similarity, corresponding sides are proportional. If in △ABC the side opposite θ is a, adjacent is b and hypotenuse is c, and in △PQR the corresponding sides are ka, kb, kc (scaled by the same factor k), then

a/c = (ka)/(kc),   a/b = (ka)/(kb),   b/c = (kb)/(kc)

• Thus the ratios a/c, a/b, b/c are unchanged by scaling. Therefore, for a given acute angle θ the ratios opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent are fixed numbers that depend only on θ.

Consequence — Definitions of primary trigonometric ratios (0° < θ < 90°):

• sin θ = (side opposite θ) / (hypotenuse)
• cos θ = (side adjacent to θ) / (hypotenuse)
• tan θ = (side opposite θ) / (side adjacent to θ) = sin θ / cos θ

Remarks: Because these ratios depend only on θ, any two right triangles sharing the same acute angle θ give the same values of sin θ, cos θ and tan θ, even if their absolute sizes differ. This is the foundational idea behind trigonometric tables and calculators.

Where this is valid: For right triangles and for acute angles (0° < θ < 90°). By extending definitions (unit circle) these ratios are defined for other angles too.

Overview
In right-angled triangles and on the unit circle, trigonometric functions are related by simple ratio and reciprocal rules. Quotient relations express one trig function as a ratio of two others; reciprocal relations express a function as the multiplicative inverse of another.

Definitions (from sides of a right triangle)
For an angle θ with opposite side = Opp, adjacent side = Adj and hypotenuse = Hyp:

• sin θ = Opp / Hyp
• cos θ = Adj / Hyp
• tan θ = Opp / Adj
• cosec θ = Hyp / Opp
• sec θ = Hyp / Adj
• cot θ = Adj / Opp

Quotient relations (derived directly)
Using the basic ratios,

• tan θ = (Opp/Hyp) / (Adj/Hyp) = sin θ / cos θ
• cot θ = (Adj/Hyp) / (Opp/Hyp) = cos θ / sin θ

Reciprocal relations
From definitions:

• cosec θ = 1 / sin θ
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ

Important derived identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ (divide sin²θ + cos²θ = 1 by cos²θ)
• 1 + cot²θ = cosec²θ (divide by sin²θ)

Domain notes / where undefined
Because of division by zero: sin θ = 0 ⇒ cosec θ undefined; cos θ = 0 ⇒ sec θ undefined; cos θ = 0 ⇒ tan θ undefined; sin θ = 0 ⇒ cot θ undefined. On the unit-circle form these correspond to specific angles (e.g. cos 90° = 0 ⇒ sec 90° undefined).

Common exact values (useful for Class 10)

How to use these relations
- To simplify expressions: replace tan θ by sin θ / cos θ or replace sec θ by 1 / cos θ.
- To solve equations: convert everything to sin and cos, use sin²θ + cos²θ = 1 and algebraic methods.
- To understand behavior: reciprocal functions (sec, cosec, cot) have vertical asymptotes where their base function (cos, sin, tan) is zero.

What they are: Pythagorean identities are fundamental trigonometric identities that relate squares of sine, cosine, tangent and their reciprocals. They follow from the Pythagorean theorem applied to a right triangle and the unit circle.

Derivation from a right triangle / unit circle: Consider a right triangle with hypotenuse 1 (unit circle). If the angle at the origin is θ, the horizontal coordinate is cosθ and the vertical coordinate is sinθ. By the Pythagorean theorem: cos²θ + sin²θ = 1.

Deriving the other identities: Divide cos²θ + sin²θ = 1 by cos²θ (where cosθ ≠ 0) to get 1 + tan²θ = sec²θ. Divide by sin²θ (where sinθ ≠ 0) to get cot²θ + 1 = csc²θ. These are the three standard Pythagorean identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Uses and remarks: These identities let you replace squares of one function by expressions in another (e.g., sin²θ = 1 − cos²θ), simplify trigonometric expressions and solve equations. Note domain restrictions: tan and sec undefined where cosθ = 0; cot and csc undefined where sinθ = 0.

Definition: Two angles are complementary if their sum is 90° (or π/2 radians). In a right triangle the two acute angles are complementary. The trigonometric ratios of complementary angles are related by simple identities: each sine equals the cosine of its complement, each tangent equals the cotangent of its complement, and so on.

