Glossary

7.4 The Other Trigonometric Functions

5 min read•june 24, 2024

Trigonometric functions are powerful tools for understanding angles and relationships in triangles. They're essential in math, physics, and engineering. These functions include sine, cosine, tangent, and their reciprocals: , , and .

Knowing exact values of trig functions for common angles is crucial. We'll explore how to calculate these values, work with reference angles beyond the first quadrant, and understand even and odd trig functions. We'll also dive into fundamental identities and using technology for trig calculations.

Trigonometric Functions

Exact values of trigonometric functions

Top images from around the web for Exact values of trigonometric functions

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

  • MrAllegretti - Trigonometric Functions - B1 View original

    Is this image relevant?

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Exact values of trigonometric functions

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

  • MrAllegretti - Trigonometric Functions - B1 View original

    Is this image relevant?

  • Trigonometric Functions and the Unit Circle | Boundless Algebra View original

    Is this image relevant?

1 of 3
  • Secant (sec⁡\sec) is the reciprocal of cosine divides 1 by the cosine value of an angle sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
    • For angle 0, sec⁡0=1cos⁡0=11=1\sec 0 = \frac{1}{\cos 0} = \frac{1}{1} = 1 since cosine of 0 is 1
    • For angle π6\frac{\pi}{6}, sec⁡π6=1cos⁡π6=132=233\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2\sqrt{3}}{3} since cosine of π6\frac{\pi}{6} is 32\frac{\sqrt{3}}{2}
  • Cosecant (csc⁡\csc) is the reciprocal of sine divides 1 by the sine value of an angle csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}
    • For angle π2\frac{\pi}{2}, csc⁡π2=1sin⁡π2=11=1\csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = \frac{1}{1} = 1 since sine of π2\frac{\pi}{2} is 1
    • For angle π3\frac{\pi}{3}, csc⁡π3=1sin⁡π3=132=233\csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2\sqrt{3}}{3} since sine of π3\frac{\pi}{3} is 32\frac{\sqrt{3}}{2}
  • Tangent (tan⁡\tan) is the ratio of sine to cosine divides the sine value by the cosine value of an angle tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
    • For angle π4\frac{\pi}{4}, tan⁡π4=sin⁡π4cos⁡π4=2222=1\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 since both sine and cosine of π4\frac{\pi}{4} are 22\frac{\sqrt{2}}{2}
    • For angle π3\frac{\pi}{3}, tan⁡π3=sin⁡π3cos⁡π3=3212=3\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} since sine of π3\frac{\pi}{3} is 32\frac{\sqrt{3}}{2} and cosine of π3\frac{\pi}{3} is 12\frac{1}{2}
  • Cotangent (cot⁡\cot) is the reciprocal of tangent divides the cosine value by the sine value of an angle cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}
    • For angle π4\frac{\pi}{4}, cot⁡π4=cos⁡π4sin⁡π4=2222=1\cot \frac{\pi}{4} = \frac{\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 since both cosine and sine of π4\frac{\pi}{4} are 22\frac{\sqrt{2}}{2}
    • For angle π6\frac{\pi}{6}, cot⁡π6=cos⁡π6sin⁡π6=3212=3\cot \frac{\pi}{6} = \frac{\cos \frac{\pi}{6}}{\sin \frac{\pi}{6}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} since cosine of π6\frac{\pi}{6} is 32\frac{\sqrt{3}}{2} and sine of π6\frac{\pi}{6} is 12\frac{1}{2}
  • These can be visualized and calculated using the

Reference angles beyond first quadrant

  • Reference angle is the acute angle formed between the terminal side of an angle and the x-axis
  • To find the reference angle, subtract the given angle from the nearest multiple of π2\frac{\pi}{2} or 90∘90^\circ
    • For angle 5π4\frac{5\pi}{4}, the nearest multiple of π2\frac{\pi}{2} is 3π2\frac{3\pi}{2}, so the reference angle is 3π2−5π4=π4\frac{3\pi}{2} - \frac{5\pi}{4} = \frac{\pi}{4}
  • The sign of the trigonometric function value depends on the quadrant of the terminal side
    1. Quadrant I (0 to π2\frac{\pi}{2}): All functions are positive
    2. Quadrant II (π2\frac{\pi}{2} to π\pi): Only sine and cosecant are positive
    3. Quadrant III (π\pi to 3π2\frac{3\pi}{2}): Only tangent and cotangent are positive
    4. Quadrant IV (3π2\frac{3\pi}{2} to 2π2\pi): Only cosine and secant are positive
  • To evaluate cos⁡(5π4)\cos(\frac{5\pi}{4}), find the reference angle π4\frac{\pi}{4} and note that 5π4\frac{5\pi}{4} is in Quadrant III where cosine is negative, so cos⁡(5π4)=−22\cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}

