Daily Olympiad: Math - Trigonometry [20260604]

Solution

Correct: A

Given sin A = 3/5. In a right-angled triangle, sin A = Perpendicular / Hypotenuse. Let the perpendicular side be 3k and the hypotenuse be 5k for some constant k. Using the Pythagorean theorem, Base² = Hypotenuse² - Perpendicular²: Base² = (5k)² - (3k)² = 25k² - 9k² = 16k². So, Base = √(16k²) = 4k. Now, tan A = Perpendicular / Base = 3k / 4k = 3/4.

Solution

Correct: B

We know that for complementary angles, sin (90° - θ) = cos θ. So, sin 27° = sin (90° - 63°) = cos 63°. Substitute this into the expression: sin² 63° + sin² 27° = sin² 63° + cos² 63°. Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the expression simplifies to 1.

Solution

Correct: A

Given √3 tan θ = 1, which implies tan θ = 1/√3. We know that tan 30° = 1/√3. Therefore, θ = 30°. Now, substitute θ = 30° into the expression sin² θ - cos² θ: sin² 30° - cos² 30° = (1/2)² - (√3/2)² = 1/4 - 3/4 = -2/4 = -1/2.

Solution

Correct: A

Given x = a cos³θ, divide by a: x/a = cos³θ. Taking the cube root on both sides: (x/a)^(1/3) = cos θ. Squaring both sides: x^(2/3)/a^(2/3) = cos²θ. Given y = a sin³θ, divide by a: y/a = sin³θ. Taking the cube root on both sides: (y/a)^(1/3) = sin θ. Squaring both sides: y^(2/3)/a^(2/3) = sin²θ. Now, add the two squared equations: x^(2/3)/a^(2/3) + y^(2/3)/a^(2/3) = cos²θ + sin²θ. Factor out 1/a^(2/3) on the left side: (x^(2/3) + y^(2/3))/a^(2/3) = cos²θ + sin²θ. Using the identity cos²θ + sin²θ = 1: (x^(2/3) + y^(2/3))/a^(2/3) = 1. Multiply by a^(2/3): x^(2/3) + y^(2/3) = a^(2/3).

Solution

Correct: C

Given tan A + cot A = 2. Square both sides of the equation: (tan A + cot A)² = 2². Expand the left side: tan² A + cot² A + 2(tan A)(cot A) = 4. We know that tan A = 1/cot A, so tan A * cot A = 1. Substitute this into the equation: tan² A + cot² A + 2(1) = 4. So, tan² A + cot² A + 2 = 4. Subtract 2 from both sides: tan² A + cot² A = 4 - 2 = 2.

Solution

Correct: A

Let the height of the observer be h_o = 1.5 m. Let the distance from the chimney be d = 28.5 m. Let the height of the chimney be H. The observer's eyes are at a height of 1.5 m from the ground. Let the horizontal distance from the observer's eye level to the chimney be the same as the distance from the base, which is 28.5 m. Let 'x' be the height of the chimney above the observer's eye level. In the right-angled triangle formed by the observer's eye, the point directly above the observer on the chimney, and the top of the chimney: tan 45° = x / 28.5. Since tan 45° = 1: 1 = x / 28.5, so x = 28.5 m. The total height of the chimney H is the height of the observer plus 'x': H = h_o + x = 1.5 m + 28.5 m = 30 m.

Solution

Correct: C

Rewrite the expression in terms of sin θ and cos θ: (1 + sin θ/cos θ + 1/cos θ)(1 + cos θ/sin θ - 1/sin θ) = ((cos θ + sin θ + 1)/cos θ)((sin θ + cos θ - 1)/sin θ) Let (sin θ + cos θ) = X. The expression becomes: = ((X + 1)/cos θ)((X - 1)/sin θ) = (X² - 1) / (sin θ cos θ) Substitute X back: ( (sin θ + cos θ)² - 1 ) / (sin θ cos θ) Expand (sin θ + cos θ)²: (sin² θ + cos² θ + 2 sin θ cos θ - 1) / (sin θ cos θ) Since sin² θ + cos² θ = 1: = (1 + 2 sin θ cos θ - 1) / (sin θ cos θ) = (2 sin θ cos θ) / (sin θ cos θ) = 2.

