7.4 The Other Trigonometric Functions

Trigonometric Functions

Beyond sine and cosine, there are four more trigonometric functions: tangent, cotangent, secant, and cosecant. Each one is built from sine and cosine using ratios or reciprocals. This section covers how to find their exact values, how to handle angles outside the first quadrant, and the key identities that tie everything together.

Exact Values of Trigonometric Functions

Since you already know sine and cosine values from the unit circle, you can find the other four trig functions using these definitions:

• Tangent is the ratio of sine to cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Cotangent is the reciprocal of tangent (or cosine over sine): $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• Secant is the reciprocal of cosine: $\sec \theta = \frac{1}{\cos \theta}$
• Cosecant is the reciprocal of sine: $\csc \theta = \frac{1}{\sin \theta}$

Notice that tangent and secant are undefined whenever $\cos \theta = 0$, and cotangent and cosecant are undefined whenever $\sin \theta = 0$.

Here are some worked examples at common angles:

• $\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$, since both sine and cosine of $\frac{\pi}{4}$ are $\frac{\sqrt{2}}{2}$
• $\tan \frac{\pi}{3} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$
• $\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$
• $\csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$
• $\cot \frac{\pi}{6} = \frac{\cos \frac{\pi}{6}}{\sin \frac{\pi}{6}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$

The pattern is always the same: start with the sine and cosine values you've memorized from the unit circle, then plug them into the appropriate ratio or reciprocal.

Reference Angles Beyond the First Quadrant

A reference angle is the acute angle formed between the terminal side of your angle and the x-axis. It tells you which first-quadrant values to use. The quadrant then tells you the sign.

To find a reference angle:

1. Determine which quadrant the angle falls in.
2. Use the appropriate formula:
   • Quadrant II: reference angle = $\pi - \theta$
   • Quadrant III: reference angle = $\theta - \pi$
   • Quadrant IV: reference angle = $2\pi - \theta$

For example, $\frac{5\pi}{4}$ is in Quadrant III, so the reference angle is $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$.

Next, determine the sign using the quadrant. A helpful mnemonic is "All Students Take Calculus" (ASTC), moving counterclockwise from Quadrant I:

1. Quadrant I ($0$ to $\frac{\pi}{2}$): All functions positive
2. Quadrant II ($\frac{\pi}{2}$ to $\pi$): Sine and cosecant positive
3. Quadrant III ($\pi$ to $\frac{3\pi}{2}$): Tangent and cotangent positive
4. Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): Cosine and secant positive

Putting it together: to evaluate $\cos\left(\frac{5\pi}{4}\right)$, you find the reference angle $\frac{\pi}{4}$, recall $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, and note that cosine is negative in Quadrant III. So $\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$.

Even vs. Odd Trigonometric Functions

Even functions satisfy $f(-\theta) = f(\theta)$. Their graphs are symmetric about the y-axis.

• Cosine and secant are even: $\cos(-\theta) = \cos(\theta)$ and $\sec(-\theta) = \sec(\theta)$

Odd functions satisfy $f(-\theta) = -f(\theta)$. Their graphs are symmetric about the origin.

• Sine, cosecant, tangent, and cotangent are all odd:
  • $\sin(-\theta) = -\sin(\theta)$
  • $\csc(-\theta) = -\csc(\theta)$
  • $\tan(-\theta) = -\tan(\theta)$
  • $\cot(-\theta) = -\cot(\theta)$

This is useful for simplifying expressions. For instance, if you need $\sin(-\frac{\pi}{6})$, you don't need the unit circle again. Since sine is odd, it's just $-\sin(\frac{\pi}{6}) = -\frac{1}{2}$.

Trigonometric Identities and Technology

Fundamental Trigonometric Identities

These identities are true for all values of $\theta$ where the expressions are defined. You'll use them constantly to simplify expressions and verify equations.

Reciprocal Identities:

• $\sin \theta = \frac{1}{\csc \theta}$
• $\cos \theta = \frac{1}{\sec \theta}$
• $\tan \theta = \frac{1}{\cot \theta}$

Quotient Identities:

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $1 + \tan^2 \theta = \sec^2 \theta$
• $1 + \cot^2 \theta = \csc^2 \theta$

The second and third Pythagorean identities are derived from the first. If you divide $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$, you get the tangent-secant version. Dividing by $\sin^2 \theta$ gives the cotangent-cosecant version. Knowing this means you only need to memorize the first one.

Technology in Trigonometric Evaluation

Scientific calculators, spreadsheets, and programming languages can all compute trig values. The most common mistake is being in the wrong angle mode, so always check that first.

Calculators:

• Make sure you're in degree mode for degree inputs and radian mode for radian inputs.
• Most calculators have $\sin$, $\cos$, and $\tan$ buttons. For $\csc$, $\sec$, and $\cot$, compute the reciprocal manually (e.g., for $\sec \theta$, calculate $1 \div \cos \theta$).

Spreadsheets (Excel, Google Sheets):

• Use =SIN(), =COS(), and =TAN(). These functions expect the angle in radians.
• For $\sin\left(\frac{\pi}{3}\right)$, enter =SIN(PI()/3).
• There are no built-in CSC, SEC, or COT functions. Use reciprocals instead: =1/COS(PI()/6) for $\sec\left(\frac{\pi}{6}\right)$.

Python:

• Use the math module: math.sin(), math.cos(), math.tan(). Angles must be in radians.
• Python's math module does not have csc, sec, or cot functions. Use reciprocals: 1/math.cos(2*math.pi/3) for $\sec\left(\frac{2\pi}{3}\right)$.