Glossary

1.1 Introduction to Trigonometric Functions

3 min read•july 25, 2024

Trigonometric functions are the building blocks of triangle math. They relate angles to side lengths, helping us understand shapes and cycles. From waves to lines, these functions pop up everywhere in math and science.

Right triangles are the key to unlocking trig's secrets. By looking at ratios of sides, we can figure out angles and lengths. This connects geometry to algebra, giving us powerful tools to solve real-world problems.

Trigonometric Functions Fundamentals

Trigonometric functions in triangles

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  • Right triangle definitions relate trig functions to triangle side ratios
    • Sine (sin) opposite side divided by hypotenuse measures vertical component
    • (cos) adjacent side divided by hypotenuse measures horizontal component
    • Tangent (tan) opposite side divided by adjacent side measures steepness
    • (csc) hypotenuse divided by opposite side reciprocal of sine
    • (sec) hypotenuse divided by adjacent side reciprocal of cosine
    • (cot) adjacent side divided by opposite side reciprocal of tangent
  • Unit circle definitions connect trig functions to coordinates
    • Sine (sin) y-coordinate of point on unit circle vertical position
    • Cosine (cos) x-coordinate of point on unit circle horizontal position
    • Tangent (tan) y/xy / x or sin⁡θ/cos⁡θ\sin \theta / \cos \theta slope of radial line
    • Cosecant (csc) 1/sin⁡θ1 / \sin \theta reciprocal of y-coordinate
    • Secant (sec) 1/cos⁡θ1 / \cos \theta reciprocal of x-coordinate
    • Cotangent (cot) x/yx / y or cos⁡θ/sin⁡θ\cos \theta / \sin \theta reciprocal of slope

Trigonometry and right triangles

  • Ratios of triangle sides link trig functions to geometry
    • Sine relates opposite side to hypotenuse vertical component
    • Cosine relates adjacent side to hypotenuse horizontal component
    • Tangent relates opposite side to adjacent side steepness or slope
  • Reciprocal relationships connect inverse functions
    • Cosecant reciprocal of sine csc⁡θ=1/sin⁡θ\csc \theta = 1 / \sin \theta
    • Secant reciprocal of cosine sec⁡θ=1/cos⁡θ\sec \theta = 1 / \cos \theta
    • Cotangent reciprocal of tangent cot⁡θ=1/tan⁡θ\cot \theta = 1 / \tan \theta
  • Pythagorean identities fundamental trig equations
    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 relates sine and cosine
    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta involves tangent and secant
    • 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta connects cotangent and cosecant

Domain and range of trigonometry

  • Sine function periodic wave
    • Domain all real numbers continuous input
    • Range [-1, 1] output bounded between -1 and 1
  • Cosine function shifted
    • Domain all real numbers continuous input
    • Range [-1, 1] output bounded between -1 and 1
  • Tangent function periodic with vertical asymptotes
    • Domain all real numbers except π/2+nπ\pi/2 + n\pi, n is integer undefined at 90°, 270°
    • Range all real numbers unbounded output
  • Cosecant function reciprocal of sine
    • Domain all real numbers except nπn\pi, n is integer undefined at 0°, 180°
    • Range (−∞,−1](-\infty, -1] and [1,∞)[1, \infty) output never between -1 and 1
  • Secant function reciprocal of cosine
    • Domain all real numbers except π/2+nπ\pi/2 + n\pi, n is integer undefined at 90°, 270°
    • Range (−∞,−1](-\infty, -1] and [1,∞)[1, \infty) output never between -1 and 1
  • Cotangent function reciprocal of tangent
    • Domain all real numbers except nπn\pi, n is integer undefined at 0°, 180°
    • Range all real numbers unbounded output

