Introduction to Trigonometry

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.