The Central Idea
The unit circle is the circle centered at the origin with radius \(1\).
Its equation is
\[ \boxed{x^2+y^2=1}. \]If an angle \(\theta\) in standard position meets the unit circle at the point
\[ P=(x,y), \]then
\[ \boxed{\cos\theta=x} \]and
\[ \boxed{\sin\theta=y}. \]The unit circle therefore converts angle measure directly into coordinates.
Why Radius \(1\) Is So Useful
In a right triangle,
\[ \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \]and
\[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}. \]On the unit circle, the hypotenuse-like radius has length \(1\).
Therefore,
\[ \cos\theta=\frac{x}{1}=x \]and
\[ \sin\theta=\frac{y}{1}=y. \]The choice of radius \(1\) removes the denominator and makes the trigonometric values equal to the coordinates themselves.
The Coordinate Plane
The unit circle is centered at
\[ (0,0). \]It intersects the coordinate axes at four fundamental points:
\[ (1,0),\qquad (0,1),\qquad (-1,0),\qquad (0,-1). \]These points correspond to quarter-turn positions around the circle.
Angles in Standard Position
An angle is in standard position when:
- its vertex is at the origin;
- its initial side lies on the positive \(x\)-axis;
- its terminal side rotates to the angle's final position.
The point where the terminal side meets the unit circle determines the trigonometric values of the angle.
Positive Rotation
By convention, counterclockwise rotation represents a positive angle.
Starting from the positive \(x\)-axis:
\[ 0^\circ \]is the starting direction,
\[ 90^\circ \]is one quarter-turn,
\[ 180^\circ \]is one half-turn,
\[ 270^\circ \]is three quarters of a turn, and
\[ 360^\circ \]completes one full revolution.
Negative Rotation
Clockwise rotation represents a negative angle.
For example,
\[ -90^\circ \]rotates one quarter-turn clockwise from the positive \(x\)-axis.
Its terminal side is the same as the terminal side of
\[ 270^\circ. \]Different angle measures can therefore end at the same point on the unit circle.
The Four Quadrants
The coordinate axes divide the plane into four quadrants.
- Quadrant I: \(x>0,\ y>0\)
- Quadrant II: \(x<0,\ y>0\)
- Quadrant III: \(x<0,\ y<0\)
- Quadrant IV: \(x>0,\ y<0\)
Because
\[ \cos\theta=x \]and
\[ \sin\theta=y, \]the coordinate signs immediately determine the signs of cosine and sine.
Sine as the \(y\)-Coordinate
If the terminal point is
\[ P=(x,y), \]then
\[ \boxed{\sin\theta=y}. \]Sine therefore measures the vertical coordinate of the point on the unit circle.
Points above the \(x\)-axis have positive sine, while points below the \(x\)-axis have negative sine.
Cosine as the \(x\)-Coordinate
For the same terminal point
\[ P=(x,y), \]we have
\[ \boxed{\cos\theta=x}. \]Cosine measures the horizontal coordinate of the point.
Points to the right of the \(y\)-axis have positive cosine, while points to the left have negative cosine.
Tangent on the Unit Circle
Recall
\[ \tan\theta=\frac{\sin\theta}{\cos\theta}. \]Since
\[ \sin\theta=y \qquad\text{and}\qquad \cos\theta=x, \]tangent becomes
\[ \boxed{\tan\theta=\frac{y}{x}} \]whenever
\[ x\neq0. \]The Six Functions from Coordinates
If
\[ P=(x,y) \]lies on the unit circle, then
\[ \boxed{ \begin{aligned} \sin\theta&=y, & \csc\theta&=\frac1y,\\[4pt] \cos\theta&=x, & \sec\theta&=\frac1x,\\[4pt] \tan\theta&=\frac{y}{x}, & \cot\theta&=\frac{x}{y}. \end{aligned} } \]Each reciprocal expression requires a nonzero denominator.
The Pythagorean Identity from the Circle
Every point on the unit circle satisfies
\[ x^2+y^2=1. \]Replace \(x\) with \(\cos\theta\) and \(y\) with \(\sin\theta\):
\[ \cos^2\theta+\sin^2\theta=1. \]Therefore,
\[ \boxed{\sin^2\theta+\cos^2\theta=1}. \]The Pythagorean trigonometric identity is built directly into the geometry of the unit circle.