Even vs odd trigonometric functions

  • Even functions are symmetric about the y-axis satisfies f(−θ)=f(θ)f(-\theta) = f(\theta)
    • Cosine and secant are even functions
      • cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta) (cosine of the negative angle equals cosine of the positive angle)
      • sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta) (secant of the negative angle equals secant of the positive angle)
  • Odd functions are symmetric about the origin satisfies f(−θ)=−f(θ)f(-\theta) = -f(\theta)
    • Sine, cosecant, tangent, and cotangent are odd functions
      • sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta) (sine of the negative angle equals the negative of sine of the positive angle)
      • csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta) (cosecant of the negative angle equals the negative of cosecant of the positive angle)
      • tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta) (tangent of the negative angle equals the negative of tangent of the positive angle)
      • cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta) (cotangent of the negative angle equals the negative of cotangent of the positive angle)

Trigonometric Identities and Technology

Fundamental trigonometric identities

  • relate the square of sine and cosine, tangent and secant, or cotangent and cosecant
    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 (square of sine plus square of cosine equals 1)
    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta (1 plus square of tangent equals square of secant)
    • 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta (1 plus square of cotangent equals square of cosecant)
  • relate sine and cosecant, cosine and secant, or tangent and cotangent
    • sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta} (sine equals the reciprocal of cosecant)
    • cos⁡θ=1sec⁡θ\cos \theta = \frac{1}{\sec \theta} (cosine equals the reciprocal of secant)
    • tan⁡θ=1cot⁡θ\tan \theta = \frac{1}{\cot \theta} (tangent equals the reciprocal of cotangent)
  • Quotient identities express tangent as the ratio of sine to cosine and cotangent as the ratio of cosine to sine
    • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} (tangent equals sine divided by cosine)
    • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta} (cotangent equals cosine divided by sine)

Technology in trigonometric evaluation

  • Most scientific calculators have buttons for trigonometric functions
    • Make sure the calculator is in the correct mode (degrees or radians)
    • For angle 150∘150^\circ, switch calculator to degree mode, press the "sin" button, enter 150, then press "=" to get sin⁡(150∘)=12\sin(150^\circ) = \frac{1}{2}
    • For angle 5π6\frac{5\pi}{6}, switch calculator to radian mode, press the "tan" button, enter 5π6\frac{5\pi}{6}, then press "=" to get tan⁡(5π6)=−3\tan(\frac{5\pi}{6}) = -\sqrt{3}
  • Spreadsheet software (Excel, Google Sheets) also have trigonometric functions
    • Use the 
      SIN()
      , 
      COS()
      , 
      TAN()
      , 
      CSC()
      , 
      SEC()
      , and 
      COT()
      functions
    • For angle π3\frac{\pi}{3}, enter 
      =SIN(PI()/3)
      into a cell to get sin⁡(π3)=32\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}
  • Programming languages (Python, JavaScript) have built-in trigonometric functions in their math libraries
    • In Python, use 
      math.sin()
      , 
      math.cos()
      , 
      math.tan()
      , 
      math.csc()
      , 
      math.sec()
      , and 
      math.cot()
    • For angle 2π3\frac{2\pi}{3}, use 
      math.cos(2*math.pi/3)
      to get cos⁡(2π3)=−12\cos(\frac{2\pi}{3}) = -\frac{1}{2}

Advanced Trigonometric Concepts

  • is an alternative way to express angles, where one radian is the angle subtended by an arc length equal to the radius of the circle
  • , such as arcsin, arccos, and arctan, allow us to find angles given trigonometric ratios
  • helps visualize their periodic nature and key features like amplitude, , and phase shifts

Key Terms to Review (26)

