This connection allows us to evaluate trigonometric functions for angles much larger than 90^o, as well as negative angles.

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The Six Functions on the Unit Circle

If a point on the unit circle is: (x, y) then:

sinθ = y

cosθ = x

and:

\( tan \theta = \frac{y}{x} \)

The reciprocal functions become:

\( csc \theta = \frac{1}{y} \)

\( sec \theta = \frac{1}{x} \)

\( cot \theta = \frac{x}{y} \)

So all six trigonometric functions can be determined from the coordinates of a point on the unit circle.

---

Example Using Coordinates

Suppose a point on the unit circle is:

\( ( \frac{3}{5}, \frac{4}{5}) \)

$\left(\frac35,\frac45\right)$

Then:

\( cos \theta = \frac{3}{5} \)

and:

\( sin \theta = \frac{4}{5} \)

Therefore:

\( tan \theta = \frac{sin \theta }{cos \theta } = \frac{ \frac{4}{5} }{ \frac{3}{5} } = \frac{4}{3} \)

The reciprocal functions are:

$\csc\theta=\frac54$$\sec\theta=\frac53$$\cot\theta=\frac34$

So:

$\boxed{ \sin\theta=\frac45,\quad \cos\theta=\frac35,\quad \tan\theta=\frac43 }$$\boxed{ \csc\theta=\frac54,\quad \sec\theta=\frac53,\quad \cot\theta=\frac34 }$

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Important Angles

Several angles appear repeatedly in trigonometry.

The most important are:

$0^\circ,\quad30^\circ,\quad45^\circ,\quad60^\circ,\quad90^\circ$

Their radian equivalents are:

$0,\quad\frac{\pi}{6},\quad\frac{\pi}{4},\quad\frac{\pi}{3},\quad\frac{\pi}{2}$

Students should become familiar with their exact sine and cosine values.

---

Exact Values

An exact value is written without rounding to a decimal.

For example:

$\sin60^\circ=\frac{\sqrt3}{2}$

is exact.

A calculator might give:

$\sin60^\circ\approx0.866$

This is an approximation.

When a question asks for an exact value, answers should normally remain as fractions and radicals.

---

Special Triangles

The exact values on the unit circle come from two important right triangles.

45°–45°–90° Triangle

Its side ratio is:

$1:1:\sqrt2$

Therefore:

$\sin45^\circ=\frac{1}{\sqrt2} =\frac{\sqrt2}{2}$

and:

$\cos45^\circ=\frac{\sqrt2}{2}$

30°–60°–90° Triangle

Its side ratio is:

$1:\sqrt3:2$

Therefore:

$\sin30^\circ=\frac12$$\cos30^\circ=\frac{\sqrt3}{2}$$\sin60^\circ=\frac{\sqrt3}{2}$$\cos60^\circ=\frac12$

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The Four Quadrants

The coordinate plane is divided into four quadrants.

The signs of sine, cosine, and tangent depend on the quadrant containing the terminal side of the angle.

Quadrant I

$\sin+,\quad\cos+,\quad\tan+$

Quadrant II

$\sin+,\quad\cos-,\quad\tan-$

Quadrant III

$\sin-,\quad\cos-,\quad\tan+$

Quadrant IV

$\sin-,\quad\cos+,\quad\tan-$

A common memory aid is ASTC:

• Quadrant I – All positive
• Quadrant II – Sine positive
• Quadrant III – Tangent positive
• Quadrant IV – Cosine positive

The reciprocal functions have the same signs as their corresponding functions:

• cosecant has the same sign as sine
• secant has the same sign as cosine
• cotangent has the same sign as tangent

---

Reference Angles

A reference angle is the positive acute angle formed between the terminal side of an angle and the

$x$

-axis.

Reference angles allow us to use familiar exact values for angles in any quadrant.

For example:

$150^\circ$

is in Quadrant II.

Its reference angle is:

$180^\circ-150^\circ=30^\circ$

We therefore use the values associated with

$30^\circ$

.

---

Example: Evaluate

$\sin150^\circ$

Reference angle:

$30^\circ$

We know:

$\sin30^\circ=\frac12$

Sine is positive in Quadrant II.

Therefore:

$\boxed{\sin150^\circ=\frac12}$

---

Example: Evaluate

$\cos150^\circ$

Again, the reference angle is:

$30^\circ$

We know:

$\cos30^\circ=\frac{\sqrt3}{2}$

But cosine is negative in Quadrant II.

Therefore:

$\boxed{\cos150^\circ=-\frac{\sqrt3}{2}}$

---

Example: Evaluate

$\tan225^\circ$

The angle

$225^\circ$

lies in Quadrant III.

Its reference angle is:

$225^\circ-180^\circ=45^\circ$

We know:

$\tan45^\circ=1$

Tangent is positive in Quadrant III.

Therefore:

$\boxed{\tan225^\circ=1}$

---

Finding Reference Angles

For angles between

$0^\circ$

and

$360^\circ$

:

Quadrant I

$\theta_R=\theta$

Quadrant II

$\theta_R=180^\circ-\theta$

Quadrant III

$\theta_R=\theta-180^\circ$

Quadrant IV

$\theta_R=360^\circ-\theta$

where

$\theta_R$

is the reference angle.

