Precalculus problems

Learn More on Precalculus

  • Precalculus problems is branch of the acclaimed Art of Problem Solving curriculum designed to challenge high-Performing middle and high school students.
  • Precalculus problems cover trigonometry, complex numbers, vectors, and matrices.
  • Calculus was developed in the latter half of the seventeenth century by two mathematicians, Namely Gottfried Leibniz and Isaac Newton. Calculus can be divided into two branches:
  • Differential Calculus and Integral Calculus. Differential calculus is used to find the rate of change of a measures; integral calculus is used to find the measures where the rate of change is known. “Functions” are defined by a formula.

Solve Precalculus problems:

1)Get the arc length corresponds to the given angle on a circle of radius 2.5. (Round your answer to three decimal places.) 45°

An ant start at the point (1, 0) on the unit circle and walks counterclockwise a distance of 4 units around the circle. Find the x and y coordinates (correct to 2 decimal places) of the last place of the ant.

Below is shown precalculus problems solutions for your better understanding:

Circumference = C = 2πR = 2π(2.5) = 5π
The arc length made by a 45º angle = (45 ⁄ 360) • C
= (45 ⁄ 360) • (5π)
= (0.625) • π
= (0.625) • π
(3 places after the decimal point) = 1.963 units
(3 significant figures)     = 1.96 units
The circle is centered at (0, 0) and has a radius = 1 and C = 2π
The ant starts at (1, 0) on the +x_axis and travels 4 units.
The angle traveled is (360º) • (4 ⁄ C) = (360º) • (4 ⁄ 2π) = 229.18312 deg
which is in the third quadrant if traveling CCW.
This angle is equivalent to (229.18312) − 180 = 49.183118 deg CCW.

2)sin x(cot x + tan x) = sec x LHS:
sin x (cos x / sin x + sin x / cos x) = (exchange to sin and cos)
cos x + sin^2 x / cos x = (distribute and simplify)
(cos^2 x / cos x + sin^2 x / cos x) = (multiply by cos / cos)
(cos^2 x + sin^2 x) / cos x = (combine into single fraction)
1 / cos x = (since cos^2 + sin^2 = 1)