Core identities (degree & radian forms):

• sin(90° − θ) = cos θ    (or sin(π/2 − x) = cos x)
• cos(90° − θ) = sin θ    (or cos(π/2 − x) = sin x)
• tan(90° − θ) = cot θ    (or tan(π/2 − x) = cot x)
• cot(90° − θ) = tan θ    (or cot(π/2 − x) = tan x)
• sec(90° − θ) = cosec θ    (or sec(π/2 − x) = csc x)
• cosec(90° − θ) = sec θ    (or csc(π/2 − x) = sec x)

Proof (using a right triangle): Consider right triangle ABC with ∠C = 90°, ∠A = θ and ∠B = 90° − θ. Let the sides be: opposite A = a, opposite B = b, hypotenuse = h. Then by definition sin A = a/h and cos A = b/h. But sin B = (opposite B)/h = b/h = cos A. Thus sin(90° − θ) = cos θ. Similar reasoning gives the other identities.

Notes: These identities hold for angles measured in degrees or radians (replace 90° with π/2). They are useful to convert problems involving one trigonometric ratio into another and simplify calculations involving complementary angles.

What are standard angles? In Class 10 trigonometry, standard angles usually mean the commonly used angles 0°, 30°, 45°, 60° and 90°. For these angles we have exact, simple trigonometric values that are learned and used frequently. These values arise from two special right triangles (30°–60°–90° and 45°–45°–90°) and the unit circle.

Derivation (special triangles):

• 45°–45°–90° triangle: take legs = 1, hypotenuse = √2. So sin45° = cos45° = 1/√2 = √2/2.
• 30°–60°–90° triangle: take hypotenuse = 2, shorter side = 1 (opposite 30°), longer side = √3 (opposite 60°). So sin30° = 1/2, cos30° = √3/2, sin60° = √3/2, cos60° = 1/2, tan30° = 1/√3 = √3/3, tan60° = √3.

Unit circle view: On the unit circle (radius 1) the coordinates of the point at angle θ are (cos θ, sin θ). The standard-angle values are just these coordinates for θ = 0°, 30°, 45°, 60°, 90°. Using symmetry and signs of coordinates you can extend values to 90°–360°.

Signs by quadrant (useful when extending values): Quadrant I (0°–90°): sin, cos, tan all +. Quadrant II (90°–180°): sin +, cos −, tan −. Quadrant III (180°–270°): sin −, cos −, tan +. Quadrant IV (270°–360°): sin −, cos +, tan −. Use the reference angle (acute angle to the x-axis) to get magnitudes and apply sign per quadrant.

Table of standard values:

Practical tips: Memorise the table and derivations (special triangles + unit circle). To find values for other standard angles (like 120°, 135°, 150°) find the corresponding acute reference angle (e.g., 180°−θ) and apply quadrant sign rules.

What it is: Evaluation and simplification of trigonometric expressions means (a) finding numerical values of trigonometric expressions for given angles and (b) reducing complex trig expressions into simpler or standard forms using identities and algebraic techniques.

Key ideas:

• Use standard values of trig ratios for special angles 0°, 30°, 45°, 60°, 90° and their multiples as building blocks.
• Convert all functions to sines and cosines when simplifying (for example tan = sin/cos, cot = cos/sin) so common factors and identities can be applied.
• Apply fundamental identities (reciprocal, Pythagorean, quotient, co-function, even–odd) to rewrite and reduce expressions.
• Use algebraic manipulation: common denominator, factorization, multiplying by conjugate (rationalization), grouping, and cancellation, being careful about domain restrictions (no division by zero).

Typical steps for simplification or verification:

1. Decide whether to evaluate numerically or to simplify symbolically.
2. If numeric evaluation, reduce angle to a known reference angle and use sign rules from quadrants; substitute exact values where possible.
3. If symbolic simplification, rewrite in terms of sin and cos, apply identities (eg sin²θ + cos²θ = 1), perform algebraic simplification, and simplify the result to a standard form.
4. Always check domain restrictions (angles where denominators vanish) and, when verifying identities, show both sides are equal for all allowable angles.

Common pitfalls: forgetting sign changes in different quadrants, dividing by zero (cos θ = 0 makes tan undefined), and assuming identities hold without checking domain.

Degrees vs radians: In Class 10 you usually work in degrees (0°–360°). In later work radians are used; the algebraic simplification techniques are the same.

Connection to geometry and unit circle: Many simplifications follow from the unit circle definition: a point on unit circle at angle θ has coordinates (cos θ, sin θ). The Pythagorean identity sin²θ + cos²θ = 1 is the circle equation.

What it means: Solving right triangle problems means finding unknown sides or angles of a triangle that has one 90° angle using trigonometric ratios and the Pythagorean theorem.