---

Evaluating All Six Functions

Suppose:

$\theta=120^\circ$

Step 1 – Find the reference angle

$180^\circ-120^\circ=60^\circ$

Step 2 – Identify the quadrant

$120^\circ$

is in Quadrant II.

Therefore:

• sine is positive
• cosine is negative
• tangent is negative

Step 3 – Find the first three functions

$\sin120^\circ=\frac{\sqrt3}{2}$$\cos120^\circ=-\frac12$$\tan120^\circ=-\sqrt3$

Step 4 – Use reciprocals

$\csc120^\circ=\frac{2}{\sqrt3} =\frac{2\sqrt3}{3}$$\sec120^\circ=-2$$\cot120^\circ=-\frac{1}{\sqrt3} =-\frac{\sqrt3}{3}$

Therefore all six functions can be determined from the reference angle and quadrant.

---

Domains and Ranges

A function's domain tells us which input values are allowed.

Its range tells us which output values are possible.

The six trigonometric functions have different domains and ranges.

---

Sine

Because sine represents the

$y$

-coordinate on the unit circle:

$-1\leq\sin\theta\leq1$

Domain

$\boxed{\text{All real numbers}}$

Range

$\boxed{[-1,1]}$

---

Cosine

Cosine represents the

$x$

-coordinate on the unit circle.

Therefore:

$-1\leq\cos\theta\leq1$

Domain

$\boxed{\text{All real numbers}}$

Range

$\boxed{[-1,1]}$

---

Tangent

Tangent can be written as:

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

Division by zero is undefined.

Therefore tangent is undefined whenever:

$\cos\theta=0$

This occurs at:

$90^\circ,\quad270^\circ,\quad450^\circ,\ldots$

or generally:

$\theta=90^\circ+180^\circ k$

where

$k$

is any integer.

Domain

All real numbers except:

$\boxed{\theta=\frac{\pi}{2}+k\pi}$

Range

$\boxed{(-\infty,\infty)}$

---

Cosecant

Since:

$\csc\theta=\frac{1}{\sin\theta}$

cosecant is undefined whenever:

$\sin\theta=0$

Therefore:

Domain

All real numbers except:

$\boxed{\theta=k\pi}$

Range

Since sine lies between

$-1$

and

$1$

, cosecant cannot have values strictly between

$-1$

and

$1$

:

$\boxed{(-\infty,-1]\cup[1,\infty)}$

---

Secant

Since:

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

secant is undefined whenever:

$\cos\theta=0$

Domain

All real numbers except:

$\boxed{\theta=\frac{\pi}{2}+k\pi}$

Range

$\boxed{(-\infty,-1]\cup[1,\infty)}$

---

Cotangent

Since:

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

cotangent is undefined whenever:

$\sin\theta=0$

Domain

All real numbers except:

$\boxed{\theta=k\pi}$

Range

$\boxed{(-\infty,\infty)}$

---

Comparing Domains and Ranges

Here:

$k\in\mathbb{Z}$

means

$k$

can be any integer.

---

Connecting Geometry and Algebra

Trigonometric functions can be understood in several different ways.

Geometrically

Using right triangles:

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$

Using the Unit Circle

Using coordinates:

$(\cos\theta,\sin\theta)$

Algebraically

Using identities such as:

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

and:

$\sin^2\theta+\cos^2\theta=1$

Graphically

Using graphs such as:

$y=\sin x$

and:

$y=\cos x$

These are not separate ideas. They are different representations of the same mathematical relationships.

---

Connecting the Unit Circle to the Sine Graph

Imagine a point moving counterclockwise around the unit circle.

Its vertical coordinate is:

$\sin\theta$

As the point travels around the circle, its vertical position changes:

$0\rightarrow1\rightarrow0\rightarrow-1\rightarrow0$

These values produce the familiar sine wave.

The sine graph is therefore closely connected to circular motion.

---

Connecting the Unit Circle to the Cosine Graph

Cosine represents the horizontal coordinate of the moving point.

As the point travels around the unit circle:

$1\rightarrow0\rightarrow-1\rightarrow0\rightarrow1$

These values create the cosine graph.

This explains why sine and cosine have the same general shape but are shifted relative to one another.

---

Periodic Behaviour

Trigonometric functions repeat their values.

This is called periodicity.

Sine and cosine repeat every:

$360^\circ$

or:

$2\pi$

Therefore:

$\sin(\theta+360^\circ)=\sin\theta$

and:

$\cos(\theta+360^\circ)=\cos\theta$

Tangent and cotangent repeat every:

$180^\circ$

or:

$\pi$

Therefore:

$\tan(\theta+180^\circ)=\tan\theta$

---

Example: Angles Greater Than 360°

Evaluate:

$\cos420^\circ$

Subtract one complete rotation:

$420^\circ-360^\circ=60^\circ$

Therefore:

$\cos420^\circ=\cos60^\circ$$\boxed{\cos420^\circ=\frac12}$

This demonstrates how periodicity allows us to simplify large angles.

