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
1751
Evaluate the expression: sin²(θ) + sin²(90° - θ)
Answer:
1
Using complementary angles, we know that sin(90° - θ) is identical to cos(θ). Therefore, sin²(90° - θ) equals cos²(θ). The expression simplifies to sin²(θ) + cos²(θ), which is the fundamental Pythagorean identity that always equals 1.
1752
Which of these functions is completely positive in the 4th quadrant?
Answer:
Cosine
In Cartesian coordinates, the fourth quadrant has positive x-values and negative y-values. Since cosine represents the x-coordinate on the unit circle, the cosine function (and its reciprocal secant) is the only primary trigonometric function that remains positive there.
1753
The angle of elevation of a tower is 45° from a point 50m away from its base. What is the height of the tower?
Answer:
50 m
Using the tangent function: tan(45°) = height / adjacent_distance. Because tan(45°) = 1, the ratio height/50 must equal 1. Thus, the height of the tower perfectly equals the distance, which is 50m.
1754
Find the value of sin(15°) / cos(15°).
Answer:
tan(15°)
By definition, the ratio of the sine of any angle to the cosine of that same angle is identically the tangent of that angle. Therefore, sin(15°) / cos(15°) rigorously simplifies to tan(15°).
1755
If tan(θ) = undefined in the interval [0, 2π], what are the values of θ?
Answer:
π/2, 3π/2
Tangent is defined as sin(θ) / cos(θ). The function becomes undefined exactly where the denominator, cos(θ), is zero. Within a single circle [0, 2π], cosine is zero at the angles 90° (π/2) and 270° (3π/2).
1756
What is the equivalent of cos(-x)?
Answer:
cos(x)
The cosine function is fundamentally an even function, which mathematically means that f(-x) must equal f(x) for all values in its domain. Consequently, cos(-x) = cos(x).
1757
Evaluate: tan(225°)
Answer:
1
The angle 225° lies in the third quadrant, where the tangent function is positive. The reference angle is 225° - 180° = 45°. Therefore, tan(225°) is equal to tan(45°), which is exactly 1.
1758
If sin(A) = 3/5 and cos(B) = 12/13, where A and B are acute angles, find sin(A + B).
Answer:
56/65
First, find missing values: cos(A) = 4/5 and sin(B) = 5/13 (using Pythagorean triples). Use the formula sin(A+B) = sin(A)cos(B) + cos(A)sin(B). This yields (3/5)*(12/13) + (4/5)*(5/13) = 36/65 + 20/65 = 56/65.
1759
Simplify: (cos²θ) / (1 - sin(θ))
Answer:
1 + sin(θ)
Using the Pythagorean identity, we substitute cos²θ with (1 - sin²θ). This gives (1 - sin²θ) / (1 - sin(θ)). The numerator is a difference of squares, factoring into (1 - sin(θ))(1 + sin(θ)). Cancelling the common factor (1 - sin(θ)) leaves exactly 1 + sin(θ).
1760
What is the equivalent of sin(360° - θ)?
Answer:
-sin(θ)
An angle of 360° - θ places the terminal side in the fourth quadrant (assuming θ is acute). In the fourth quadrant, the sine function is strictly negative. The reference angle is θ, so sin(360° - θ) = -sin(θ).