Table of Contents

1. Introduction
2. Angles
3. Rays and Quadrants
4. Angle and Arc Length Relationship
- - 4.1 How Angles Relate to Arc Length
  - 4.3 Examples of Calculating Arc Length
5. Co-Terminal and Reference Angles
6. The Unit Circle
7. Sine, Cosine, and Tangent
8. Frequently Asked Questions:

3.2 Sine, Cosine, and Tangent in AP Precalculus

Utkarsh
January 16, 2025

Table of Contents

1. Introduction
2. Angles
3. Rays and Quadrants
4. Angle and Arc Length Relationship
- - 4.1 How Angles Relate to Arc Length
  - 4.3 Examples of Calculating Arc Length
5. Co-Terminal and Reference Angles
6. The Unit Circle
7. Sine, Cosine, and Tangent
8. Frequently Asked Questions:

1. Introduction

Sine, Cosine, and Tangent are fundamental trigonometric tools in mathematics, with applications ranging from solving right triangles to modeling periodic phenomena like sound waves, light, and even ocean tides. These functions form the foundation of trigonometry and are crucial for understanding the relationships between angles and sides of triangles. In AP Precalculus, mastering these functions is essential for tackling more advanced topics in calculus and beyond.

The terms sine, cosine, and tangent may seem abstract at first, but they originate from simple geometric relationships in right triangles. Each function represents a specific ratio of sides relative to a given angle. For example:

Sine (sin) of an angle is the ratio of the opposite side to the hypotenuse.
Cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse.
Tangent (tan) of an angle is the ratio of the opposite side to the adjacent side.

2. Angles

2.1 What Are Angles?

An angle is formed when two rays share a common endpoint, called the vertex. The starting ray is known as the initial side, and the ray that moves away to form the angle is the terminal side. The size of an angle measures the amount of rotation from the initial side to the terminal side, and this rotation can be measured in different units.

2.2 Degrees vs. Radians

Angles can be measured in degrees or radians, with each unit offering unique advantages:

Degrees: This unit divides a full circle into 360 equal parts. A right angle is 90°, and a straight angle is 180°. Degrees are commonly used in everyday contexts, such as navigation or architecture.
Radians: This unit relates angles to the length of an arc on a circle. A full circle corresponds to $2\pi$ radians, a straight angle is $\pi$ radians, and a right angle is $\frac{\pi}{2}$ radians. Radians are preferred in mathematics and science because they simplify many equations and calculations.

The relationship between degrees and radians is:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$ and $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

2.3 Converting Between Degrees and Radians (with Examples)

Converting between degrees and radians is straightforward using the formulas above. Here are a few examples:

Convert $90^\circ$ to radians: $90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians.}$
Convert $\frac{\pi}{3}$ radians to degrees: $\frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ$ $.$

3. Rays and Quadrants

Angles are not only about their size but also about their position. To fully understand trigonometric functions, it’s essential to know how angles are formed and how their position in a coordinate plane affects their properties. This involves understanding rays and quadrants.

3.1 Initial Ray and Terminal Ray

When an angle is drawn on a coordinate plane, it is formed by two rays:

Initial Ray (or Initial Side): This is the starting position of the angle, typically aligned with the positive x-axis.
Terminal Ray (or Terminal Side): This is the ray that rotates away from the initial ray to form the angle.

The rotation direction determines the sign of the angle:

Counterclockwise Rotation: Positive angle (e.g., $90^\circ, 45^\circ, \frac{\pi}{3}$ ).
Clockwise Rotation: Negative angle (e.g., $-90^\circ, -45^\circ, -\frac{\pi}{3}$ ).

3.2 Standard Position of an Angle

An angle is said to be in standard position when:

Its vertex is located at the origin of the coordinate plane.
Its initial side lies along the positive x-axis.

3.3 Quadrants and Their Role in Angle Measurement

The coordinate plane is divided into four regions, called quadrants, based on the signs of the x and y coordinates. Each quadrant affects the sign of trigonometric functions differently:

Quadrant I: Both $x > 0$ and $y > 0$ , so all trigonometric functions ( $⁡\sin, \cos, \tan$ ) are positive.
Quadrant II: $x < 0, y > 0$ , so $⁡\sin$ is positive, but ⁡ $\cos$ and $\tan$ are negative.
Quadrant III: Both $x < 0$ and $y < 0$ , so $\tan$ is positive, while $⁡\sin$ and ⁡ $\cos$ are negative.
Quadrant IV: $x > 0, y < 0$ , so $\cos$ is positive, but $⁡\sin$ and $\tan$ are negative.

4. Angle and Arc Length Relationship

The relationship between an angle and the length of the arc it subtends is a key concept in trigonometry and AP Precalculus. This relationship provides a natural way to measure angles in radians and lays the groundwork for understanding the unit circle and trigonometric functions.