The Axis Angles
Four important angles place the terminal side directly on a coordinate axis:
\[ 0^\circ,\qquad 90^\circ,\qquad 180^\circ,\qquad 270^\circ. \]A full turn returns to
\[ 360^\circ, \]which has the same terminal point as \(0^\circ\).
At \(0^\circ\)
The terminal point is
\[ (1,0). \]Therefore,
\[ \boxed{\cos0^\circ=1} \]and
\[ \boxed{\sin0^\circ=0}. \]Tangent is
\[ \tan0^\circ=\frac01=0. \]At \(90^\circ\)
The terminal point is
\[ (0,1). \]Therefore,
\[ \boxed{\cos90^\circ=0} \]and
\[ \boxed{\sin90^\circ=1}. \]Tangent would require
\[ \frac10, \]so
\[ \boxed{\tan90^\circ\text{ is undefined}}. \]At \(180^\circ\)
The terminal point is
\[ (-1,0). \]Thus,
\[ \boxed{\cos180^\circ=-1} \]and
\[ \boxed{\sin180^\circ=0}. \]Therefore,
\[ \boxed{\tan180^\circ=0}. \]At \(270^\circ\)
The terminal point is
\[ (0,-1). \]Therefore,
\[ \boxed{\cos270^\circ=0} \]and
\[ \boxed{\sin270^\circ=-1}. \]Since cosine is zero,
\[ \boxed{\tan270^\circ\text{ is undefined}}. \]At \(360^\circ\)
One full counterclockwise revolution returns to
\[ (1,0). \]Therefore,
\[ \boxed{\cos360^\circ=1}, \] \[ \boxed{\sin360^\circ=0}, \]and
\[ \boxed{\tan360^\circ=0}. \]The values repeat because the terminal point repeats.
The \(45^\circ\) Point
At \(45^\circ\), the horizontal and vertical coordinates are equal.
Let
\[ x=y. \]Since the point lies on the unit circle,
\[ x^2+y^2=1. \]Therefore,
\[ 2x^2=1, \] \[ x^2=\frac12, \]and in Quadrant I,
\[ x=y=\frac{\sqrt2}{2}. \]So the terminal point is
\[ \boxed{ \left( \frac{\sqrt2}{2}, \frac{\sqrt2}{2} \right) }. \]The \(30^\circ\) Point
From the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle,
\[ \cos30^\circ=\frac{\sqrt3}{2} \]and
\[ \sin30^\circ=\frac12. \]Therefore, the unit-circle point is
\[ \boxed{ \left( \frac{\sqrt3}{2}, \frac12 \right) }. \]The \(60^\circ\) Point
Similarly,
\[ \cos60^\circ=\frac12 \]and
\[ \sin60^\circ=\frac{\sqrt3}{2}. \]Therefore, the unit-circle point is
\[ \boxed{ \left( \frac12, \frac{\sqrt3}{2} \right) }. \]Three Fundamental Quadrant-I Points
The most important exact points in Quadrant I are
\[ \boxed{ \begin{aligned} 30^\circ&: \left(\frac{\sqrt3}{2},\frac12\right),\\[4pt] 45^\circ&: \left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right),\\[4pt] 60^\circ&: \left(\frac12,\frac{\sqrt3}{2}\right). \end{aligned} } \]These coordinate patterns reappear throughout the rest of the circle.
The Coordinate Pattern
Notice how the \(x\)- and \(y\)-coordinates exchange as we move from \(30^\circ\) to \(60^\circ\):
\[ \left(\frac{\sqrt3}{2},\frac12\right) \]becomes
\[ \left(\frac12,\frac{\sqrt3}{2}\right). \]This reflects the complementary relationship
\[ \sin\theta=\cos(90^\circ-\theta). \]Signs in Quadrant I
In Quadrant I,
\[ x>0,\qquad y>0. \]Therefore,
\[ \cos\theta>0 \]and
\[ \sin\theta>0. \]Since tangent is \(y/x\),
\[ \tan\theta>0. \]All six trigonometric functions are positive in Quadrant I.