Solution

Correct: A

Given sin θ + cos θ = √2 cos θ. Divide the entire equation by cos θ (assuming cos θ ≠ 0): (sin θ / cos θ) + (cos θ / cos θ) = (√2 cos θ / cos θ) tan θ + 1 = √2 Subtract 1 from both sides: tan θ = √2 - 1.

Solution

Correct: B

The expression is a product of tangents of angles from 1° to 89°. We use the complementary angle identity: tan (90° - θ) = cot θ. Consider pairs of terms from the beginning and end: tan 1° * tan 89° = tan 1° * tan (90° - 1°) = tan 1° * cot 1° = 1. tan 2° * tan 88° = tan 2° * tan (90° - 2°) = tan 2° * cot 2° = 1. This pattern continues up to tan 44° * tan 46° = 1. The middle term is tan 45°, which is 1. So, the product is (tan 1° tan 89°)(tan 2° tan 88°)...(tan 44° tan 46°) * tan 45°. = (1)(1)...(1) * 1 = 1.

Solution

Correct: C

Given 4 tan θ = 3, which implies tan θ = 3/4. To simplify the expression (4 sin θ - cos θ) / (4 sin θ + cos θ), divide both the numerator and the denominator by cos θ: = ((4 sin θ / cos θ) - (cos θ / cos θ)) / ((4 sin θ / cos θ) + (cos θ / cos θ)) = (4 tan θ - 1) / (4 tan θ + 1) Now, substitute tan θ = 3/4: = (4 * (3/4) - 1) / (4 * (3/4) + 1) = (3 - 1) / (3 + 1) = 2 / 4 = 1/2.

Solution

Correct: A

We know the trigonometric identity: sec² θ - tan² θ = 1. This identity can be factored as a difference of squares: (sec θ - tan θ)(sec θ + tan θ) = 1. Given that sec θ + tan θ = p, substitute this into the factored identity: (sec θ - tan θ)(p) = 1. To find sec θ - tan θ, divide both sides by p: sec θ - tan θ = 1/p.

Solution

Correct: A

From sin (A + B) = 1, we know that sin 90° = 1. So, A + B = 90° --- (Equation 1). From cos (A - B) = √3/2, we know that cos 30° = √3/2. So, A - B = 30° --- (Equation 2). Add Equation 1 and Equation 2: (A + B) + (A - B) = 90° + 30° 2A = 120° A = 60°. Substitute A = 60° into Equation 1: 60° + B = 90° B = 90° - 60° B = 30°. Thus, A = 60° and B = 30°.

Solution

Correct: A

Let the length of the ladder be the hypotenuse (c) of a right-angled triangle, c = 10 m. Let the height of the window from the ground be one of the legs (a) of the triangle, a = 8 m. Let the distance of the foot of the ladder from the base of the wall be the other leg (b). Using the Pythagorean theorem: a² + b² = c². 8² + b² = 10² 64 + b² = 100 b² = 100 - 64 b² = 36 b = √36 = 6 m. So, the distance of the foot of the ladder from the base of the wall is 6 m.

Solution

Correct: A

The given expression is (1 + sin θ)(1 - sin θ) / (1 + cos θ)(1 - cos θ). Using the algebraic identity (a + b)(a - b) = a² - b²: Numerator: (1 + sin θ)(1 - sin θ) = 1² - sin² θ = cos² θ. Denominator: (1 + cos θ)(1 - cos θ) = 1² - cos² θ = sin² θ. So, the expression simplifies to cos² θ / sin² θ. We know that cos θ / sin θ = cot θ. Therefore, cos² θ / sin² θ = (cot θ)². Given cot θ = 7/8. So, (cot θ)² = (7/8)² = 49/64.