4.1 How Angles Relate to Arc Length

When an angle is drawn in standard position on a circle, the arc length is the curved distance between the points where the initial and terminal sides intersect the circle. The arc length depends on two factors:

The radius of the circle ( $r$ )
The measure of the angle in radians ( $\theta$ )

The formula connecting these quantities is:

$\text{Arc Length (s)} = r \cdot \theta$

Here:

$s$ is the arc length.
$r$ is the radius of the circle.
$\theta$ is the angle in radians.

4.3 Examples of Calculating Arc Length

Here are some practical examples to illustrate the relationship:

Given Radius and Angle in Radians:
A circle has a radius of $5 \, \text{units}$ . Find the arc length subtended by an angle of $2 \, \text{radians}$ .

$s = r \cdot \theta = 5 \cdot 2 = 10 \, \text{units.}$

Given Radius and Angle in Degrees:
A circle has a radius of $4 \, \text{units}$ . Find the arc length subtended by an angle of $60^\circ$ .

First, convert $60^\circ$ to radians: $\theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \, \text{radians.}$

Then, calculate the arc length: $s = r \cdot \theta = 4 \cdot \frac{\pi}{3} = \frac{4\pi}{3} \, \text{units.}$

Finding the Angle Given Arc Length and Radius:
A circle has a radius of $7 \, \text{units}$ , and the arc length is $21 \, \text{units}$ . Find the angle subtended by the arc.
Using the formula $s = r \cdot \theta$ : $\theta = \frac{s}{r} = \frac{21}{7} = 3 \, \text{radians.}$

5. Co-Terminal and Reference Angles

5.1 Co-Terminal Angles

Definition: Co-terminal angles are angles that share the same initial side and terminal side but may have different measures. They are created by adding or subtracting full rotations (multiples of $360^\circ$ or $2\pi$ radians) to a given angle.

Formula for Co-Terminal Angles:

In degrees: $\text{Co-terminal angle} = \theta + 360^\circ \cdot k$
In radians: $\text{Co-terminal angle} = \theta + 2\pi \cdot k$

Here, $k$ is any integer (positive, negative, or zero).

Examples of Co-Terminal Angles:

For $\theta = 45^\circ: \text{Co-terminal angles: } 45^\circ + 360^\circ = 405^\circ, \; 45^\circ - 360^\circ = -315^\circ.$
For $\theta = \frac{\pi}{4}: \text{Co-terminal angles: } \frac{\pi}{4} + 2\pi = \frac{9\pi}{4}, \; \frac{\pi}{4} - 2\pi = -\frac{7\pi}{4}.$

5.2 Reference Angles

Definition: A reference angle is the acute angle (between $0^\circ$ and $90^\circ$ , or 0 and $\frac{\pi}{2}$ radians) formed between the terminal side of a given angle and the x-axis. Reference angles are always positive and help determine trigonometric values of angles in all quadrants.

How to Find the Reference Angle:

Quadrant I: The reference angle is the angle itself ( $\theta_{\text{ref}} = \theta$ ).
Quadrant II: Subtract the angle from $180^\circ$ (or $\pi radians$ ): $\theta_{\text{ref}} = 180^\circ - \theta \; \text{or} \; \pi - \theta.$
Quadrant III: Subtract $180^\circ$ (or $\pi$ ) from the angle: $\theta_{\text{ref}} = \theta - 180^\circ \; \text{or} \; \theta - \pi.$
Quadrant IV: Subtract the angle from $360^\circ$ (or $2\pi$ ): $\theta_{\text{ref}} = 360^\circ - \theta \; \text{or} \; 2\pi - \theta$ .

5.3 Examples of Reference Angles:

In Degrees:
- For $\theta = 150^\circ$ (Quadrant II): $\theta_{\text{ref}} = 180^\circ - 150^\circ = 30^\circ$ .
- For $\theta = 210^\circ$ (Quadrant III): $\theta_{\text{ref}} = 210^\circ - 180^\circ = 30^\circ$ .
- For $\theta = 330^\circ$ (Quadrant IV): $\theta_{\text{ref}} = 360^\circ - 330^\circ = 30^\circ$ .
In Radians:
- For $\theta = \frac{5\pi}{6}$ (Quadrant II): $\theta_{\text{ref}} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$ .
- For $\theta = \frac{7\pi}{4}$ (Quadrant IV): $\theta_{\text{ref}} = 2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$ .

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

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6. The Unit Circle

6.1 What Is the Unit Circle?

The unit circle is a circle centered at the origin $(0, 0)$ in the Cartesian coordinate plane with a radius of $1$ . Its equation is:

$x^2 + y^2 = 1$

Here:

$x$ represents the horizontal coordinate.
$y$ represents the vertical coordinate.

