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Introduction to Trigonometry

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the fundamental trigonometric ratios—sine, cosine, tangent, cosecant, secant, and cotangent—defined for acute angles in a right triangle. It establishes the relationship between the sides and angles of a right triangle, which is the core idea of trigonometry. The chapter also covers the values of these ratios for specific angles (0°, 30°, 45°, 60°, 90°) and proves basic trigonometric identities (sin²A + cos²A = 1, sec²A = 1 + tan²A, cosec²A = 1 + cot²A). The one big idea a student must walk away with is that trigonometric ratios depend only on the angle, not on the size of the triangle, and can be used to find unknown sides or angles in right triangles.
Key Concepts (12)
Trigonometry
The study of relationships between the sides and angles of a triangle.
e.g. Using trigonometry, a student can find the height of the Qutub Minar without actually measuring it, by imagining a right triangle formed with the ground and the line of sight.
Trigonometric Ratios
Ratios of the sides of a right triangle with respect to its acute angles.
e.g. In a right triangle ABC, right-angled at B, sin A = side opposite to angle A / hypotenuse = BC/AC.
Sine of angle A (sin A)
In a right triangle, sine of angle A is the ratio of the side opposite to angle A to the hypotenuse.
e.g. If in triangle ABC, BC = 4 and AC = 5, then sin A = 4/5.
Cosine of angle A (cos A)
In a right triangle, cosine of angle A is the ratio of the side adjacent to angle A to the hypotenuse.
e.g. If in triangle ABC, AB = 3 and AC = 5, then cos A = 3/5.
Tangent of angle A (tan A)
In a right triangle, tangent of angle A is the ratio of the side opposite to angle A to the side adjacent to angle A.
e.g. If BC = 4 and AB = 3, then tan A = 4/3.
Cosecant of angle A (cosec A)
The reciprocal of sine A. cosec A = 1/sin A = hypotenuse / side opposite to angle A.
e.g. If sin A = 4/5, then cosec A = 5/4.
Secant of angle A (sec A)
The reciprocal of cosine A. sec A = 1/cos A = hypotenuse / side adjacent to angle A.
e.g. If cos A = 3/5, then sec A = 5/3.
Cotangent of angle A (cot A)
The reciprocal of tangent A. cot A = 1/tan A = side adjacent to angle A / side opposite to angle A.
e.g. If tan A = 4/3, then cot A = 3/4.
Trigonometric Identity
An equation involving trigonometric ratios of an angle that is true for all values of the angle(s) involved.
e.g. sin²A + cos²A = 1 is true for all A such that 0° ≤ A ≤ 90°.
Side opposite to angle A
The side that faces angle A in a right triangle; it does not form the angle.
e.g. In triangle ABC with right angle at B, side BC is opposite to angle A.
Side adjacent to angle A
The side that forms angle A along with the hypotenuse, excluding the hypotenuse itself.
e.g. In triangle ABC with right angle at B, side AB is adjacent to angle A.
Hypotenuse
The longest side of a right triangle, opposite the right angle.
e.g. In triangle ABC, right-angled at B, side AC is the hypotenuse.
Formulas & Equations (16)
sin A = side opposite to angle A / hypotenuse = BC/AC
Reads as: Sine of angle A equals side opposite to angle A divided by hypotenuse
Variables: BC = side opposite to angle A, AC = hypotenuse (both in same unit, e.g., cm)
Use when: When you need to find the sine ratio given sides, or find a side given the sine and another side.
⚠ Students often confuse the opposite side with the adjacent side, especially when the triangle is not drawn in standard orientation.
cos A = side adjacent to angle A / hypotenuse = AB/AC
Reads as: Cosine of angle A equals side adjacent to angle A divided by hypotenuse
Variables: AB = side adjacent to angle A, AC = hypotenuse (same unit)
Use when: When you need to find the cosine ratio given sides, or find a side given the cosine and another side.
⚠ Students sometimes write cos A = opposite/hypotenuse, confusing it with sine.
tan A = side opposite to angle A / side adjacent to angle A = BC/AB
Reads as: Tangent of angle A equals side opposite to angle A divided by side adjacent to angle A
Variables: BC = side opposite to angle A, AB = side adjacent to angle A (same unit)
Use when: When you need to find the tangent ratio given sides, or find a side given the tangent and another side.
⚠ Students often forget that tan A = sin A / cos A, and instead try to compute it directly from sides incorrectly.
cosec A = 1/sin A = hypotenuse / side opposite to angle A = AC/BC
Reads as: Cosecant of angle A equals one divided by sine A, or hypotenuse divided by opposite side
Variables: AC = hypotenuse, BC = side opposite to angle A (same unit)
Use when: When you need the reciprocal of sine, or when given cosec to find sin.
⚠ Students write cosec A = 1/cos A, confusing it with secant.
sec A = 1/cos A = hypotenuse / side adjacent to angle A = AC/AB
Reads as: Secant of angle A equals one divided by cosine A, or hypotenuse divided by adjacent side
Variables: AC = hypotenuse, AB = side adjacent to angle A (same unit)
Use when: When you need the reciprocal of cosine, or when given sec to find cos.