Signs of trigonometric functions

  • Quadrant I (0° to 90°) positive quadrant
    • All trigonometric functions positive
  • Quadrant II (90° to 180°) sine positive quadrant
    • Sine and cosecant positive vertical components positive
    • Cosine, tangent, secant, and cotangent negative
  • Quadrant III (180° to 270°) tangent positive quadrant
    • Tangent and cotangent positive slope-related functions positive
    • Sine, cosine, cosecant, and secant negative
  • Quadrant IV (270° to 360°) cosine positive quadrant
    • Cosine and secant positive horizontal components positive
    • Sine, tangent, cosecant, and cotangent negative
  • Mnemonic device All Students Take Calculus
    • A All functions positive (Quadrant I)
    • S Sine and its reciprocal (cosecant) positive (Quadrant II)
    • T Tangent and its reciprocal (cotangent) positive (Quadrant III)
    • C Cosine and its reciprocal (secant) positive (Quadrant IV)

Key Terms to Review (25)

30 degrees: 30 degrees is a specific angle measurement commonly encountered in trigonometry, representing one-sixth of a full rotation. This angle is fundamental in various trigonometric calculations, allowing for the evaluation of sine, cosine, and tangent values, as well as serving as a building block for understanding angles in different contexts such as the unit circle and right triangles.
45 degrees: 45 degrees is an angle that is exactly halfway between 0 degrees and 90 degrees, making it one of the most commonly encountered angles in trigonometry. This angle is significant because it has specific sine, cosine, and tangent values, and it plays a crucial role in understanding the behavior of trigonometric functions in various contexts.
90 degrees: 90 degrees is a specific angle measurement that is fundamental in trigonometry and represents a quarter of a complete rotation or one-fourth of a circle. This angle plays a critical role in defining the properties and behaviors of sine and cosine functions, which are crucial in analyzing periodic phenomena, and it serves as a reference point for evaluating various trigonometric functions, identities, and equations.
Amplitude: Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position in trigonometric functions, particularly in sine and cosine graphs. This term is crucial when understanding how these functions oscillate and can be transformed, impacting their height and overall shape. It plays a significant role in determining the vertical stretch or compression of the graph, influencing how the wave interacts with shifts and periodicity.
Arccosine: Arccosine is the inverse function of cosine, denoted as $$\text{arccos}(x)$$, which returns the angle whose cosine is a given number. This function is essential for finding angles in right triangles when the length of the adjacent side and hypotenuse are known, and it connects to various aspects of trigonometric relationships, helping to solve problems involving angles and distances in both theoretical and practical applications.
Arcsine: Arcsine is the inverse function of the sine function, denoted as $$\arcsin(x)$$ or sometimes $$\sin^{-1}(x)$$. It gives the angle whose sine is the given number, allowing you to find an angle when you know the ratio of the opposite side to the hypotenuse in a right triangle. This connection makes it crucial for solving various problems related to angles, triangles, and trigonometric equations.
Arctangent: Arctangent is the inverse function of the tangent function, allowing you to find an angle when you know the ratio of the opposite side to the adjacent side in a right triangle. This function is essential for determining angles based on specific trigonometric ratios and connects deeply with concepts like inverse trigonometric functions, right triangle relationships, and evaluating trigonometric functions. It is denoted as $$ ext{arctan}(x)$$ or sometimes $$ an^{-1}(x)$$, providing a way to express angles in relation to their tangent values.
Cos(x): The term cos(x) refers to the cosine function, a fundamental trigonometric function that relates the angle x to the ratio of the adjacent side to the hypotenuse in a right triangle. This function is essential in various applications, such as modeling periodic phenomena, and plays a crucial role in defining other trigonometric functions like sine and tangent. Cosine is also key in understanding wave patterns, oscillations, and rotations in mathematics and physics.
Cosecant: Cosecant is a trigonometric function that is defined as the reciprocal of the sine function. This means that for any angle, the cosecant can be calculated by taking 1 divided by the sine of that angle. It plays a crucial role in understanding various aspects of trigonometry, including relationships in right triangles, circular functions, and fundamental identities.
Cosine: Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. This relationship holds true in various contexts, from defining circular functions on the unit circle to solving complex triangles, making it essential in understanding angles and their measures.
Cosine graph: A cosine graph is a periodic function that represents the cosine values of angles on the Cartesian plane, typically oscillating between -1 and 1. The graph has a distinctive wave-like shape, with its peaks and troughs occurring at regular intervals, reflecting the cosine function's periodic nature and its relationship to angles measured in radians or degrees.