Cofunction Theorem: The cofunction theorem is a fundamental principle in trigonometry that relates the values of certain trigonometric functions to the values of other trigonometric functions. It establishes a connection between the trigonometric functions and their reciprocal functions, allowing for efficient calculations and understanding of the relationships between these functions.
Cofunctions: Cofunctions, in the context of trigonometry, are pairs of trigonometric functions that have a special relationship where one function's value is determined by the other function's value. These cofunctions are closely related and often used together in various trigonometric identities and applications.
Cosecant: The cosecant, often abbreviated as csc, is one of the fundamental trigonometric functions. It represents the reciprocal of the sine function, meaning it is the ratio of the hypotenuse to the opposite side of a right triangle. The cosecant function is particularly useful in understanding the properties and applications of trigonometry, which are essential in various mathematical and scientific disciplines.
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 right triangle and the angles formed within it.
Decomposition: Decomposition involves breaking down a complex fraction into simpler partial fractions. This is often used to simplify integration or solve algebraic equations more easily.
Even function: An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.
Even Function: An even function is a mathematical 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.
Graphing Trigonometric Functions: Graphing trigonometric functions involves visually representing the periodic and cyclic nature of trigonometric functions on a coordinate plane. This process allows for the analysis of the functions' properties, behaviors, and applications in various contexts.
Heaviside method: The Heaviside method, also known as the cover-up method, is a technique for finding partial fraction decompositions. It simplifies the process by covering up parts of the equation to quickly find coefficients.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions, allowing one to find the angle given the value of a trigonometric ratio. They are essential for solving trigonometric equations and understanding the behavior of periodic functions.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, odd functions exhibit symmetry about the origin.
Odd Function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that the graph of an odd function is symmetric about the origin, with the graph being a reflection across both the $x$-axis and the $y$-axis.
Partial fraction: Partial fractions are a way to decompose a rational function into a sum of simpler fractions. This technique is useful for integrating rational functions or solving certain types of algebraic equations.
Period: The period of a function is the distance or interval along the independent variable axis over which the function's shape or pattern repeats itself. It is a fundamental concept in the study of periodic functions, such as trigonometric functions, and is essential for understanding their properties and graphs.
Pythagorean Identities: Pythagorean identities are fundamental trigonometric equations that relate the values of the sine, cosine, and tangent functions. They are derived from the Pythagorean theorem and provide important connections between the different trigonometric functions.
Quadrantal Angles: Quadrantal angles are special angles that are multiples of 90 degrees, such as 0°, 90°, 180°, and 270°. These angles are significant in the context of trigonometric functions, as they represent the cardinal directions on the unit circle and have unique properties.
Quadratic: A quadratic is a polynomial of degree 2, typically in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratics have a characteristic parabolic graph that opens upwards if $a > 0$ and downwards if $a < 0$.
Radian Measure: Radian measure is a way of expressing angles in terms of the ratio of the length of the arc subtended by the angle to the radius of the circle. It is a fundamental concept in trigonometry that provides a more natural and versatile way of working with angles compared to the more familiar degree measure.
Reciprocal functions: Reciprocal functions are mathematical functions defined as the multiplicative inverse of a given function. Specifically, if a function is represented as $$f(x)$$, its reciprocal function is given by $$g(x) = \frac{1}{f(x)}$$. These functions exhibit unique properties in the context of trigonometric functions, particularly with the sine, cosine, and tangent functions, as they help illustrate relationships among various trigonometric identities.
Reciprocal Identities: Reciprocal identities are a set of fundamental trigonometric identities that express the relationship between the reciprocal trigonometric functions, such as secant, cosecant, and cotangent, and the primary trigonometric functions, such as sine, cosine, and tangent. These identities provide a way to convert between the different trigonometric functions and are essential for verifying and simplifying trigonometric expressions.
Restricted Domain: The restricted domain of a function refers to the limited range of input values for which the function is defined. It represents the subset of the domain where the function can be evaluated without resulting in undefined or invalid outputs.
Secant: A secant is a line that intersects a circle at two distinct points. It is one of the fundamental trigonometric functions, along with sine, cosine, tangent, and others, that describe the relationships between the sides and angles of a right triangle.
Simplifying Trigonometric Expressions: Simplifying trigonometric expressions involves reducing the complexity of trigonometric functions, such as sine, cosine, tangent, and their inverse functions, by applying various trigonometric identities and algebraic manipulations. This process aims to express the expression in a more concise and manageable form, often with fewer trigonometric functions or simpler arguments.
Trigonometric Ratios: Trigonometric ratios are the fundamental mathematical relationships between the sides and angles of a right triangle. These ratios, including sine, cosine, and tangent, are essential for understanding and applying right triangle trigonometry, the behavior of other trigonometric functions, inverse trigonometric functions, and solving trigonometric equations.
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 concept in trigonometry, as it provides a visual representation of the relationships between the trigonometric functions and the angles they represent.
Vertical Asymptotes: Vertical asymptotes are vertical lines that a graph approaches but never touches. They represent the values of the independent variable where a function is undefined or has a vertical discontinuity.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide