AP Precalculus Practice Test: Unit 3 Question #10 The Point on Unit Circle Given a Radius and Angle

Please subscribe!    / nickperich   My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May. *AP Precalculus Practice Test: Unit 3 Question #10* *The Point on the Unit Circle Given a Radius and Angle* --- *Key Concepts and Vocabulary* 1. **Unit Circle**: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The point on the unit circle for a given angle corresponds to the coordinates \((x, y)\), where \(x = \cos(\theta)\) and \(y = \sin(\theta)\). 2. **Radius**: The radius of the circle determines the distance from the origin to any point on the circle. In the case of the unit circle, the radius is always 1, but for circles with other radii, the coordinates are scaled by the radius. 3. **Angle**: The angle \(\theta\) is measured from the positive \(x\)-axis, and it defines the location of the point on the circle. The point's coordinates can be determined using the trigonometric functions cosine and sine. 4. **Trigonometric Functions**: The cosine and sine functions for an angle \(\theta\) are defined as: \[ \cos(\theta) = x \quad \text{and} \quad \sin(\theta) = y \] --- *Question Setup* We are tasked with finding the coordinates of a point on the unit circle given a specific radius and angle. For example, suppose we are given a radius of 3 and an angle of \(\frac{\pi}{4}\). --- *Step-by-Step Solution* 1. **Identify the Radius and Angle**: We are given a radius of 3 and an angle of \(\frac{\pi}{4}\). The point on the circle corresponding to this angle will have the coordinates \((x, y)\), where: \[ x = 3 \cdot \cos\left(\frac{\pi}{4}\right) \quad \text{and} \quad y = 3 \cdot \sin\left(\frac{\pi}{4}\right) \] 2. **Use Known Values for Cosine and Sine**: From the unit circle, we know: \[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \quad \text{and} \quad \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] 3. **Calculate the Coordinates**: Multiply the cosine and sine values by the radius (3): \[ x = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \quad \text{and} \quad y = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \] 4. **Final Answer**: The coordinates of the point on the circle are: \[ \left( \frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2} \right) \] --- *Final Answer* \[ \boxed{\left( \frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2} \right)} \] --- *Purpose of the Question* This problem assesses your ability to: Use trigonometric functions to find the coordinates of a point on a circle, given the radius and angle. Apply knowledge of the unit circle to scale the coordinates for circles with radii other than 1. Correctly use the cosine and sine functions to determine the \(x\)- and \(y\)-coordinates of a point on the circle. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study