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3.3 Sine and Cosine Function Values in AP Precalculus

Sine and Cosine Function Values in AP Precalculus
Table of Contents

1. Introduction to Sine and Cosine Functions

Sine (⁡\sin) and cosine (\cos) are fundamental trigonometric functions that form the backbone of everything you need to know about AP Precalculus and beyond. These functions arise naturally in the study of right triangles, circles, and oscillatory phenomena, making them essential tools for understanding angles, distances, and periodic behavior.

The Connection Between Sine, Cosine, and the Unit Circle

At their core, sine and cosine are tied to the geometry of the unit circle — a circle with a radius of 1, centered at the origin on the Cartesian coordinate plane. Any angle \theta drawn from the positive x-axis corresponds to a unique point on the unit circle. The coordinates of this point are (\cos\theta, \sin\theta), where:

  • Cosine (\cos\theta) represents the horizontal (x-coordinate) distance of the point from the origin.
  • Sine (\sin\theta) represents the vertical (y-coordinate) distance of the point from the origin.

By defining sine and cosine using the unit circle, we extend their applicability beyond acute angles (as found in right triangles) to any real number angle, whether positive or negative, measured in radians or degrees.

The Periodicity of Sine and Cosine

One of the remarkable properties of sine and cosine is their periodicity. Both functions repeat their values after a complete revolution of 2\pi radians (360 degrees). This means:

  • \sin(\theta + 2\pi) = \sin\theta
  • cos⁡θ\cos(\theta + 2\pi) = \cos\theta

This periodic nature makes these functions invaluable in describing oscillatory motions like waves, pendulums, and circular motion.

Symmetry and Quadrants

Sine and cosine exhibit unique symmetry properties, which help in evaluating their values across different quadrants:

  • Cosine: An even function, meaning \cos(-\theta) = \cos\theta.
  • Sine: An odd function, meaning \sin(-\theta) = -\sin\theta.

In addition, the signs of sine and cosine depend on the quadrant of the angle θ\theta:

  • Quadrant I: Both \sin\theta and \cos\theta are positive.
  • Quadrant II: \sin\theta is positive, but \cos\theta is negative.
  • Quadrant III: Both \sin\theta and \cos\theta are negative.
  • Quadrant IV: \sin\theta is negative, but \cos\theta is positive.

In AP Precalculus, you’ll not only learn how to calculate sine and cosine values for elementary and multiple angles but also explore their behavior in complex scenarios like determining the intersection points of terminal rays with circles. This knowledge sets the stage for more advanced studies in calculus, physics, and engineering.

2. Elementary Angles and Their Sine and Cosine Values

Elementary angles are specific angles commonly used in trigonometry, expressed in both degrees and radians. They are critical for understanding the behavior of sine and cosine functions, and their values often serve as reference points in more complex problems.

The Unit Circle and Its Role

The unit circle, with the equation x^2 + y^2 = 1, plays a central role in defining sine and cosine. Each angle \theta, measured from the positive x-axis, corresponds to a unique point (\cos\theta, \sin\theta) on the circle. Elementary angles are specific angles at which these trigonometric values are well-defined and often memorized.

Elementary Angles in Radians

In trigonometry, radians are the preferred unit of angle measurement. Here are the elementary angles and their corresponding radian measures:

  • 0^\circ = 0 radians
  • 30^\circ = \frac{\pi}{6} radians
  • 45^\circ = \frac{\pi}{4} radians
  • 60^\circ = \frac{\pi}{3} radians
  • 90^\circ = \frac{\pi}{2} radians

Each of these angles lies in the first quadrant of the unit circle. However, their counterparts in other quadrants can be determined using symmetry properties.

