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Honors Pre-Calculus
Table of Contents
Unit 5 Overview: Trigonometric Functions
5.1 Angles
5.2 Unit Circle: Sine and Cosine Functions
5.3 The Other Trigonometric Functions
5.4 Right Triangle Trigonometry

📏honors pre-calculus review

5.3 The Other Trigonometric Functions

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Trigonometric functions are essential tools in math, helping us understand angles and ratios in triangles. They're not just for geometry - these functions pop up in physics, engineering, and even music theory. Let's dive into the world of sines, cosines, and tangents!

We'll explore exact values, reference angles, and even-odd properties of trig functions. We'll also look at important identities and how to use your calculator effectively. Understanding these concepts will give you a solid foundation for more advanced math.

Trigonometric Functions

Exact values of trigonometric functions

  • Tangent ($\tan$) represents the ratio of the opposite side to the adjacent side in a right triangle
    • $\tan(\pi/6) = 1/\sqrt{3}$ (30° angle in a 30-60-90 triangle)
    • $\tan(\pi/4) = 1$ (45° angle in an isosceles right triangle)
    • $\tan(\pi/3) = \sqrt{3}$ (60° angle in a 30-60-90 triangle)
  • Cotangent ($\cot$) is the reciprocal of tangent, representing the ratio of the adjacent side to the opposite side
    • $\cot(\pi/6) = \sqrt{3}$ (30° angle in a 30-60-90 triangle)
    • $\cot(\pi/4) = 1$ (45° angle in an isosceles right triangle)
    • $\cot(\pi/3) = 1/\sqrt{3}$ (60° angle in a 30-60-90 triangle)
  • Secant ($\sec$) is the reciprocal of cosine, representing the ratio of the hypotenuse to the adjacent side
    • $\sec(\pi/6) = 2/\sqrt{3}$ (30° angle in a 30-60-90 triangle)
    • $\sec(\pi/4) = \sqrt{2}$ (45° angle in an isosceles right triangle)
    • $\sec(\pi/3) = 2$ (60° angle in a 30-60-90 triangle)
  • Cosecant ($\csc$) is the reciprocal of sine, representing the ratio of the hypotenuse to the opposite side
    • $\csc(\pi/6) = 2$ (30° angle in a 30-60-90 triangle)
    • $\csc(\pi/4) = \sqrt{2}$ (45° angle in an isosceles right triangle)
    • $\csc(\pi/3) = 2/\sqrt{3}$ (60° angle in a 30-60-90 triangle)
  • The unit circle is a useful tool for visualizing and calculating these exact values

Reference angles in trigonometry

  • Reference angles are acute angles (between 0 and π/2) used to determine the values of trigonometric functions in other quadrants
    • Angles in the first quadrant (0 to π/2) are their own reference angles
    • Angles in other quadrants have reference angles found by reflecting the angle about the x or y-axis until it lies in the first quadrant
  • Tangent and cotangent have the same sign as their reference angle in the first and third quadrants (positive) and the opposite sign in the second and fourth quadrants (negative)
  • Secant has the same sign as its reference angle in the first and fourth quadrants (positive) and the opposite sign in the second and third quadrants (negative)
  • Cosecant has the same sign as its reference angle in the first and second quadrants (positive) and the opposite sign in the third and fourth quadrants (negative)

Even vs odd trigonometric functions

  • Even functions are symmetric about the y-axis, meaning $f(-x) = f(x)$
    • Cosine ($\cos$) and secant ($\sec$) are even functions
    • Reflect the graph of an even function over the y-axis, and it will look the same
  • Odd functions are symmetric about the origin, meaning $f(-x) = -f(x)$
    • Sine ($\sin$), tangent ($\tan$), cosecant ($\csc$), and cotangent ($\cot$) are odd functions
    • Reflect the graph of an odd function over the origin (rotate 180°), and it will look the same

Trigonometric Identities

Fundamental trigonometric identities

  • Reciprocal identities express the relationship between a trigonometric function and its reciprocal
    • $\sin(x) = 1/\csc(x)$ (sine and cosecant are reciprocals)
    • $\cos(x) = 1/\sec(x)$ (cosine and secant are reciprocals)
    • $\tan(x) = 1/\cot(x)$ (tangent and cotangent are reciprocals)
  • Pythagorean identities are derived from the Pythagorean theorem and relate the squares of trigonometric functions
    • $\sin^2(x) + \cos^2(x) = 1$ (relates sine and cosine)
    • $1 + \tan^2(x) = \sec^2(x)$ (relates tangent and secant)
    • $1 + \cot^2(x) = \csc^2(x)$ (relates cotangent and cosecant)
  • Quotient identities express one trigonometric function as the quotient of two others
    • $\tan(x) = \sin(x)/\cos(x)$ (tangent is sine divided by cosine)
    • $\cot(x) = \cos(x)/\sin(x)$ (cotangent is cosine divided by sine)

Calculator use for trigonometric functions

  • Ensure the calculator is set to the correct angle mode (degrees or radians)
    • Most calculators have a DRG or DEG/RAD button to switch between modes
    • Radians are often used in calculus and more advanced mathematics
  • Use the appropriate trigonometric function buttons to evaluate expressions
    • $\sin$, $\cos$, $\tan$ buttons for sine, cosine, and tangent
    • $\csc$, $\sec$, $\cot$ buttons (if available) or reciprocal buttons (1/x) with $\sin$, $\cos$, $\tan$ for cosecant, secant, and cotangent
  • Be aware of the calculator's limitations for very large or very small angles
    • Results may be displayed as an error or in scientific notation
    • Use identities or reference angles to simplify calculations when necessary

