Daily Olympiad: Math - Trigonometry [20260915]

Solution

Correct: A

Given sin A = 3/5. In a right-angled triangle, sin A = Opposite / Hypotenuse. So, let the Opposite side be 3k and the Hypotenuse be 5k for some positive k. Using the Pythagoras theorem, (Adjacent)² + (Opposite)² = (Hypotenuse)². So, (Adjacent)² + (3k)² = (5k)². (Adjacent)² + 9k² = 25k². (Adjacent)² = 25k² - 9k² = 16k². Adjacent = √(16k²) = 4k. Now, tan A = Opposite / Adjacent = 3k / 4k = 3/4.

Solution

Correct: A

We know the standard trigonometric values: sin 30° = 1/2 and cos 60° = 1/2. Substitute these values into the expression: 2 × (1/2) × (1/2) = 1 × (1/2) = 1/2.

Solution

Correct: B

We use the complementary angle identity: sin θ = cos (90° - θ). Comparing sin 70° = cos x with this identity, we can write sin 70° as cos (90° - 70°) = cos 20°. So, cos 20° = cos x. Since x is an acute angle, we can equate the angles: x = 20°.

Solution

Correct: A

We use the fundamental trigonometric identity: sin²A + cos²A = 1. Substitute the given value of cos A: sin²A + (12/13)² = 1. sin²A + 144/169 = 1. sin²A = 1 - 144/169. sin²A = (169 - 144) / 169. sin²A = 25/169. sin A = √(25/169) = 5/13. Since A is an acute angle, sin A must be positive.

Solution

Correct: C

We use the identity sin²θ + cos²θ = 1, which can be rearranged to 1 - sin²θ = cos²θ. Substitute this into the numerator of the expression: (cos²θ) / cos²θ. This simplifies to 1.

Solution

Correct: B

Let the length of the ladder be 'L'. The distance from the wall to the foot of the ladder is 5m, which is the Adjacent side to the angle of 60°. The length of the ladder is the Hypotenuse. We use the cosine function: cos(angle) = Adjacent / Hypotenuse. cos 60° = 5 / L. We know cos 60° = 1/2. So, 1/2 = 5 / L. Cross-multiplying gives L = 2 × 5 = 10 m.

Solution

Correct: A

If tan θ = 1, then θ must be 45° (since tan 45° = 1). Now substitute θ = 45° into the expression (sin θ + cos θ). We know sin 45° = 1/√2 and cos 45° = 1/√2. So, sin 45° + cos 45° = 1/√2 + 1/√2 = 2/√2. This simplifies to √2 (by multiplying numerator and denominator by √2: (2√2)/(√2 * √2) = 2√2/2 = √2).

Solution

Correct: B

We know the standard trigonometric values: sin 45° = 1/√2 and cos 30° = √3/2. Substitute these values into the expression: (1/√2)² + (√3/2)² = (1/2) + (3/4). To add these fractions, find a common denominator, which is 4. So, (2/4) + (3/4) = 5/4.

Solution

Correct: C

We know the complementary angle identity: cot θ = tan (90° - θ). So, cot 30° = tan (90° - 30°) = tan 60°. The given equation becomes tan (A + B) = tan 60°. Since (A + B) is an acute angle, we can equate the angles: A + B = 60°.

Solution

Correct: A

First, find the hypotenuse AC using the Pythagoras theorem: AC² = AB² + BC² = 8² + 6² = 64 + 36 = 100. So, AC = √100 = 10 cm. For angle C, the Opposite side is AB = 8 cm, the Adjacent side is BC = 6 cm, and the Hypotenuse is AC = 10 cm. The definition of sec C is Hypotenuse / Adjacent = AC / BC = 10 / 6.

Solution

Correct: C

We use the identity 1 + tan²A = sec²A. Substitute this into the expression: (sec²A) cos²A. We also know that sec A is the reciprocal of cos A, i.e., sec A = 1/cos A, so sec²A = 1/cos²A. Therefore, (1/cos²A) × cos²A = 1.

Solution

Correct: A

Let θ be the angle of elevation of the sun. The height of the pole is the Opposite side (10m), and the length of the shadow is the Adjacent side (10√3 m). We use the tangent function: tan θ = Opposite / Adjacent. tan θ = 10 / (10√3) = 1/√3. We know that tan 30° = 1/√3. Therefore, θ = 30°.

Solution

Correct: A

We know that cosec A is the reciprocal of sin A, i.e., cosec A = 1 / sin A. Given cosec A = 2. So, 2 = 1 / sin A. Rearranging this, we get sin A = 1/2.

Solution

Correct: C

We use the standard trigonometric values: tan 45° = 1, cot 45° = 1, sin 90° = 1, cos 0° = 1. Substitute these values into the expression: (1 + 1) / (1 - 1) = 2 / 0. Division by zero is undefined.

Solution

Correct: B

Substitute the given expressions for x and y into x² + y²: x² + y² = (r cos θ)² + (r sin θ)². This expands to r² cos²θ + r² sin²θ. Factor out r²: r²(cos²θ + sin²θ). We know the identity cos²θ + sin²θ = 1. So, the expression simplifies to r² × 1 = r².

Solution

Correct: A

We use the complementary angle identity: tan θ = cot (90° - θ). So, we can write tan 2A as cot (90° - 2A). The given equation becomes cot (90° - 2A) = cot (A - 18°). Equating the angles (since cot function is one-to-one for acute angles): 90° - 2A = A - 18°. Collect terms with A on one side and constants on the other: 90° + 18° = A + 2A. 108° = 3A. A = 108° / 3 = 36°.

Solution

Correct: B

An identity is an equation that is true for all permissible values of the variable. (A) sin θ = cos θ is only true for specific angles (e.g., θ = 45°), not for all θ. (B) sin²θ + cos²θ = 1 is a fundamental trigonometric identity, true for all values of θ. (C) tan θ = 1 is only true for specific angles (e.g., θ = 45°). (D) sin θ = 1/2 is only true for specific angles (e.g., θ = 30°). Therefore, sin²θ + cos²θ = 1 is the correct identity.

Solution

Correct: B

We use two fundamental trigonometric identities: 1) sec²θ - tan²θ = 1 2) cosec²θ - cot²θ = 1 Substitute these values into the expression: 1 / 1 = 1.

Solution

Correct: A

We use the identity sin²θ + cos²θ = 1. Substitute sin θ = m/n: (m/n)² + cos²θ = 1. m²/n² + cos²θ = 1. cos²θ = 1 - m²/n². To subtract, find a common denominator: cos²θ = (n² - m²) / n². Since θ is acute, cos θ is positive. So, cos θ = √((n² - m²) / n²) = √(n² - m²) / n.

Solution

Correct: B

Let 'L' be the length of the string. This is the Hypotenuse of the right-angled triangle formed. The height of the kite is 60m, which is the Opposite side to the angle of inclination of 30°. We use the sine function: sin(angle) = Opposite / Hypotenuse. sin 30° = 60 / L. We know sin 30° = 1/2. So, 1/2 = 60 / L. Cross-multiplying gives L = 2 × 60 = 120 m.