Sine and Cosine Function Values in AP Precalculus

Sine and Cosine Values for Elementary Angles

The values of sine and cosine for elementary angles are derived from the geometry of the unit circle and are summarized below:

Angle (in Radians)\sin\theta\cos\theta
001
\frac{\pi}{6}\frac{1}{2}\frac{\sqrt{3}}{2}
\frac{\pi}{4}\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}
\frac{\pi}{3}\frac{\sqrt{3}}{2}\frac{1}{2}
\frac{\pi}{2}10

Extension to Negative and Larger Angles

The values of sine and cosine repeat periodically, with a period of 2\pi. For negative angles or angles greater than 2\pi, these values can be determined using periodicity:

  • \sin(\theta + 2\pi) = \sin\theta
  • \cos(\theta + 2\pi) = \cos\theta
  • \sin(-\theta) = -\sin\theta
  • \cos(-\theta) = \cos\theta

For example:

  • \sin(-\frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}
  • \cos(-\frac{\pi}{6}) = \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}

Symmetry and Signs in Different Quadrants

The signs of sine and cosine depend on the quadrant of the angle:

  • Quadrant I: Both \sin\theta and \cos\theta are positive.
  • Quadrant II: \sin\theta is positive, \cos\theta is negative.
  • Quadrant III: Both are negative.
  • Quadrant IV: \sin\theta is negative, \cos\theta is positive.

Using these properties, you can easily determine the sine and cosine values for any angle.

3. Sine and Cosine Values at Multiples of Elementary Angles

Understanding sine and cosine values at multiples of elementary angles is crucial for solving problems in trigonometry, particularly in AP Precalculus. These multiples extend the applicability of trigonometric functions to angles across all quadrants, including both positive and negative rotations.

Multiples of Elementary Angles

Multiples of elementary angles can be expressed as:

  • Positive multiples, such as 2\theta, 3\theta, \dots
  • Negative multiples, such as -\theta, -2\theta, \dots

These angles can also exceed 2\pi radians (or 360^\circ), representing rotations around the unit circle. Trigonometric periodicity simplifies the computation of sine and cosine for such angles.

Using Periodicity

Sine and cosine are periodic functions:

  • \sin(\theta + 2\pi n) = \sin\theta
  • \cos(\theta + 2\pi n) = \cos\theta

This periodicity means that to find the sine or cosine of any angle, you can subtract or add multiples of 2\pi to bring it within the interval [0, 2\pi], simplifying calculations.

Sign Changes in Multiples

Multiples of elementary angles may shift the values of sine and cosine depending on their positions in the unit circle:

  • Quadrant I: 0 \leq \theta < \frac{\pi}{2}, both \sin\theta and \cos\theta are positive.
  • Quadrant II: \frac{\pi}{2} \leq \theta < \pi, \sin\theta is positive, \cos\theta is negative.
  • Quadrant III: 2\pi \leq \theta < \frac{3\pi}{2}, both \sin\theta and \cos\theta are negative.
  • Quadrant IV: \frac{3\pi}{2} \leq \theta < 2\pi, \sin\theta is negative, \cos\theta is positive.

Examples of Multiples

  1. Half Turns (\pi, 2\pi, 3\pi, \dots):
    • \sin\pi = 0, \cos\pi = -1
    • \sin2\pi = 0, \cos2\pi = 1
  2. Quarter Turns (\frac{\pi}{2}, \frac{3\pi}{2}):
    • \sin\frac{\pi}{2} = 1, \cos\frac{\pi}{2} = 0
    • \sin\frac{3\pi}{2} = -1, \cos\frac{3\pi}{2} = 0
  3. Negative Angles:
    • \sin(-\theta) = -\sin\theta
    • \cos(-\theta) = \cos\theta

    For example:

    • \sin(-\frac{\pi}{4}) = -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2}
    • \cos(-\frac{\pi}{4}) = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}
  4. Multiples of π6\frac{\pi}{6} and π3\frac{\pi}{3}:
    • \sin(2 \times \frac{\pi}{6}) = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}
    • \cos(2 \times \frac{\pi}{6}) = \cos\frac{\pi}{3} = \frac{1}{2}

Special Cases: Angles Greater than 2\pi

For angles exceeding 2\pi, reduce the angle by subtracting 2\pi until it lies within [0, 2\pi]. For example:

  • \sin(5\pi) = \sin(5\pi - 2\pi \times 2) = \sin\pi = 0
  • \cos(5\pi) = \cos\pi = -1

Download this free AP Precalculus worksheet with solutions for targeted practice and a better understanding of concepts

Start preparing effectively now!

