All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
1881
Which of the following identities is correct?
Answer:
1 + tan²(θ) = sec²(θ)
Starting with the Pythagorean identity sin²(θ) + cos²(θ) = 1, if we divide the entire equation by cos²(θ), we get (sin²(θ)/cos²(θ)) + 1 = 1/cos²(θ). This simplifies to tan²(θ) + 1 = sec²(θ).
1882
What is the exact value of cos(45°)?
Answer:
1/√2
In an isosceles right triangle with angles 45-45-90, the two legs are equal. If the legs are 1, the hypotenuse is √2 by the Pythagorean theorem. The cosine is adjacent/hypotenuse, which is 1/√2 (or √2/2 when rationalized).
1883
If sin(θ) = 3/5, what is the value of csc(θ)?
Answer:
5/3
The cosecant function, csc(θ), is the reciprocal of the sine function. Therefore, csc(θ) = 1 / sin(θ). If sin(θ) = 3/5, then calculating the reciprocal gives csc(θ) = 5/3.
1884
What is the value of sin²(θ) + cos²(θ)?
Answer:
1
This is the fundamental Pythagorean identity in trigonometry. For any angle θ, the sum of the square of its sine and the square of its cosine always equals 1, derived directly from the Pythagorean theorem (a² + b² = c²) divided by c².
1885
Which of the following is equivalent to tan(θ)?
Answer:
sin(θ)/cos(θ)
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Since sine is opposite/hypotenuse and cosine is adjacent/hypotenuse, dividing sine by cosine yields (opposite/hypotenuse) / (adjacent/hypotenuse) = opposite/adjacent = tan(θ).
1886
What is the value of sin(30°)?
Answer:
1/2
The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. For a 30-degree angle in a standard 30-60-90 triangle, the opposite side is exactly half the length of the hypotenuse. Thus, sin(30°) = 1/2.
1887
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Answer:
75(√3 - 1) m
Step 1: Let the distances of the ships from the lighthouse be x and y (x > y). Step 2: For the closer ship (45°), tan(45°) = 75 / y ⇒ y = 75 m. For the farther ship (30°), tan(30°) = 75 / x ⇒ 1 / √3 = 75 / x ⇒ x = 75√3 m. Step 3: Distance between ships = x - y = 75√3 - 75 = 75(√3 - 1) m.
1888
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
Answer:
7(√3 + 1) m
Step 1: Let the horizontal distance be x. Angle of depression to the foot is 45°, so tan(45°) = 7 / x ⇒ x = 7 m. Step 2: Let the height of the tower above the building be h'. Angle of elevation is 60°, so tan(60°) = h' / 7 ⇒ √3 = h' / 7 ⇒ h' = 7√3. Step 3: Total height of the tower = h' + 7 = 7√3 + 7 = 7(√3 + 1) m.
1889
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
Answer:
10√3 m
Step 1: Let the height of the tower be h. The distance from the foot is 30 m. Step 2: tan(30°) = h / 30. Step 3: 1 / √3 = h / 30, so h = 30 / √3 = 10√3 m.
1890
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
Answer:
8√3 m
Step 1: Let the standing part be h1 and the broken part be h2. Distance = 8 m. Step 2: tan(30°) = h1 / 8 ⇒ h1 = 8 / √3. cos(30°) = 8 / h2 ⇒ √3 / 2 = 8 / h2 ⇒ h2 = 16 / √3. Step 3: Total height = h1 + h2 = (8 / √3) + (16 / √3) = 24 / √3 = 8√3 m.