Signs in Quadrant II
In Quadrant II,
\[ x<0,\qquad y>0. \]Therefore,
\[ \cos\theta<0, \qquad \sin\theta>0. \]Hence,
\[ \tan\theta=\frac{y}{x}<0. \]Cosecant has the same sign as sine, secant has the same sign as cosine, and cotangent has the same sign as tangent.
Signs in Quadrant III
In Quadrant III,
\[ x<0,\qquad y<0. \]Therefore,
\[ \cos\theta<0, \qquad \sin\theta<0. \]But
\[ \tan\theta=\frac{y}{x}>0 \]because a negative divided by a negative is positive.
Signs in Quadrant IV
In Quadrant IV,
\[ x>0,\qquad y<0. \]Therefore,
\[ \cos\theta>0, \qquad \sin\theta<0, \]and
\[ \tan\theta<0. \]A Sign Summary
\[ \begin{array}{c|ccc} \text{Quadrant} & \sin\theta & \cos\theta & \tan\theta\\ \hline \text{I} & + & + & +\\ \text{II} & + & - & -\\ \text{III} & - & - & +\\ \text{IV} & - & + & - \end{array} \]The reciprocal functions always have the same sign as their partners:
\[ \csc\leftrightarrow\sin, \qquad \sec\leftrightarrow\cos, \qquad \cot\leftrightarrow\tan. \]Exact Values beyond Quadrant I
Points elsewhere on the unit circle often have the same coordinate magnitudes as familiar Quadrant-I points, but their signs change according to the quadrant.
For example, a point with coordinate magnitudes
\[ \frac{\sqrt3}{2} \qquad\text{and}\qquad \frac12 \]might appear as
\[ \left( -\frac{\sqrt3}{2}, \frac12 \right) \]in Quadrant II.
The next lesson will formalize this using reference angles.
Example: A Quadrant-II Point
Suppose
\[ P= \left( -\frac{\sqrt3}{2}, \frac12 \right). \]Then
\[ \boxed{\cos\theta=-\frac{\sqrt3}{2}} \]and
\[ \boxed{\sin\theta=\frac12}. \]Tangent is
\[ \tan\theta = \frac{1/2}{-\sqrt3/2} = -\frac1{\sqrt3} = \boxed{-\frac{\sqrt3}{3}}. \]Example: A Quadrant-III Point
Suppose
\[ P= \left( -\frac{\sqrt2}{2}, -\frac{\sqrt2}{2} \right). \]Then
\[ \boxed{\cos\theta=-\frac{\sqrt2}{2}} \]and
\[ \boxed{\sin\theta=-\frac{\sqrt2}{2}}. \]Therefore,
\[ \tan\theta = \frac{-\sqrt2/2}{-\sqrt2/2} = \boxed{1}. \]Example: A Quadrant-IV Point
Suppose
\[ P= \left( \frac12, -\frac{\sqrt3}{2} \right). \]Then
\[ \boxed{\cos\theta=\frac12} \]and
\[ \boxed{\sin\theta=-\frac{\sqrt3}{2}}. \]Thus,
\[ \tan\theta = \frac{-\sqrt3/2}{1/2} = \boxed{-\sqrt3}. \]Unit-Circle Coordinates Must Satisfy the Circle Equation
Suppose someone proposes the point
\[ \left(\frac35,\frac45\right). \]Check:
\[ \left(\frac35\right)^2+ \left(\frac45\right)^2 = \frac9{25}+\frac{16}{25} = 1. \]Therefore the point lies on the unit circle.
Example: Rejecting an Invalid Point
Consider
\[ \left(\frac34,\frac34\right). \]Then
\[ \left(\frac34\right)^2+ \left(\frac34\right)^2 = \frac9{16}+\frac9{16} = \frac{18}{16} \neq1. \]Therefore this point does not lie on the unit circle.
Finding a Missing Coordinate
Suppose a point on the unit circle has
\[ x=\frac35 \]and lies in Quadrant I.
