Introduction to Trigonometry

REVISION SUMMARY: Introduction to Trigonometry (NCERT Class 10)

1. Chapter at a Glance

• Trigonometry studies relationships between sides and acute angles of a right triangle.
• Six trigonometric ratios (sin A, cos A, tan A, cosec A, sec A, cot A) are defined using opposite side, adjacent side and hypotenuse with respect to an acute angle.
• Values of trigonometric ratios of an acute angle remain the same regardless of the lengths of the sides of the triangle (established via AA similarity).
• Standard values of trigonometric ratios are defined for 0°, 30°, 45°, 60° and 90°.
• Three fundamental trigonometric identities hold: sin²A + cos²A = 1, 1 + tan²A = sec²A and 1 + cot²A = cosec²A (valid in the domains where the expressions are defined).
• If one trigonometric ratio of an acute angle is known, the other five ratios can be found using Pythagoras theorem and definitions.
• Trigonometric ratios help determine unknown sides/angles in right triangles when at least one side and one acute angle (or another side) are given.

2. Definitions, Theorems and Results

Definitions (exactly as framed): - In right △ABC right-angled at B, for acute ∠A:
sin A = BC/AC (opposite/hypotenuse),
cos A = AB/AC (adjacent/hypotenuse),
tan A = BC/AB (opposite/adjacent),
cosec A = AC/BC = 1/sin A,
sec A = AC/AB = 1/cos A,
cot A = AB/BC = 1/tan A. - sin 0° = 0, cos 0° = 1; tan 0° = 0, cot 0° undefined, sec 0° = 1, cosec 0° undefined.
- sin 90° = 1, cos 90° = 0; tan 90° undefined, cot 90° = 0, sec 90° undefined, cosec 90° = 1.

Standard results (no proof required beyond NCERT derivation): - tan A = sin A / cos A and cot A = cos A / sin A. - Trigonometric ratios of ∠A are identical in any right triangle containing the same acute angle (via AA similarity of △PAM ~ △CAB, etc.). - sin A and cos A are always ≤ 1; sec A and cosec A are always ≥ 1 (hypotenuse is longest side).

Trigonometric identities (proved in text; students must reproduce proofs in exams): - cos²A + sin²A = 1 (0° ≤ A ≤ 90°). - 1 + tan²A = sec²A (0° ≤ A < 90°). - 1 + cot²A = cosec²A (0° < A ≤ 90°).

3. Formula Sheet

Standard values table (NCERT Table 8.1)
∠A | 0° | 30° | 45° | 60° | 90°
sin A | 0 | 1/2 | 1/√2 | √3/2 | 1
cos A | 1 | √3/2 | 1/√2 | 1/2 | 0
tan A | 0 | 1/√3 | 1 | √3 | not defined
cosec A | not defined | 2 | √2 | 2/√3 | 1
sec A | 1 | 2/√3 | √2 | 2 | not defined
cot A | not defined | √3 | 1 | 1/√3 | 0

4. Solved-Example Patterns

Type 1: Find remaining ratios when one ratio is given
Draw right triangle, assign sides in the given ratio (using k), apply Pythagoras to find third side, then write all six ratios.

Type 2: Prove two acute angles equal if their trigonometric ratios are equal
Assume sin B = sin Q (or cos A = cos B), form two right triangles, show corresponding sides proportional, apply AA similarity (Theorem 6.4) to conclude ∠B = ∠Q.

Type 3: Evaluate expressions using standard values or identities
Substitute exact values from the table or apply sin²A + cos²A = 1, 1 + tan²A = sec²A, etc., and simplify.

Type 4: Verify or prove trigonometric identities
Start from LHS, convert all terms to sin/cos (or tan/sec), apply Pythagoras-derived identities, reduce to RHS. Keep domain restrictions in mind.

Type 5: Find sides/angles in a right triangle using ratios
Select the ratio linking known and unknown quantities (e.g., tan C for adjacent side), solve; use Pythagoras as alternative check.

Type 6: Solve equations involving trigonometric ratios of (A ± B)
Convert to known standard angles (30°, 45°, 60°), solve simultaneous linear equations for A and B.

5. Common Mistakes and Exam Pitfalls

• Writing sin A = opposite/adjacent or confusing opposite and adjacent sides when angle changes.
• Forgetting that tan 90°, sec 90°, cosec 0° and cot 0° are undefined; attempting to evaluate them.
• Omitting the “k” multiplier when sides are in ratio and then forgetting to cancel k later.
• Using sin²A + cos²A = 1 outside 0°–90° or ignoring that 1 + tan²A = sec²A fails at 90°.
• Sign errors when taking square roots (AB must be positive).
• Assuming sin(A + B) = sin A + sin B or similar false expansions.
• Missing that cos A or sin A ≤ 1 always; claiming sec A < 1 for some acute A.
• Not stating the domain when writing identities or claiming an identity holds for A = 0°/90° without checking.