⚠ Students write sec A = 1/sin A, confusing it with cosecant.
cot A = 1/tan A = side adjacent to angle A / side opposite to angle A = AB/BC
Reads as: Cotangent of angle A equals one divided by tangent A, or adjacent side divided by opposite side
Variables: AB = side adjacent to angle A, BC = side opposite to angle A (same unit)
Use when: When you need the reciprocal of tangent, or when given cot to find tan.
⚠ Students write cot A = adjacent/hypotenuse, which is actually cos A.
tan A = sin A / cos A
Reads as: Tangent of angle A equals sine A divided by cosine A
Variables: sin A and cos A are dimensionless ratios
Use when: When you need to derive tan from sin and cos, or simplify expressions involving both.
⚠ Students forget this identity and try to compute tan separately, wasting time.
cot A = cos A / sin A
Reads as: Cotangent of angle A equals cosine A divided by sine A
Variables: cos A and sin A are dimensionless ratios
Use when: When you need to derive cot from sin and cos, or simplify expressions.
⚠ Students often write cot A = sin A / cos A, which is actually tan A.
sin²A + cos²A = 1
Reads as: Sine squared A plus cosine squared A equals one
Variables: A is an acute angle (0° ≤ A ≤ 90°)
Use when: To find sin A if cos A is known (or vice versa), or to prove other identities.
⚠ Students forget that this is sin²A, not sin(A²), and sometimes write sin A² + cos A² = 1.
1 + tan²A = sec²A
Reads as: One plus tangent squared A equals secant squared A
Variables: A is an acute angle (0° ≤ A < 90°; not defined for A = 90°)
Use when: To find sec A if tan A is known, or to prove identities involving sec and tan.
⚠ Students write 1 + tan²A = cosec²A, confusing sec and cosec.
cot²A + 1 = cosec²A
Reads as: Cotangent squared A plus one equals cosecant squared A
Variables: A is an acute angle (0° < A ≤ 90°; not defined for A = 0°)
Use when: To find cosec A if cot A is known, or to prove identities involving cosec and cot.
⚠ Students write cot²A + 1 = sec²A, confusing cosec and sec.
sin 0° = 0, cos 0° = 1, tan 0° = 0
Reads as: Sine zero degrees equals zero, cosine zero degrees equals one, tangent zero degrees equals zero
Variables: Angle in degrees
Use when: When evaluating trigonometric expressions at 0°.
⚠ Students think sin 0° = 1 or cos 0° = 0.
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
Reads as: Sine thirty degrees equals one-half, cosine thirty degrees equals root three over two, tangent thirty degrees equals one over root three
Variables: Angle in degrees
Use when: When evaluating trigonometric expressions at 30°.
⚠ Students confuse sin 30° and cos 30° values.
sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
Reads as: Sine forty-five degrees equals one over root two, cosine forty-five degrees equals one over root two, tangent forty-five degrees equals one
Variables: Angle in degrees
Use when: When evaluating trigonometric expressions at 45°.
⚠ Students forget that sin 45° = cos 45° = 1/√2, not 1/2.
sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
Reads as: Sine sixty degrees equals root three over two, cosine sixty degrees equals one-half, tangent sixty degrees equals root three
Variables: Angle in degrees
Use when: When evaluating trigonometric expressions at 60°.
⚠ Students confuse sin 60° and sin 30° values.
sin 90° = 1, cos 90° = 0, tan 90° = not defined
Reads as: Sine ninety degrees equals one, cosine ninety degrees equals zero, tangent ninety degrees is not defined
Variables: Angle in degrees
Use when: When evaluating trigonometric expressions at 90°.
⚠ Students think tan 90° = ∞ is acceptable; CBSE expects 'not defined'.
How It Works
Finding trigonometric ratios given one ratio
  1. Step 1: Draw a right triangle and label the sides based on the given ratio. For example, if tan A = 4/3, let BC = 4k and AB = 3k.
  2. Step 2: Use Pythagoras theorem to find the third side. For the example, AC² = (4k)² + (3k)² = 25k², so AC = 5k.
  3. Step 3: Write all six trigonometric ratios using their definitions with the side lengths found.
  4. Step 4: Simplify by canceling k where possible.
💡 Always take the positive square root for side lengths since they are lengths. Also, remember that sin A and cos A are always ≤ 1, but sec A and cosec A are ≥ 1.
Proving trigonometric identities
  1. Step 1: Start with the Left-Hand Side (LHS) of the identity.
  2. Step 2: Convert all trigonometric ratios into sin and cos if needed, using tan = sin/cos, cot = cos/sin, sec = 1/cos, cosec = 1/sin.
  3. Step 3: Simplify the expression by combining fractions, using algebraic identities like (a+b)(a-b) = a² - b².
  4. Step 4: Use the fundamental identities sin²A + cos²A = 1, 1 + tan²A = sec²A, or cot²A + 1 = cosec²A to simplify further.
  5. Step 5: Continue simplifying until the LHS equals the Right-Hand Side (RHS).
💡 Do not cross-multiply or bring terms from RHS to LHS. Work on one side only (usually LHS) until it matches the other side. Also, ensure the angles are acute and expressions are defined.
Finding unknown sides or angles in a right triangle
  1. Step 1: Identify which sides are given and which angle is known or to be found.
