Unit Circle with Secant, Cosecant, and Cotangent Revision Notes for HSC SSCE Mathematics Advanced

Unit Circle with Secant, Cosecant, and Cotangent

Understanding reciprocal trigonometric functions

The unit circle provides a powerful way to understand the three reciprocal trigonometric functions: secant, cosecant, and cotangent. These functions are defined for any circle centered at the origin, not just the unit circle.

Definitions for a circle of radius r

When we have a circle of radius $r$ centered at the origin, and we measure an angle $\theta$ anticlockwise from the positive $x$ -axis to reach a point $(x, y)$ on the circle, we define three reciprocal trigonometric functions:

Secant is the reciprocal of the cosine function. The formula is:

$\sec \theta = \frac{r}{x}$

where:

$\sec \theta$ is the secant of angle $\theta$
$r$ is the radius of the circle
$x$ is the $x$ -coordinate of the point on the circle

Cosecant is the reciprocal of the sine function. The formula is:

$\text{cosec } \theta = \frac{r}{y}$

where:

$\text{cosec } \theta$ is the cosecant of angle $\theta$
$r$ is the radius of the circle
$y$ is the $y$ -coordinate of the point on the circle

Cotangent is the reciprocal of the tangent function. The formula is:

$\cot \theta = \frac{x}{y}$

where:

$\cot \theta$ is the cotangent of angle $\theta$
$x$ is the $x$ -coordinate of the point on the circle
$y$ is the $y$ -coordinate of the point on the circle

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The radius can be calculated using the distance formula: $r = \sqrt{x^2 + y^2}$ . This relationship is fundamental to understanding how these functions work on circles of different sizes.

Simplified formulas for the unit circle

For the unit circle (where $r = 1$ ), these formulas become simpler:

$\sec \theta = \frac{1}{x}$

$\text{cosec } \theta = \frac{1}{y}$

$\cot \theta = \frac{x}{y}$

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These simplified formulas are particularly useful because we often work with the unit circle when finding exact values of trigonometric functions. With $r = 1$ , the calculations become much more straightforward, making it easier to work with special angles and memorize key values.

When are these functions undefined?

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Each reciprocal function becomes undefined when its denominator equals zero. This is a crucial concept to understand:

Secant ( $\sec \theta = \frac{1}{x}$ ) is undefined when $x = 0$ , which occurs at $\theta = 90°$ and $\theta = 270°$ (or $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ in radians)
Cosecant ( $\text{cosec } \theta = \frac{1}{y}$ ) is undefined when $y = 0$ , which occurs at $\theta = 0°$ and $\theta = 180°$ (or $0$ and $\pi$ in radians)
Cotangent ( $\cot \theta = \frac{x}{y}$ ) is undefined when $y = 0$ , which occurs at $\theta = 0°$ and $\theta = 180°$ (or $0$ and $\pi$ in radians)

Always check whether the denominator is zero before calculating these functions!

Exact values for $\frac{\pi}{6}$ multiples

When working with angles that are integer multiples of $\frac{\pi}{6}$ (such as $0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}$ , etc.), we can find exact values for the reciprocal trigonometric functions using the unit circle coordinates.

Unit circle coordinates for $\frac{\pi}{6}$ and $\frac{\pi}{3}$ multiples

These special angles correspond to 30° and 60° intervals. The diagram below shows the key coordinates around the unit circle:

Key coordinates to remember include:

At $\frac{\pi}{6}$ (30°): $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
At $\frac{\pi}{3}$ (60°): $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
At $\frac{\pi}{2}$ (90°): $(0, 1)$

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Notice that the coordinates for $\frac{\pi}{6}$ and $\frac{\pi}{3}$ are related - the $x$ and $y$ values are swapped! This reflects the complementary nature of these angles. Memorizing one set helps you remember the other.

Worked example: Finding reciprocal trig values

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Worked Example: Finding sec θ, cosec θ, and cot θ for a point on the unit circle

Let's find the exact values of $\sec \theta$ , $\text{cosec } \theta$ , and $\cot \theta$ for an angle $\theta$ on the unit circle that passes through the point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ .

Finding $\sec \theta$ :

We use the formula $\sec \theta = \frac{1}{x}$ with $x = \frac{\sqrt{3}}{2}$ .

$\sec \theta = \frac{1}{x}$

Substitute $x = \frac{\sqrt{3}}{2}$ :

$= \frac{1}{\frac{\sqrt{3}}{2}}$

Simplify by multiplying by the reciprocal:

$= \frac{2}{\sqrt{3}}$

Rationalise the denominator by multiplying numerator and denominator by $\sqrt{3}$ :

$= \frac{2\sqrt{3}}{3}$

Finding $\text{cosec } \theta$ :

We use the formula $\text{cosec } \theta = \frac{1}{y}$ with $y = \frac{1}{2}$ .

$\text{cosec } \theta = \frac{1}{y}$

Substitute $y = \frac{1}{2}$ :

$= \frac{1}{\frac{1}{2}}$

Simplify:

$= 2$

Finding $\cot \theta$ :

We use the formula $\cot \theta = \frac{x}{y}$ with $x = \frac{\sqrt{3}}{2}$ and $y = \frac{1}{2}$ .

$\cot \theta = \frac{x}{y}$

Substitute $x = \frac{\sqrt{3}}{2}$ and $y = \frac{1}{2}$ :

$= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$

Simplify by multiplying by the reciprocal:

$= \sqrt{3}$

Additional worked examples for $\frac{\pi}{6}$ multiples

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Worked Example: Finding $\sec\left(\frac{\pi}{3}\right)$

From the unit circle, at $\frac{\pi}{3}$ , we have $x = \frac{1}{2}$ .