Key ideas:

• Label the triangle: hypotenuse (the side opposite 90°), opposite (side opposite the angle of interest), adjacent (other non‑hypotenuse side).
• Use trigonometric ratios for an acute angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
• Use the Pythagorean theorem when two sides are known: hypotenuse^2 = opposite^2 + adjacent^2.
• If you need the angle, use inverse trigonometric functions: θ = sin^(-1)(...), cos^(-1)(...), or tan^(-1)(...).
• Use special triangles to shortcut calculations: 45°-45°-90° (legs equal, hypotenuse = leg·√2) and 30°-60°-90° (sides 1 : √3 : 2).

Step-by-step method:

1. Draw the figure and mark the right angle and the known values.
2. Identify the angle of interest θ and label opposite, adjacent, hypotenuse.
3. Choose the appropriate relation (sin, cos, tan or Pythagoras).
4. Set up the equation and solve for the unknown (algebra or inverse trig as required).
5. Check units and reasonableness (e.g., sin and cos values must lie between 0 and 1).

Common pitfalls: mixing up opposite/adjacent, using hypotenuse instead of adjacent in tan/cos, forgetting to use inverse trig when solving for angles, and not checking calculator mode (degrees vs radians).

What this topic means
"Proofs and Reasoning" in Introduction to Trigonometry explains why trigonometric ratios (sin, cos, tan, etc.) are well defined for an acute angle and how basic trig identities follow from geometry (right triangles and the Pythagorean theorem). It teaches how to set up logical steps to derive identities and to solve height-and-distance problems.

1. Why trig ratios are well defined (proof using similar triangles)
Consider two right triangles that have one acute angle equal to θ. By AA (angle–angle) similarity the triangles are similar, so corresponding sides are proportional. If in one triangle the sides adjacent, opposite and hypotenuse are a, b, c and in the other are ka, kb, kc (k>0), then opposite/hypotenuse = a/c = (ka)/(kc). Hence sinθ = opposite/hypotenuse is independent of triangle size and depends only on θ. The same argument holds for cosθ and tanθ. This justifies the definitions:

• sinθ = opposite/hypotenuse
• cosθ = adjacent/hypotenuse
• tanθ = opposite/adjacent

2. Complementary angle relations (geometric reasoning)
In a right triangle, the two acute angles add to 90°. If one acute angle is θ, the other is 90° − θ. The side opposite θ is the side adjacent to 90° − θ, and vice versa. Therefore sinθ = cos(90° − θ) and cosθ = sin(90° − θ). Similarly tanθ = cot(90° − θ).

3. Pythagorean identity (proof from Pythagoras)
In a right triangle with sides opposite = p, adjacent = q, hypotenuse = r, Pythagoras gives p^2 + q^2 = r^2. Divide both sides by r^2 to get (p/r)^2 + (q/r)^2 = 1, i.e. sin^2θ + cos^2θ = 1. This is the fundamental identity from which other identities follow (see below).

4. Deriving quotient and reciprocal identities
From definitions: tanθ = (opposite/hypotenuse)/(adjacent/hypotenuse) = sinθ/cosθ. Reciprocals: cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ.

5. Derived identities by algebraic reasoning
Divide sin^2θ + cos^2θ = 1 by cos^2θ to obtain tan^2θ + 1 = sec^2θ. Divide by sin^2θ to obtain 1 + cot^2θ = cosec^2θ. These are obtained by algebraic manipulation based on the basic Pythagorean identity.

6. Reasoning in problem solving (strategy)
When solving heights and distances: (a) draw a clear right triangle, (b) label the angle and known sides, (c) choose the trigonometric ratio matching the angle and known side, (d) solve algebraically, and (e) check units and reasonableness.

Why the proofs matter
They guarantee that trig ratios are consistent (do not depend on triangle size) and provide tools to derive many identities used in algebraic simplification and applied problems.

In Class 10 Trigonometry the Exercises and Examples focus on using right triangles to define trigonometric ratios (sine, cosine, tangent and their reciprocals) for acute angles and applying identities to solve numerical and word problems. Typical tasks include calculating trig ratios from given side lengths, finding missing sides or angles, using Pythagorean identity to obtain other ratios, and solving height & distance problems using angle of elevation/depression.

Core ideas:

• Given a right triangle with an acute angle θ, define: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
• Use reciprocal relations: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
• Use Pythagorean identity: sin^2θ + cos^2θ = 1 to find one ratio from another.
• Apply complementary-angle relations: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, etc.
• Translate real situations (height of a pole, ladder, slope) into right triangles and apply trig ratios to compute unknown lengths or angles.

Strategy for exercises:

• Sketch the figure and mark the right angle, the angle θ, hypotenuse, opposite and adjacent sides.
• Choose the appropriate trig ratio that relates known and unknown quantities.
• If only one trig ratio is given, use reciprocal/quadratic identities to find others (careful about signs; for Class 10 angles are acute so positive).
• Check units and round final numerical answers to suitable accuracy.