Trigonometry and Coordinate Geometry Quick Test Quiz

Sharpen your understanding of trigonometry and coordinate geometry with this quick quiz, covering core concepts such as identities, equations, and geometric interpretations on the Cartesian plane. Perfect for students and enthusiasts looking to reinforce foundational skills and problem-solving strategies in mathematics.

1. Identifying a Correct Trigonometric Identity

   Which of the following is a fundamental trigonometric identity involving sine and cosine for any angle θ?

   1. tan²θ + 1 = secθ
   2. cosθ / sinθ = tanθ
   3. sinθ - cosθ = 1
   4. sin²θ + cos²θ = 1

   Explanation: The equation sin²θ + cos²θ = 1 is known as the Pythagorean identity and holds for all real values of θ. The option sinθ - cosθ = 1 is incorrect as there is no such identity. Tan²θ + 1 = secθ is an incorrect form; the correct version is tan²θ + 1 = sec²θ. Cosθ / sinθ = tanθ is also incorrect because it should be cotθ, not tanθ.

2. Distance Formula Application

   What is the distance between the points (3, 4) and (0, 0) on a coordinate plane?

   1. 5
   2. 7
   3. 25
   4. 1

   Explanation: The distance between two points (x₁, y₁) and (x₂, y₂) is found using the formula √[(x₂ - x₁)² + (y₂ - y₁)²]. Substituting the values, √[(3−0)² + (4−0)²] = √[9 + 16] = √25 = 5. Seven and twenty-five are incorrect values from misapplication of the formula, while one is much too small relative to the coordinates provided.

3. Understanding Slope

   If a line passes through the points (2, 3) and (4, 7), what is the slope of the line?

   1. 4
   2. 0.5
   3. 2
   4. 1

   Explanation: The slope (m) is calculated as (y₂−y₁)/(x₂−x₁), which gives (7−3)/(4−2) = 4/2 = 2. The value one is obtained by incorrectly subtracting only numerators or denominators. Four is the difference in y-values without dividing by the x-difference, and 0.5 results from swapping the subtraction order. Only two is mathematically correct for these coordinates.

4. Evaluating Trigonometric Ratios

   What is the value of cos 60°?

   1. 0.866
   2. 1
   3. 0.5
   4. 0

   Explanation: Cos 60° is exactly 0.5, a commonly memorized value in trigonometry. Zero is the value for cos 90°, not 60°. The value 0.866 corresponds to sin 60°, not cosine. The number one is correct for cos 0° or cos 360°, but not 60°.

5. Equation of a Circle

   Which equation represents a circle with center at (0, 0) and radius 4 on the coordinate plane?

   1. x + y = 4
   2. x² + y² = 16
   3. x² - y² = 16
   4. x² + y² = 8

   Explanation: The standard equation for a circle centered at (0, 0) with radius r is x² + y² = r². Substituting r = 4 gives x² + y² = 16. The second option is an equation of a hyperbola, not a circle. The third is the equation of a straight line, while the fourth has the wrong radius squared, representing a smaller circle.