6.2 Angles on the Unit Circle

Standard Position: An angle is in standard position if its vertex is at the origin, its initial side lies along the positive $x$ -axis, and its terminal side rotates counterclockwise.
Radians and Degrees: Angles on the unit circle can be measured in both degrees and radians, with key conversions: $360^\circ = 2\pi \, \text{radians}, \; 180^\circ = \pi \, \text{radians}$ .

6.3 Coordinates and Trigonometric Functions

For any angle \theta in standard position, the terminal side intersects the unit circle at a point $(x, y)$ , where:

$x = \cos\theta$ : The cosine of the angle is the $x$ -coordinate.
$y = \sin\theta$ : The sine of the angle is the $y$ -coordinate.
$\tan\theta = \frac{\sin\theta}{\cos\theta}$ : The tangent is the ratio of the sine to the cosine (defined when $\cos\theta \neq 0$ ).

6.5 Key Points on the Unit Circle

The coordinates for some common angles are:

$0^\circ (0)$ → $(1, 0)$
$90^\circ (\pi/2)$ → $(0, 1)$
$180^\circ (\pi)$ → $(-1, 0)$
$270^\circ (3\pi/2)$ → $(0, -1)$

Other angles, like $30^\circ, 45^\circ$ , and $60^\circ$ , correspond to well-known coordinate pairs involving $\sqrt{3}/2, \sqrt{2}/2$ , and $1/2$ .

7. Sine, Cosine, and Tangent

Sine, cosine, and tangent are the fundamental trigonometric functions that relate the angles of a triangle to the lengths of its sides. In AP Precalculus, these functions are explored not only in the context of right triangles but also as continuous functions on the unit circle.

7.1 Defining Sine, Cosine, and Tangent

In a right triangle:

Sine ( $\sin$ ): The ratio of the length of the side opposite the angle to the hypotenuse. $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
Cosine ( $\cos$ ): The ratio of the length of the side adjacent to the angle to the hypotenuse. $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
Tangent ( $\tan$ ): The ratio of the length of the side opposite the angle to the side adjacent to the angle. $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

7.2 Sine, Cosine, and Tangent on the Unit Circle

For any angle $\theta$ in standard position:

$\sin\theta$ : The $y$ -coordinate of the point where the terminal side of the angle intersects the unit circle.
$\cos\theta$ : The $x$ -coordinate of the same point.
$\tan\theta$ : The ratio of the sine to the cosine: $\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \text{provided } \cos\theta \neq 0$ .

7.3 Periodicity and Symmetry

Periodicity:
- Sine and cosine repeat every $360^\circ$ ( $2\pi$ radians)= $\sin(\theta + 2\pi) = \sin\theta, \quad \cos(\theta + 2\pi) = \cos\theta$
- Tangent repeats every $180^\circ$ ( $\pi$ radians): $\tan(\theta + \pi) = \tan\theta$
Symmetry:
- $\sin(-\theta) = -\sin\theta$ : Odd function.
- $\cos(-\theta) = \cos\theta$ : Even function.
- $\tan(-\theta) = -\tan\theta$ : Odd function.

7.4 Key Values for Sine, Cosine, and Tangent

Using the unit circle, we identify the values of sine, cosine, and tangent for key angles:

Angle ( $\theta$ )	$\sin\theta$	$\cos\theta$	$\tan\theta = \frac{\sin\theta}{\cos\theta}$
$0^\circ (0)$	$0$	$1$	$0$
$30^\circ (\pi/6)$	$1/2$	$\sqrt{3}/2$	$1/\sqrt{3}$
$45^\circ (\pi/4)$	$1/\sqrt{2}$	$1/\sqrt{2}$	$1$
$60^\circ (\pi/3)$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$90^\circ (\pi/2)$	$1$	$0$	Undefined ( $\cos\theta = 0$ )

7.5 Example Problem

Problem: Calculate $\sin 120^\circ, \cos 120^\circ$ , and $\tan 120^\circ$ .

Step 1: Locate the Angle.
$120^\circ$ is in Quadrant II.
Step 2: Find the Reference Angle.
Reference angle = $180^\circ - 120^\circ = 60^\circ$ .
Step 3: Apply the Unit Circle.
In Quadrant II:
- $\sin 120^\circ = \sin 60^\circ = \sqrt{3}/2$ (positive in Quadrant II).
- $\cos 120^\circ = -\cos 60^\circ = -1/2$ (negative in Quadrant II).
- $\tan 120^\circ = \frac{\sin 120^\circ}{\cos 120^\circ} = \frac{\sqrt{3}/2}{-1/2} = -\sqrt{3}$ .

8. Frequently Asked Questions:

Q 1: How many students take AP Precalculus?

In its inaugural year, over 200,000 students across all states enrolled in the course, with more than 175,000 taking the exam.

Q 2: Does AP Precalculus help with the SAT?

Not particularly. AP Precalculus is primarily designed to prepare students for calculus, while calculus concepts are not included in the SAT.

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!

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