7.4 The Other Trigonometric Functions

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: cosecant, secant, and cotangent.

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

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• Secant ($\sec$) is the reciprocal of cosine divides 1 by the cosine value of an angle $\sec \theta = \frac{1}{\cos \theta}$
  • For angle 0, $\sec 0 = \frac{1}{\cos 0} = \frac{1}{1} = 1$ since cosine of 0 is 1
  • For angle $\frac{\pi}{6}$, $\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2\sqrt{3}}{3}$ since cosine of $\frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$
• Cosecant ($\csc$) is the reciprocal of sine divides 1 by the sine value of an angle $\csc \theta = \frac{1}{\sin \theta}$
  • For angle $\frac{\pi}{2}$, $\csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = \frac{1}{1} = 1$ since sine of $\frac{\pi}{2}$ is 1
  • For angle $\frac{\pi}{3}$, $\csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2\sqrt{3}}{3}$ since sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$
• Tangent ($\tan$) is the ratio of sine to cosine divides the sine value by the cosine value of an angle $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  • For angle $\frac{\pi}{4}$, $\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$ since both sine and cosine of $\frac{\pi}{4}$ are $\frac{\sqrt{2}}{2}$
  • For angle $\frac{\pi}{3}$, $\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$ since sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$ and cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$
• Cotangent ($\cot$) is the reciprocal of tangent divides the cosine value by the sine value of an angle $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$
  • For angle $\frac{\pi}{4}$, $\cot \frac{\pi}{4} = \frac{\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$ since both cosine and sine of $\frac{\pi}{4}$ are $\frac{\sqrt{2}}{2}$
  • For angle $\frac{\pi}{6}$, $\cot \frac{\pi}{6} = \frac{\cos \frac{\pi}{6}}{\sin \frac{\pi}{6}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$ since cosine of $\frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$ and sine of $\frac{\pi}{6}$ is $\frac{1}{2}$
• These trigonometric ratios can be visualized and calculated using the unit circle

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 $\frac{\pi}{2}$ or $90^\circ$
  • For angle $\frac{5\pi}{4}$, the nearest multiple of $\frac{\pi}{2}$ is $\frac{3\pi}{2}$, so the reference angle is $\frac{3\pi}{2} - \frac{5\pi}{4} = \frac{\pi}{4}$
• The sign of the trigonometric function value depends on the quadrant of the terminal side
  1. Quadrant I (0 to $\frac{\pi}{2}$): All functions are positive
  2. Quadrant II ($\frac{\pi}{2}$ to $\pi$): Only sine and cosecant are positive
  3. Quadrant III ($\pi$ to $\frac{3\pi}{2}$): Only tangent and cotangent are positive
  4. Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): Only cosine and secant are positive
• To evaluate $\cos(\frac{5\pi}{4})$, find the reference angle $\frac{\pi}{4}$ and note that $\frac{5\pi}{4}$ is in Quadrant III where cosine is negative, so $\cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$

Even vs odd trigonometric functions

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

Trigonometric Identities and Technology

Fundamental trigonometric identities

• Pythagorean identities relate the square of sine and cosine, tangent and secant, or cotangent and cosecant
  • $\sin^2 \theta + \cos^2 \theta = 1$ (square of sine plus square of cosine equals 1)
  • $1 + \tan^2 \theta = \sec^2 \theta$ (1 plus square of tangent equals square of secant)
  • $1 + \cot^2 \theta = \csc^2 \theta$ (1 plus square of cotangent equals square of cosecant)
• Reciprocal identities relate sine and cosecant, cosine and secant, or tangent and cotangent
  • $\sin \theta = \frac{1}{\csc \theta}$ (sine equals the reciprocal of cosecant)
  • $\cos \theta = \frac{1}{\sec \theta}$ (cosine equals the reciprocal of secant)
  • $\tan \theta = \frac{1}{\cot \theta}$ (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
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$ (tangent equals sine divided by cosine)
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$ (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 $150^\circ$, switch calculator to degree mode, press the "sin" button, enter 150, then press "=" to get $\sin(150^\circ) = \frac{1}{2}$
  • For angle $\frac{5\pi}{6}$, switch calculator to radian mode, press the "tan" button, enter $\frac{5\pi}{6}$, then press "=" to get $\tan(\frac{5\pi}{6}) = -\sqrt{3}$
• Spreadsheet software (Excel, Google Sheets) also have trigonometric functions
  • Use the SIN(), COS(), TAN(), CSC(), SEC(), and COT() functions
  • For angle $\frac{\pi}{3}$, enter =SIN(PI()/3) into a cell to get $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
• 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 $\frac{2\pi}{3}$, use math.cos(2*math.pi/3) to get $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$

Advanced Trigonometric Concepts

• Radian measure 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
• Inverse trigonometric functions, such as arcsin, arccos, and arctan, allow us to find angles given trigonometric ratios
• Graphing trigonometric functions helps visualize their periodic nature and key features like amplitude, period, and phase shifts