4. Finding Intersection Points of Terminal Rays with the Circle

Intersection points lie on the circumference of the circle and can be determined using the angle’s terminal ray and the circle’s radius.

Understanding Terminal Rays and the Circle

  1. Terminal Ray: A ray originating from the center of the circle and extending outward, making a specific angle (\theta) with the positive x-axis.
  2. Circle: Defined by its radius r and center (0, 0). The equation of the circle is: x^2 + y^2 = r^2
  3. Intersection Point: The point where the terminal ray intersects the circle, defined as (x, y).

For a given angle \theta and radius <span class="katex">r, the intersection point can be expressed as:

x = r \cos\theta, \quad y = r \sin\theta

Sine and Cosine Function Values in AP Precalculus

Steps to Find Intersection Points

  1. Identify the Angle (\theta):
    • The angle \theta is measured from the positive x-axis in the counterclockwise direction.
    • For negative angles, the measurement is clockwise.
  2. Determine the Coordinates: Use the parametric equations:x = r \cos\theta y = r \sin\theta
  3. Verify Using the Circle Equation: Substitute x = r \cos\theta and y = r \sin\theta into the circle equation to verify:(r \cos\theta)^2 + (r \sin\theta)^2 = r^2 (\cos^2\theta + \sin^2\theta) = r^2Since \cos^2\theta + \sin^2\theta = 1, the point lies on the circle.

Example 1: Basic Calculation

  • Given: r = 5, \theta = \frac{\pi}{3}.
  • Find the intersection point. x = 5 \cos\frac{\pi}{3} = 5 \times \frac{1}{2} = 2.5 y = 5 \sin\frac{\pi}{3} = 5 \times \frac{\sqrt{3}}{2} \approx 4.33
  • Intersection point: (2.5, 4.33).

Example 2: Negative Angle

  • Given: r = 3, \theta = -\frac{\pi}{4}.
  • Find the intersection point. x = 3 \cos(-\frac{\pi}{4}) = 3 \times \frac{\sqrt{2}}{2} \approx 2.12 y = 3 \sin(-\frac{\pi}{4}) = 3 \times -\frac{\sqrt{2}}{2} \approx -2.12
  • Intersection point: (2.12, -2.12).

5. Determining Angle and Radius from Intersection Coordinates

Relationship Between Coordinates, Radius, and Angle

If a point (x, y) lies on the circumference of a circle with center (0, 0), the following relationships hold:

  1. Radius r :r = \sqrt{x^2 + y^2}This is derived from the equation of the circle: x^2 + y^2 = r^2.
  2. Angle \theta: The angle \theta can be determined using the inverse trigonometric functions:\tan\theta = \frac{y}{x}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)

Steps to Determine Angle and Radius

  1. Calculate the Radius (r): Use the distance formula:r = \sqrt{x^2 + y^2}This ensures the point lies on a circle with radius r
  2. Find the Angle (\theta):
    • Compute \theta using: \theta = \tan^{-1}\left(\frac{y}{x}\right)
    • Adjust \theta based on the quadrant of the point (x, y):
      • Quadrant I: \theta is as calculated.
      • Quadrant II: \theta = \pi - \tan^{-1}\left(\frac{|y|}{|x|}\right).
      • Quadrant III: \theta = \pi + \tan^{-1}\left(\frac{|y|}{|x|}\right).
      • Quadrant IV: \theta = 2\pi - \tan^{-1}\left(\frac{|y|}{|x|}\right).
  3. Verify the Results: Ensure that the computed x and y satisfy:x = r \cos\theta, \quad y = r \sin\theta