  2. Step 2: Choose the appropriate trigonometric ratio that relates the known and unknown sides. For example, if opposite side and hypotenuse are involved, use sin.
  3. Step 3: Set up the equation using the ratio and the known angle value from the table (e.g., sin 30° = 1/2).
  4. Step 4: Solve the equation for the unknown side or angle.
  5. Step 5: If finding an angle, use the inverse relationship (e.g., if sin R = 1/2, then R = 30°).
💡 Always check if the triangle is right-angled. The hypotenuse is always opposite the right angle. For angles, use the standard values from Table 8.1.
Exam Traps — Don't Make These Mistakes
⚠ Confusing opposite and adjacent sides
✗ In triangle ABC, right-angled at B, for angle A, students write sin A = AB/AC (adjacent/hypotenuse) instead of BC/AC.
→ The side opposite to angle A is BC, not AB. AB is adjacent to angle A. The definition of sin A is opposite/hypotenuse, not adjacent/hypotenuse.
✓ sin A = BC/AC (side opposite to angle A divided by hypotenuse).
⚠ Misusing reciprocal ratios
✗ cosec A = 1/cos A, sec A = 1/sin A, cot A = 1/cos A.
→ The reciprocals are: cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A. Students often swap sin and cos.
✓ cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A.
⚠ Forgetting that sin A and cos A are ≤ 1
✗ sin θ = 4/3 for some angle θ.
→ In a right triangle, the hypotenuse is the longest side, so opposite/hypotenuse ≤ 1 and adjacent/hypotenuse ≤ 1. 4/3 > 1, so it is impossible.
✓ sin θ is always ≤ 1. For example, sin θ = 3/4 is possible, but sin θ = 4/3 is not.
⚠ Treating sin A as product of sin and A
✗ sin A means sin × A, and they try to cancel 'sin' in equations.
→ sin A is an abbreviation for 'the sine of angle A'. 'sin' separated from A has no meaning. It is not a product.
✓ sin A is a single symbol representing a ratio. Do not separate 'sin' from 'A'.
⚠ Incorrect values for standard angles
✗ sin 30° = √3/2, cos 60° = √3/2, tan 45° = √3.
→ The standard values are: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1. Students often confuse values across angles.
✓ Memorize Table 8.1: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, etc.
⚠ Not rationalizing denominators
✗ sin 45° = 1/√2, but they leave it as 1/√2 in the final answer when the question expects rationalized form.
→ While 1/√2 is technically correct, CBSE often expects rationalized denominators like √2/2 in final answers, especially in evaluation questions.
✓ Write sin 45° = √2/2, cos 45° = √2/2, etc., by rationalizing.
Compare & Contrast
Definition in a right triangle
sin A
cos A
sin A = opposite/hypotenuse, while cos A = adjacent/hypotenuse. For a given acute angle, sin A and cos A are different unless A = 45°.
Reciprocal relationship
tan A
cot A
tan A = opposite/adjacent, while cot A = adjacent/opposite. They are reciprocals of each other: tan A × cot A = 1.
Reciprocal relationship with cos and sin
sec A
cosec A
sec A = 1/cos A = hypotenuse/adjacent, while cosec A = 1/sin A = hypotenuse/opposite. They are not reciprocals of each other.
Trigonometric identity derivation
sin²A + cos²A = 1
1 + tan²A = sec²A
The first identity is derived by dividing Pythagoras theorem by hypotenuse², while the second is derived by dividing by adjacent side². They are equivalent but expressed in different ratios.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total approximately 11 marks
Most asked: Trigonometric ratios of specific angles (0°, 30°, 45°, 60°, 90°) · Trigonometric identities (sin²A + cos²A = 1, 1 + tan²A = sec²A, cot²A + 1 = cosec²A) · Finding other trigonometric ratios given one ratio · Proving trigonometric identities
Don't skip: Proving trigonometric identities (5-mark question); Finding all trigonometric ratios given one ratio (3-mark question); Evaluating expressions using standard angle values (2-mark question)
One-Line Revision (read aloud before the exam)
  1. Trigonometry is the study of relationships between the sides and angles of a triangle.
  2. In a right triangle, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.
  3. cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A.
  4. tan A = sin A / cos A, and cot A = cos A / sin A.
  5. The values of trigonometric ratios depend only on the angle, not on the size of the triangle.
  6. sin 0° = 0, cos 0° = 1, tan 0° = 0.
  7. sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.
  8. sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1.
  9. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
  10. sin 90° = 1, cos 90° = 0, tan 90° is not defined.
  11. sin²A + cos²A = 1 is true for all A where 0° ≤ A ≤ 90°.
  12. 1 + tan²A = sec²A is true for all A where 0° ≤ A < 90°.
  13. cot²A + 1 = cosec²A is true for all A where 0° < A ≤ 90°.
  14. The value of sin A or cos A is always less than or equal to 1.
  15. The value of sec A or cosec A is always greater than or equal to 1.
  16. sin A is not the product of sin and A; it is an abbreviation for the sine of angle A.
  17. If one trigonometric ratio is known, all others can be found using Pythagoras theorem and identities.