Advanced Trigonometric Concepts

Periodic Functions and Graphing Techniques

  • Trigonometric functions are periodic, repeating their values at regular intervals
  • The period of a function is the smallest positive value of p for which f(x + p) = f(x) for all x
  • Graphing techniques for trigonometric functions:
    • Identify the amplitude, period, and vertical shift
    • Use transformations to sketch graphs of more complex trigonometric functions
    • Apply domain and range restrictions to graph portions of trigonometric functions

Inverse Trigonometric Functions

  • Inverse trigonometric functions "undo" the effect of the original trigonometric functions
  • Common inverse functions include arcsin (sin^(-1)), arccos (cos^(-1)), and arctan (tan^(-1))
  • The domain and range of inverse trigonometric functions are restricted to ensure they are functions
  • Inverse trigonometric functions are useful for solving equations involving trigonometric functions

Key Terms to Review (21)

Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions, allowing us to determine the angle given the ratio of the sides of a right triangle. They are essential in understanding and solving various trigonometric equations and problems.
Asymptotes: An asymptote is a straight line that a curve approaches but never touches. It provides important information about the behavior and characteristics of a function as it approaches its limits.
Tangent: A tangent is a line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry, geometry, and calculus, with applications across various mathematical and scientific disciplines.
Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of the coordinate plane. It is a fundamental tool in trigonometry that helps visualize and understand the relationships between angles, their trigonometric functions, and the coordinates of points on the circle.
Graphing Techniques: Graphing techniques refer to the various methods and approaches used to visually represent mathematical functions, relationships, and data on a coordinate plane. These techniques enable the effective communication and analysis of complex information through the use of graphs, plots, and other visual representations.
Domain and Range: The domain and range are fundamental concepts in mathematics that describe the set of input values and output values, respectively, for a given function. These terms are particularly relevant in the context of trigonometric functions, as they help define the possible values and behaviors of these functions.
Secant: A secant is a line that intersects a curve or circle at two distinct points. It is a fundamental concept in trigonometry, geometry, and calculus, with applications in various fields of mathematics and physics.
Cofunction Theorem: The cofunction theorem is a fundamental concept in trigonometry that establishes a relationship between certain trigonometric functions. It states that the trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant are related in a specific way, allowing for the substitution of one function with another in various trigonometric identities and equations.
Restricted Domain: The restricted domain of a function refers to the limited range of input values for which the function is defined. This concept is particularly important in the context of trigonometric functions and their inverse functions.
Quadrantal Angles: Quadrantal angles are special angles that are multiples of 90 degrees, or $\pi/2$ radians. These angles are significant in the study of trigonometry as they have unique properties and relationships with the trigonometric functions.
Reference Angle: The reference angle is the acute angle formed between a given angle and the nearest coordinate axis in the unit circle. It represents the smallest positive angle that has the same trigonometric function values as the given angle.
Reciprocal Identities: Reciprocal identities are a set of trigonometric identities that relate the reciprocals of the trigonometric functions. These identities provide a way to express one trigonometric function in terms of another, which can be useful in solving trigonometric equations and simplifying trigonometric expressions.
Period: The period of a function refers to the distance or interval along the x-axis over which the function repeats itself. It represents the length of one complete cycle of the function's graph. The period is a fundamental characteristic of periodic functions, such as the sine and cosine functions, and is crucial in understanding and analyzing the behavior of these functions.
Cotangent: The cotangent is one of the fundamental trigonometric functions, defined as the reciprocal of the tangent function. It represents the ratio of the adjacent side to the opposite side of a right triangle, providing a way to describe the relationship between the sides of a triangle and the angles within it.
Pythagorean Identities: Pythagorean identities are fundamental trigonometric identities that describe the relationships between the trigonometric functions of a given angle. These identities are derived from the Pythagorean theorem and are essential in solving trigonometric equations and simplifying trigonometric expressions.
Reciprocal Functions: A reciprocal function is a type of function where the input and output values are reciprocals of each other. In other words, if the input is $x$, the output is $1/x$. Reciprocal functions are closely related to the other trigonometric functions and are important for understanding their properties and applications.
Quotient Identities: Quotient identities are a set of trigonometric identities that relate the trigonometric functions by expressing one function as a ratio or quotient of two other functions. These identities are particularly useful in simplifying trigonometric expressions and solving trigonometric equations.
Periodic Functions: A periodic function is a function that repeats its values at regular intervals. This means that the function's values follow a pattern that is reproduced at fixed periods or intervals. Periodic functions are an important concept in various areas of mathematics, including trigonometry, Fourier analysis, and the study of waves and oscillations.
Cosecant: The cosecant (csc) is one of the six fundamental trigonometric functions, defined as the reciprocal of the sine function. It represents the ratio of the hypotenuse to the opposite side of a right triangle, and is used to describe the relationship between the sides and angles of a triangle.
Odd Functions: An odd function is a mathematical function where the graph is symmetric about the origin, meaning that for any input x, the function value f(x) is the negative of the function value for the input -x. In other words, f(-x) = -f(x) for all x in the domain of the function.
Even Functions: An even function is a function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.