$\sec\left(\frac{\pi}{3}\right) = \frac{1}{x}$

$= \frac{1}{\frac{1}{2}}$

$= 2$

Therefore, $\sec\left(\frac{\pi}{3}\right) = 2$ .

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Worked Example: Finding $\text{cosec}\left(\frac{\pi}{6}\right)$

From the unit circle, at $\frac{\pi}{6}$ , we have $y = \frac{1}{2}$ .

$\text{cosec}\left(\frac{\pi}{6}\right) = \frac{1}{y}$

$= \frac{1}{\frac{1}{2}}$

$= 2$

Therefore, $\text{cosec}\left(\frac{\pi}{6}\right) = 2$ .

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Worked Example: Finding $\cot\left(\frac{\pi}{2}\right)$

From the unit circle, at $\frac{\pi}{2}$ , we have $x = 0$ and $y = 1$ .

$\cot\left(\frac{\pi}{2}\right) = \frac{x}{y}$

$= \frac{0}{1}$

$= 0$

Therefore, $\cot\left(\frac{\pi}{2}\right) = 0$ .

Exact values for $\frac{\pi}{4}$ multiples

Angles that are integer multiples of $\frac{\pi}{4}$ (such as $\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$ , etc.) also have exact values for the reciprocal trigonometric functions.

Unit circle coordinates for $\frac{\pi}{4}$ multiples

These special angles correspond to 45° intervals. The diagram below shows the coordinates:

Key coordinates to remember:

At $\frac{\pi}{4}$ (45°): $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
At $\frac{3\pi}{4}$ (135°): $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
At $\frac{5\pi}{4}$ (225°): $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
At $\frac{7\pi}{4}$ (315°): $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

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For angles that are multiples of $\frac{\pi}{4}$ , both coordinates have the same absolute value: $\frac{\sqrt{2}}{2}$ . The signs change depending on the quadrant, but the magnitude remains constant. This pattern makes these angles particularly easy to work with.

Worked examples for $\frac{\pi}{4}$ multiples

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Worked Example: Finding $\sec\left(\frac{\pi}{4}\right)$

From the unit circle, at $\frac{\pi}{4}$ , we have $x = \frac{\sqrt{2}}{2}$ .

$\sec\left(\frac{\pi}{4}\right) = \frac{1}{x}$

Substitute $x = \frac{\sqrt{2}}{2}$ :

$= \frac{1}{\frac{\sqrt{2}}{2}}$

Simplify:

$= \frac{2}{\sqrt{2}}$

Rationalise:

$= \frac{2\sqrt{2}}{2} = \sqrt{2}$

Therefore, $\sec\left(\frac{\pi}{4}\right) = \sqrt{2}$ .

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Worked Example: Finding $\text{cosec}\left(\frac{\pi}{4}\right)$

From the unit circle, at $\frac{\pi}{4}$ , we have $y = \frac{\sqrt{2}}{2}$ .

$\text{cosec}\left(\frac{\pi}{4}\right) = \frac{1}{y}$

Substitute $y = \frac{\sqrt{2}}{2}$ :

$= \frac{1}{\frac{\sqrt{2}}{2}}$

Simplify and rationalise:

$= \sqrt{2}$

Therefore, $\text{cosec}\left(\frac{\pi}{4}\right) = \sqrt{2}$ .

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Worked Example: Finding $\cot\left(\frac{3\pi}{4}\right)$

From the unit circle, at $\frac{3\pi}{4}$ , we have $x = -\frac{\sqrt{2}}{2}$ and $y = \frac{\sqrt{2}}{2}$ .

$\cot\left(\frac{3\pi}{4}\right) = \frac{x}{y}$

Substitute $x = -\frac{\sqrt{2}}{2}$ and $y = \frac{\sqrt{2}}{2}$ :

$= \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$

Simplify:

$= -1$

Therefore, $\cot\left(\frac{3\pi}{4}\right) = -1$ .

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When working with angles in the second quadrant (like $\frac{3\pi}{4}$ ), remember that the $x$ -coordinate is negative while the $y$ -coordinate is positive. This affects the signs of your final answers, particularly for cotangent.

Exam tips

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Helpful strategies for solving reciprocal trigonometric function problems:

Identify the coordinates from the unit circle for the given angle
Apply the appropriate formula: $\sec \theta = \frac{1}{x}$ , $\text{cosec } \theta = \frac{1}{y}$ , or $\cot \theta = \frac{x}{y}$
Substitute the values carefully, paying attention to signs
Rationalise denominators when they contain square roots
Check for undefined values by ensuring denominators are not zero

Remember: Always verify that your denominator is not zero before calculating, as this will result in an undefined value.

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Key Points to Remember:

Secant is the reciprocal of cosine: $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{x}$ on the unit circle
Cosecant is the reciprocal of sine: $\text{cosec } \theta = \frac{1}{\sin \theta} = \frac{1}{y}$ on the unit circle
Cotangent is the reciprocal of tangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{x}{y}$ on the unit circle
These functions are undefined when their denominators equal zero: sec θ when $x = 0$ , cosec θ when $y = 0$ , and cot θ when $y = 0$
Use unit circle coordinates for special angles (multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ ) to find exact values efficiently
When simplifying, remember to rationalise denominators that contain square roots

Unit Circle with Secant, Cosecant, and Cotangent (HSC SSCE Mathematics Advanced): Revision Notes