MCQs — 10 Questions with Answers

Q1. A student measures angle A in a right triangle ABC (right angle at B) using two different right triangles: one where AB = 3, BC = 4, and another where AB = 6, BC = 8. She calculates sin A in both triangles and gets different values. Why does this happen?

  • A. The sine ratio depends only on the angle, not the size of the triangle, so the values must be equal ✓
  • B. The sine ratio changes based on the actual lengths of the sides, regardless of the angle
  • C. Similar triangles preserve angle measures but not trigonometric ratios
  • D. The two triangles have different angles A, so different sine values are expected

Answer: A — By the AA similarity criterion and proportionality of similar triangles, sin A = BC/AC is constant for a given angle regardless of triangle size; the student made a calculation error, not a conceptual one. Option B wrongly suggests trigonometric ratios depend on absolute side lengths rather than their proportion.

Q2. In right triangle ABC with right angle at C, a student observes that sin A × cosec A = 1. Her teacher asks her to verify this is true for ALL acute angles. Which reasoning correctly explains why this must always be true?

  • A. Because sin A and cosec A are reciprocals by definition, their product is always 1 ✓
  • B. Because sin A = BC/AB and cosec A = AB/BC, and these ratios multiply to 1
  • C. Because the angle A remains the same in both ratios, so the sides must cancel
  • D. Because sin and cosec are complementary functions, like sine and cosine

Answer: A — Cosec A is defined as 1/sin A, making them reciprocals by definition, so their product is always 1. Option B confuses the hypotenuse with another side in the definition of cosec A.

Q3. A surveyor uses the relationship tan A = sin A / cos A to calculate the height of a building. A student claims this relationship is merely a rearrangement and doesn't reveal anything new about angle A. Is this claim correct?