---

Negative Angles

Negative angles are measured clockwise from the positive

$x$

-axis.

For example:

$-45^\circ$

ends in Quadrant IV.

Therefore:

$\sin(-45^\circ)=-\frac{\sqrt2}{2}$

while:

$\cos(-45^\circ)=\frac{\sqrt2}{2}$

This gives the useful relationships:

$\boxed{\sin(-\theta)=-\sin\theta}$

and:

$\boxed{\cos(-\theta)=\cos\theta}$

Sine is an odd function, while cosine is an even function.

---

A Complete Example

Evaluate exactly:

$\sec315^\circ$

Step 1 – Identify the quadrant

$315^\circ$

lies in Quadrant IV.

Step 2 – Find the reference angle

$360^\circ-315^\circ=45^\circ$

Step 3 – Determine cosine

Cosine is positive in Quadrant IV:

$\cos315^\circ=\frac{\sqrt2}{2}$

Step 4 – Take the reciprocal

$\sec315^\circ= \frac{1}{\frac{\sqrt2}{2}}$$=\frac{2}{\sqrt2}$$=\sqrt2$

Therefore:

$\boxed{\sec315^\circ=\sqrt2}$

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Another Complete Example

Evaluate exactly:

$\csc240^\circ$

Reference angle:

$240^\circ-180^\circ=60^\circ$

The angle lies in Quadrant III, where sine is negative.

Therefore:

$\sin240^\circ=-\frac{\sqrt3}{2}$

Take the reciprocal:

$\csc240^\circ=-\frac{2}{\sqrt3}$

Rationalize:

$\boxed{\csc240^\circ=-\frac{2\sqrt3}{3}}$

---

A Strategy for Exact Trigonometric Values

When asked to find an exact value:

Step 1: Determine the quadrant.

Step 2: Find the reference angle.

Step 3: Recall the exact value for

$30^\circ$

,

$45^\circ$

, or

$60^\circ$

.

Step 4: Determine whether the answer should be positive or negative.

Step 5: Use a reciprocal if the question asks for

$\csc$

,

$\sec$

, or

$\cot$

.

Step 6: Simplify the answer exactly rather than converting to a decimal.

---

Common Misconceptions

SOH-CAH-TOA is not limited to the whole study of trigonometry.

It is extremely useful for right triangles, but the unit circle extends trigonometric functions to much broader sets of angles.

Tangent is not always defined.

Since:

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

it is undefined whenever:

$\cos\theta=0$

Sine and cosine cannot be larger than 1 or smaller than -1.

Their values are coordinates on a circle of radius 1.

Reference angles are always positive acute angles.

They describe the angle between the terminal side and the

$x$

-axis.

Exact values should not automatically be converted to decimals.

For example:

$\frac{\sqrt3}{2}$

is usually preferred to:

$0.866$

when an exact value is requested.

---

Did You Know?

Trigonometric graphs appear throughout science because many natural processes are periodic.

Sine and cosine functions can model:

• sound waves
• light waves
• alternating electrical current
• ocean tides
• vibrations
• rotating objects
• seasonal cycles
• pendulum motion

The unit circle, triangle ratios, and sine-wave graphs may look like different topics at first, but they are all representations of the same underlying trigonometric relationships.

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Key Terms

Trigonometric function – A function relating angles to ratios or coordinates.

Unit circle – A circle of radius 1 centred at the origin.

Sine – The

$y$

-coordinate of a point on the unit circle.

Cosine – The

$x$

-coordinate of a point on the unit circle.

Tangent – The ratio

$\sin\theta/\cos\theta$

.

Cosecant – The reciprocal of sine.

Secant – The reciprocal of cosine.

Cotangent – The reciprocal of tangent.

Reference angle – The positive acute angle between an angle's terminal side and the

$x$

-axis.

Exact value – A value expressed exactly using fractions, integers, or radicals rather than a rounded decimal.

Domain – The possible inputs of a function.

Range – The possible outputs of a function.

Period – The interval after which a function repeats its values.

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Key Takeaways

• There are six trigonometric functions:

$\boxed{\sin,\cos,\tan,\csc,\sec,\cot}$

• On the unit circle:

$\boxed{(\cos\theta,\sin\theta)}$

• The reciprocal relationships are:

$\csc\theta=\frac1{\sin\theta}$$\sec\theta=\frac1{\cos\theta}$$\cot\theta=\frac1{\tan\theta}$

• Exact values for$30^\circ$,$45^\circ$, and$60^\circ$can be determined using special triangles.
• Reference angles allow exact values to be found in all four quadrants.
• The signs of trig functions depend on the quadrant.
• Sine and cosine have range:

$[-1,1]$

• Tangent and cotangent have all real numbers as their range but have restrictions in their domains.
• Secant and cosecant have range:

$(-\infty,-1]\cup[1,\infty)$

• Sine, cosine, secant, and cosecant repeat every$2\pi$.
• Tangent and cotangent repeat every$\pi$.
• Trigonometric ideas can be represented geometrically, algebraically, graphically, and through the unit circle.
• Understanding the connections between these representations provides the foundation for more advanced work with trigonometric identities, equations, functions, and calculus.