Example 1: Point in Quadrant I

  • Given: (x, y) = (3, 4).
  • Find r: r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
  • Find \theta: \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 \, \text{radians}

Example 2: Point in Quadrant III

  • Given: (x, y) = (-5, -5).
  • Find r: r = \sqrt{x^2 + y^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
  • Find \theta: \tan\theta = \frac{y}{x} = \frac{-5}{-5} = 1 Since the point is in Quadrant III: \theta = \pi + \tan^{-1}(1) = \pi + \frac{\pi}{4} = \frac{5\pi}{4}

6. Problems

Sine and Cosine Function Values in AP Precalculus

This problem requires determining the coordinates of point Q on a circle of radius 15, with the given triangle P Q O being an isosceles right triangle. Let’s break it down step by step:

Step 1: Understand the Geometry

  1. The circle is centered at O with a radius r = 15.
  2. Q lies on the circle, so its coordinates (x, y) must satisfy the equation of the circle: x^2 + y^2 = r^2 = 15^2 = 225.
  3. Triangle P Q O is an isosceles right triangle, meaning the two legs OQ and PQ are of equal length, and the angle \angle POQ is \frac{\pi}{4} radians.
  2. dius r, the coordinates of a point at an angle \theta (measured counterclockwise from the positive x-axis) are:(x, y) = (r \cos\theta, r \sin\theta).
  3. Since Q is in the fourth quadrant:
    • The cosine (⁡\cos) is positive.
    • The sine (⁡\sin) is negative. Therefore, the coordinates of Q are:

    Q = \left( r \cos\left(\frac{\pi}{4}\right), -r \sin\left(\frac{\pi}{4}\right) \right).

Step 3: Plug in the Values

  1. Substituting r = 15 and θ=π4\theta = \frac{\pi}{4}:Q = \left( 15 \cos\left(\frac{\pi}{4}\right), -15 \sin\left(\frac{\pi}{4}\right) \right).
  2. Recall the trigonometric values for \frac{\pi}{4}:\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.
  3. Substituting these values:Q = \left( 15 \cdot \frac{\sqrt{2}}{2}, -15 \cdot \frac{\sqrt{2}}{2} \right).
  4. Simplify:Q = \left( \frac{15\sqrt{2}}{2}, -\frac{15\sqrt{2}}{2} \right).

Step 4: Match with the Options

The correct option corresponds to the fourth quadrant, where x > 0 and y < 0. From the choices:

  • Correct answer: (C) \left( 15\cos\frac{\pi}{4}, -15\sin\frac{\pi}{4} \right).

Frequently Asked Questions:

Que :1) How many units does AP Precalculus cover?

AP Precalculus is divided into 3 main units:

  1. Unit 1: Polynomial and Rational Functions
    Focuses on key features, transformations, and applications of polynomial and rational functions.
  2. Unit 2: Exponential and Logarithmic Functions
    Includes modeling with exponential growth and decay and understanding logarithmic functions.
  3. Unit 3: Trigonometric and Polar Functions
    Explores trigonometric ratios, unit circle properties, and polar coordinates.

PS: Unit 4 is not tested in AP

Que : 2) How long is the AP Precalculus Exam?

The AP Precalculus Exam is 3 hours long.

  • Section 1: Multiple Choice (Graphing Calculator not Required)
    • 28 questions
    • 80 minutes
  • Section 2: Multiple Choice (Graphing Calculator Required)
    • 12 questions
    • 40 minutes
  • Section 3: Free Response (Graphing Calculator Required)
    • 2 questions
    • 30 minutes
  • Section 3: Free Response (Graphing Calculator not Required)
  • 2 questions
  • 30 minutes

Keep an eye on the official College Board web page for current information regarding the Exam!

Need personalized guidance? Book a private online AP Precalculus tutor now!

 

About Colin Phillips
About Colin Phillips

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