  • A. Yes, it's just a rearrangement with no additional insight
  • B. No, it shows that tangent can be expressed in terms of sine and cosine, revealing how these three ratios are interconnected ✓
  • C. No, because tangent is always larger than both sine and cosine
  • D. Yes, but only when angle A is greater than 45°

Answer: B — The relationship tan A = sin A / cos A is algebraically derived from definitions but reveals a fundamental interconnection between the three primary ratios, not merely a rearrangement. Option A misses the conceptual significance of how ratios relate.

Q4. Two students draw right triangles with angle A = 60°. One student uses a 3-4-5 triangle to find sin 60°, while another uses a different right triangle. They get different numerical values. What is the most likely explanation?

  • A. One student made an error; for a fixed angle, sin A has a unique value regardless of triangle size
  • B. The 3-4-5 triangle cannot have a 60° angle, so one student's triangle is impossible ✓
  • C. Trigonometric ratios vary based on the specific triangle chosen
  • D. One student calculated sin 60° correctly while the other calculated a different ratio by mistake

Answer: B — A 3-4-5 right triangle has angles approximately 37°, 53°, and 90°, so it cannot contain a 60° angle; this eliminates one calculation as impossible. Option A assumes both calculations are valid, which they cannot be.

Q5. Assertion (A): For any acute angle A in a right triangle, cot A = cos A / sin A. Reason (R): Cotangent is defined as the reciprocal of tangent, and tangent equals sin A / cos A. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both statements are true: cot A = 1/tan A = 1/(sin A/cos A) = cos A/sin A, and R correctly explains why A is true. The logical chain directly connects the definition of cot to the formula.

Q6. Assertion (A): If sin A = 3/5 for some acute angle A, then cos A must equal 4/5. Reason (R): In any right triangle, sin²A + cos²A = 1 (the Pythagorean identity). Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — If sin A = 3/5, then sin²A = 9/25, so cos²A = 1 − 9/25 = 16/25, giving cos A = 4/5 (positive for acute A). The Pythagorean identity (R) directly justifies this conclusion (A).

Q7. A student claims: 'Since sin A and cos A are both ratios of sides in a right triangle, they must always be less than 1.' Is this claim always true, sometimes true, or never true?

  • A. Always true, because both numerator and denominator are sides of the triangle, and the numerator is never longer than the hypotenuse ✓
  • B. Sometimes true, depending on which angle is chosen
  • C. Never true, because larger triangles have larger sine and cosine values
  • D. Always true except when the angle is 90°

Answer: A — For acute angles, sin A = opposite/hypotenuse < 1 and cos A = adjacent/hypotenuse < 1 always hold because the hypotenuse is the longest side in a right triangle. Option C confuses absolute side lengths with ratios.

Q8. When you extend point P on the hypotenuse AC (or beyond) and draw perpendiculars to form new right triangles, the trigonometric ratios of angle A remain unchanged. This is true because:

  • A. The angle A is the same in every triangle formed
  • B. All the new triangles are similar to the original triangle ABC, and similarity preserves the ratios of corresponding sides ✓
  • C. Trigonometric ratios do not depend on the size of the triangle
  • D. The hypotenuse length stays constant in all triangles

Answer: B — The AA similarity criterion guarantees that all triangles formed with the same angle A are similar to ABC, and similarity preserves proportional side ratios. Option A states a true fact but does not explain why ratios remain unchanged.

Q9. A right triangle has sides in the ratio 1 : √3 : 2 (opposite to angles 30°, 60°, and 90° respectively). A student uses this to claim that sin 30° = 1/2. Is this reasoning correct?

  • A. Yes, the ratio of sides directly gives the sine value ✓
  • B. Yes, but only for the 30-60-90 triangle
  • C. No, because the actual side lengths matter, not just the ratio
  • D. Yes, and this same approach works for any acute angle in any right triangle

Answer: A — Sin 30° = opposite/hypotenuse = 1/2 follows directly from the side ratio, and this value holds for ANY 30-60-90 triangle (by similarity). Option C wrongly suggests trigonometric ratios depend on absolute side lengths.

Q10. A student observes that in her right triangle, sin A < cos A. She concludes that angle A must be less than 45°. Is her conclusion logically justified?