Cotangent: Cotangent is a trigonometric function that represents the ratio of the adjacent side to the opposite side in a right triangle, and is defined as the reciprocal of the tangent function. It can also be understood on the unit circle, where it is associated with the angle's x-coordinate divided by its y-coordinate. This function is vital for exploring relationships in right triangles, understanding identities, and analyzing circular functions.
Modeling periodic phenomena: Modeling periodic phenomena refers to the use of mathematical functions, particularly trigonometric functions like sine and cosine, to represent and analyze events that repeat at regular intervals. This concept is fundamental in various fields including physics, engineering, and biology, where understanding the patterns and cycles of oscillating systems is crucial for predicting behavior over time.
Periodicity: Periodicity refers to the repeating nature of functions, specifically how certain trigonometric functions return to the same value at regular intervals. In trigonometry, this concept is crucial because it helps in understanding the behavior of sine, cosine, and other trigonometric functions over their defined intervals, influencing how we interpret their graphs and solve equations.
Pythagorean Identity: The Pythagorean Identity is a fundamental relationship in trigonometry that expresses the square of the sine function plus the square of the cosine function as equal to one, represented as $$ ext{sin}^2(x) + ext{cos}^2(x) = 1$$. This identity is crucial in connecting the concepts of angle measures and the unit circle, forming a basis for deriving other identities and solving various trigonometric problems.
Radians: Radians are a unit of angular measure used in mathematics, particularly in trigonometry, where angles are measured based on the radius of a circle. One radian is defined as the angle formed when the arc length is equal to the radius of the circle. Radians are essential for understanding the relationships between angles and side lengths in right triangles and are also foundational for defining trigonometric functions like sine and cosine.
Reciprocal Identities: Reciprocal identities are fundamental relationships in trigonometry that express the relationship between the primary trigonometric functions and their reciprocals. They illustrate how each trigonometric function can be defined in terms of another function, highlighting the interconnectedness of these functions. Understanding these identities is essential for solving various trigonometric problems, including graphing secant and cosecant functions, evaluating functions in right triangles, and working with fundamental identities.
Reference Angle: A reference angle is the smallest angle formed by the terminal side of a given angle and the x-axis, always measured as a positive acute angle. This concept is crucial for understanding how angles relate to trigonometric functions across different quadrants, as it helps simplify calculations by connecting them to familiar angles in the first quadrant.
Sec(x): Secant, denoted as sec(x), is a trigonometric function defined as the reciprocal of the cosine function. In terms of a right triangle, it represents the ratio of the length of the hypotenuse to the length of the adjacent side. Secant is not only essential in trigonometry but also helps establish relationships between various trigonometric functions, particularly in solving triangles and understanding periodic behavior in various contexts.
Secant: Secant is a trigonometric function defined as the reciprocal of cosine. In a right triangle, it is calculated as the ratio of the hypotenuse to the adjacent side. The secant function also has a significant connection to the unit circle, where it corresponds to the x-coordinate of a point on the circle and helps to describe circular functions.
Sin(x): The function sin(x) represents the sine of an angle x, which is a fundamental trigonometric function that relates the angle to the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. This function plays a vital role in various applications, such as wave motion, oscillations, and circular motion, and serves as a building block for understanding other trigonometric functions like cosine and tangent.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. This relationship extends beyond right triangles to various contexts, such as the unit circle and periodic functions, where sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.
Sine Wave: A sine wave is a smooth, periodic oscillation that represents the graph of the sine function, which is fundamental in trigonometry. It demonstrates how a quantity varies in a regular and predictable manner over time, reflecting the relationship between angles and lengths in a right triangle. Sine waves are important in various fields such as physics, engineering, and music, as they model oscillatory behaviors like sound waves and alternating currents.
Solving Triangles: Solving triangles involves finding the unknown sides and angles of a triangle when given certain known values. This process often requires the use of trigonometric functions, which relate the angles to the lengths of the sides, as well as inverse trigonometric functions to find angles from known ratios. The concepts of triangle solutions are also closely tied to identities that simplify expressions for double or half angles, making it easier to work with various triangle types.
Tangent: The tangent function, often abbreviated as 'tan', relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. This function is crucial in various branches of mathematics, especially in solving triangles and modeling periodic phenomena.
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