Solution

Correct: B

Given sin θ - cos θ = 0, which implies sin θ = cos θ. Since sin θ = cos θ, we can divide by cos θ (assuming cos θ ≠ 0) to get tan θ = 1. For acute angles, θ = 45°. Now, we need to find the value of sin⁴ θ + cos⁴ θ. Substitute θ = 45°: sin⁴ 45° + cos⁴ 45° = (sin 45°)⁴ + (cos 45°)⁴ = (1/√2)⁴ + (1/√2)⁴ = (1/2)² + (1/2)² = 1/4 + 1/4 = 2/4 = 1/2. Alternatively, from sin θ = cos θ, square both sides to get sin² θ = cos² θ. We also know sin² θ + cos² θ = 1. Substitute cos² θ = sin² θ into the identity: sin² θ + sin² θ = 1. 2 sin² θ = 1, so sin² θ = 1/2. Since cos² θ = sin² θ, then cos² θ = 1/2. Now, sin⁴ θ + cos⁴ θ = (sin² θ)² + (cos² θ)² = (1/2)² + (1/2)² = 1/4 + 1/4 = 1/2.

Solution

Correct: B

We use the complementary angle identity: cos θ = sin (90° - θ). So, cos 72° = cos (90° - 18°) = sin 18°. Substitute this into the expression: sin 18° / cos 72° = sin 18° / sin 18° = 1 (assuming sin 18° ≠ 0, which is true).

Solution

Correct: A

Given x = a cos θ, we can write cos θ = x/a. Given y = b sin θ, we can write sin θ = y/b. We know the fundamental trigonometric identity: sin² θ + cos² θ = 1. Substitute the expressions for sin θ and cos θ into the identity: (y/b)² + (x/a)² = 1 y²/b² + x²/a² = 1 To combine the terms on the left side, find a common denominator (a²b²): (a²y² + b²x²) / (a²b²) = 1 Multiply both sides by a²b²: a²y² + b²x² = a²b².

Solution

Correct: B

We know that 1/sec A is equivalent to cos A. The cosine function, cos A, has a defined range of values. For any angle A, the value of cos A always lies between -1 and 1, inclusive. That is, -1 ≤ cos A ≤ 1. Therefore, the maximum value that cos A (or 1/sec A) can attain is 1.

Solution

Correct: B

Given sin A = 1/2. We know that sin 30° = 1/2, so A = 30°. Now substitute A = 30° into the expression 3 cot² A + 3. We know cot 30° = √3. So, 3 cot² 30° + 3 = 3(√3)² + 3 = 3(3) + 3 = 9 + 3 = 12. Alternatively, we can use the identity 1 + cot² A = cosec² A, which means cot² A = cosec² A - 1. Substitute this into the expression: 3 cot² A + 3 = 3(cosec² A - 1) + 3 = 3 cosec² A - 3 + 3 = 3 cosec² A. Since sin A = 1/2, then cosec A = 1/sin A = 1/(1/2) = 2. So, 3 cosec² A = 3(2)² = 3(4) = 12.

Solution

Correct: A

We use the fundamental trigonometric identities: 1 + tan² A = sec² A 1 + cot² A = cosec² A Substitute these identities into the given expression: (1 + tan² A) / (1 + cot² A) = sec² A / cosec² A. Now, express sec A and cosec A in terms of sin A and cos A: sec A = 1/cos A, so sec² A = 1/cos² A. cosec A = 1/sin A, so cosec² A = 1/sin² A. Substitute these back into the expression: (1/cos² A) / (1/sin² A) = (1/cos² A) * (sin² A / 1) = sin² A / cos² A. Since sin A / cos A = tan A, then sin² A / cos² A = tan² A.