  • A. Yes, because sin A = cos A exactly when A = 45°, so smaller angles give smaller sine values
  • B. Yes, because sin A < cos A implies opposite side < adjacent side, which happens only when A < 45° ✓
  • C. No, because the relationship between sine and cosine depends on the triangle's size, not angle A
  • D. No, because sin A and cos A are unrelated ratios that cannot be directly compared

Answer: B — If sin A < cos A, then opposite/hypotenuse < adjacent/hypotenuse, so opposite < adjacent; in a right triangle this occurs precisely when A < 45°. Option A incorrectly assumes sine increases monotonically with angle size.

Flashcards

In right triangle ABC with acute angle A, define sine of angle A.

sin A = (side opposite to angle A) / (hypotenuse) = BC/AC.

What is the relationship between tan A, sin A, and cos A?

tan A = sin A / cos A, which equals (opposite/hypotenuse) ÷ (adjacent/hypotenuse) = opposite/adjacent.

Name the reciprocal relationship for sine, cosine, and tangent.

cosec A = 1/sin A, sec A = 1/cos A, and cot A = 1/tan A.

Why do trigonometric ratios remain constant even if triangle size changes?

Triangles with the same angle are similar; their corresponding sides are proportional, keeping ratios equal.

In right triangle ABC with right angle at B, which side is the hypotenuse?

AC is the hypotenuse because it is the side opposite the right angle.

For acute angle A in right triangle ABC, define cosine of angle A.

cos A = (side adjacent to angle A) / (hypotenuse) = AB/AC.

What does 'opposite side' mean for angle A in a right triangle?

The side opposite to angle A is the side that does not form angle A and faces it.

If sin A = 1/3, what can you say about the sides BC and AC?

The sides BC and AC are in ratio 1:3, meaning if BC = k, then AC = 3k for some positive k.

Which trigonometric ratios are independent, and which are dependent?

sin A, cos A, and tan A are independent; cosec A, sec A, and cot A are their reciprocals and dependent.

When you change from angle A to angle C in the same right triangle, what happens to 'opposite' and 'adjacent'?

The sides that were opposite to A become adjacent to C, and vice versa, because the angle position changes.

Important Board Questions

In right triangle ABC with right angle at B, side AB = 5 cm and side BC = 12 cm. Find sin A and cos A. [2 marks]

Use Pythagoras theorem to find AC (hypotenuse = 13 cm). Then apply definitions: sin A = opposite/hypotenuse = BC/AC = 12/13 and cos A = adjacent/hypotenuse = AB/AC = 5/13.

If sin A = 3/5 in a right triangle, find the values of cos A and tan A without using a calculator. Explain your reasoning. [3 marks]

Set opposite = 3k and hypotenuse = 5k for some positive k. Use Pythagoras to find adjacent = 4k. Then cos A = adjacent/hypotenuse = 4/5 and tan A = opposite/adjacent = 3/4. Explain that k cancels out in ratios.

A student stands 15 metres away from the base of a tower. The angle of elevation to the top of the tower is A. If tan A = 4/3, find the height of the tower. Also explain why the trigonometric ratio remains the same if we move to a different distance and measure a different angle. [5 marks]

Use tan A = height/distance; 4/3 = height/15, so height = 20 metres. For the second part, explain using similar triangles: when angle A is fixed, any right triangle with angle A has proportional sides, so the ratio opposite/adjacent is always 4/3 regardless of triangle size. Similar triangles have equal trigonometric ratios for the same angle.

Assertion-Reason Questions for Introduction to Trigonometry

Q1. Assertion (A):

In a right triangle, sin²A + cos²A = 1 for any acute angle A.

Reason (R):

The sum of squares of sine and cosine of an angle equals one because (BC)² + (AB)² = (AC)² by Pythagoras theorem.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion holds. sin²A + cos²A = (BC/AC)² + (AB/AC)² = (BC² + AB²)/AC² = AC²/AC² = 1 by Pythagoras theorem. This is the fundamental trigonometric identity.

Q2. Assertion (A):

If tan A = 1, then sin A = cos A.

Reason (R):

tan A = sin A/cos A, so when tan A = 1, we have sin A/cos A = 1, which means sin A = cos A.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the assertion. The relationship tan A = sin A/cos A is the fundamental definition. When tan A = 1, dividing both sides by cos A gives sin A = cos A, confirming the assertion.

Q3. Assertion (A):

The value of trigonometric ratios of an angle depends on the size of the right triangle.

Reason (R):

Similar right triangles with the same acute angle have proportional sides, so their trigonometric ratios are equal.

Answer: D — A is false but R is true — The assertion is false. Trigonometric ratios depend only on the angle, not on the size of the triangle. The reason correctly states that similar triangles have equal trigonometric ratios, which contradicts the assertion. Since the reason is true but the assertion is false, the answer is D.

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Case-Based Questions (CBQ) for Introduction to Trigonometry

Case Study:

A student is observing a tall building from ground level. She stands 50 meters away from the base of the building and measures the angle of elevation to the top of the building as 60°. The angle of elevation is the angle formed between the horizontal ground and her line of sight to the top. Using trigonometric ratios, she wants to find the height of the building without actually climbing it. The ground distance, building height, and line of sight form a right triangle where the right angle is at the base of the building.

Q1. In this scenario, what is the angle between the ground and the line of sight to the top?

  • A. 30°
  • B. 60° ✓
  • C. 45°
  • D. 90°

Answer: B

Q2. Which trigonometric ratio should be used to find the height if the distance and angle of elevation are known?

  • A. sin 60°
  • B. cos 60°
  • C. tan 60° ✓
  • D. cot 60°

Answer: C

Q3. If tan 60° = √3, what is the height of the building?

  • A. 50 meters
  • B. 50√3 meters ✓
  • C. 50/√3 meters
  • D. 100 meters

Answer: B

Q4. What would be the angle of elevation if the student moved 50√3 meters away from the building while maintaining the same height?

  • A. 30° ✓
  • B. 45°
  • C. 60°
  • D. 90°

Answer: A

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Previous Year Questions (PYQ) for Introduction to Trigonometry

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 1 mark

The value of sin 90° cos 90° – sin² 60° is:

  • 1/4
  • 3/4
  • –3/4
  • 5/4

CBSE 2026 · Set 31/4/1 · 1 mark

If sin θ = 1/7, then tan θ is:

  • 1/(4√3)
  • 1/(2√3)
  • (4√3)/7
  • 6/7

CBSE 2026 · Set 31/4/1 · 2 marks

If sin (A – B) = 1/2 and tan (A + B) = √3, 0° ≤ A + B < 90°, A > B then find the values of A and B.

CBSE 2026 · Set 31/4/1 · 3 marks

Prove that (cot A – cos A)/(cot A + cos A) = (sec A – tan A)/(sec A + tan A)

CBSE 2026 · Set 31/4/2 · 1 mark

The value of tan 30° tan 60° – sin 90° cos 90° is:

  • 1
  • 0
  • 2
  • not defined

CBSE 2026 · Set 31/4/2 · 1 mark

If sec θ = 3, then the value of tan θ is:

  • √(3 – 1)
  • √(3 – 1)
  • √2
  • 2√2

CBSE 2026 · Set 31/4/2 · 2 marks

If sin 2A = √3/2 and 2 tan B + 1 = 3, then find the value of (A + B).

CBSE 2026 · Set 31/4/2 · 3 marks

Prove that (sec θ + tan θ)² – 1 / (sec θ + tan θ)² + 1 = sin θ.

CBSE 2026 · Set 31/4/3 · 1 mark

The value of 2 tan 45° sin² 60° – cos 90° is:

  • –1/4
  • 3/2
  • 3/4
  • 1/4

CBSE 2026 · Set 31/4/3 · 2 marks

If sin A = 1/2 and tan B = √3, then verify that cos (A + B) = cos A cos B – sin A sin B.

CBSE 2026 · Set 31/5/1 · 1 mark

If value of cot θ is 5, then sin θ equals

  • 1/√6
  • √6
  • 5/√6
  • 1/2

CBSE 2026 · Set 31/5/1 · 1 mark

Assertion (A): For an acute angle θ, cos θ is always less than 1. Reason (R): In a right-angled triangle, hypotenuse is the longest side and cos θ = Base/Hypotenuse.

  • Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A)
  • Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A)
  • Assertion (A) is true, but Reason (R) is false
  • Assertion (A) is false, but Reason (R) is true

See all 169 board PYQs for this chapter — open PYQ tab →

Class 10 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 10 Mathematics Previous Year Questions, Chapter-wise

All 1107 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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