Use
\[ x^2+y^2=1. \]Then
\[ \frac9{25}+y^2=1, \] \[ y^2=\frac{16}{25}. \]Since Quadrant I has positive \(y\),
\[ \boxed{y=\frac45}. \]The Quadrant Determines the Sign
If the same \(x\)-coordinate
\[ x=\frac35 \]belongs to a point in Quadrant IV, then
\[ y^2=\frac{16}{25} \]still gives the magnitude
\[ |y|=\frac45. \]But Quadrant IV has negative \(y\), so
\[ \boxed{y=-\frac45}. \]Finding Trigonometric Values from a Point
Suppose
\[ P=\left(-\frac5{13},\frac{12}{13}\right). \]Then
\[ \boxed{\cos\theta=-\frac5{13}}, \] \[ \boxed{\sin\theta=\frac{12}{13}}, \]and
\[ \boxed{\tan\theta=-\frac{12}{5}}. \]Finding Reciprocal Functions from a Point
For
\[ P=\left(-\frac5{13},\frac{12}{13}\right), \]the reciprocal functions are
\[ \boxed{\sec\theta=-\frac{13}{5}}, \] \[ \boxed{\csc\theta=\frac{13}{12}}, \]and
\[ \boxed{\cot\theta=-\frac5{12}}. \]Undefined Values on the Unit Circle
Coordinate definitions make undefined trigonometric values easy to see.
Since
\[ \tan\theta=\frac{y}{x}, \]tangent is undefined when
\[ x=0. \]Since
\[ \sec\theta=\frac1x, \]secant is also undefined when \(x=0\).
Likewise, cosecant and cotangent are undefined when \(y=0\).
Where Tangent Is Undefined
On the unit circle, \(x=0\) at
\[ (0,1) \]and
\[ (0,-1). \]These correspond to
\[ 90^\circ \]and
\[ 270^\circ \]during one full revolution.
Therefore tangent is undefined at those quarter-turn positions.
Where Cotangent Is Undefined
Cotangent is
\[ \cot\theta=\frac{x}{y}. \]It is undefined when
\[ y=0. \]During one revolution, this occurs at
\[ 0^\circ, \qquad 180^\circ, \qquad 360^\circ. \]Where Secant Is Undefined
Secant is
\[ \sec\theta=\frac1x. \]Therefore secant is undefined where \(x=0\), including
\[ 90^\circ \]and
\[ 270^\circ. \]Where Cosecant Is Undefined
Cosecant is
\[ \csc\theta=\frac1y. \]Therefore cosecant is undefined where \(y=0\), including
\[ 0^\circ, \qquad 180^\circ, \qquad 360^\circ. \]Angles Can Continue beyond One Revolution
Rotation does not stop after
\[ 360^\circ. \]For example,
\[ 450^\circ=360^\circ+90^\circ. \]After one full revolution, the additional \(90^\circ\) places the terminal side at the same position as \(90^\circ\).
Therefore,
\[ \sin450^\circ=\sin90^\circ=1 \]and
\[ \cos450^\circ=\cos90^\circ=0. \]Coterminal Angles
Angles that share the same initial and terminal sides are called coterminal angles.
For example,
\[ 30^\circ, \qquad 390^\circ, \qquad -330^\circ \]all end at the same unit-circle point.
Therefore they have the same sine and cosine values.
Generating Coterminal Angles
In degree measure, adding or subtracting full revolutions produces coterminal angles:
\[ \boxed{\theta+360^\circ k}, \]where \(k\) is any integer.
A later lesson will express the same idea naturally in radians.
Periodicity Begins with Repeated Position
If two angles differ by one complete revolution, they end at the same point on the unit circle.
Therefore,
\[ \boxed{\sin(\theta+360^\circ)=\sin\theta} \]and
\[ \boxed{\cos(\theta+360^\circ)=\cos\theta}. \]This repeating behavior is the geometric source of trigonometric periodicity.
Tangent Repeats More Frequently
Tangent is
\[ \frac{y}{x}. \]A half-turn changes
\[ (x,y) \]to
\[ (-x,-y). \]Therefore,
\[ \frac{-y}{-x}=\frac{y}{x}. \]So tangent already repeats after
\[ 180^\circ. \]This will become important when trigonometric graphs are studied.
Symmetry across the \(x\)-Axis
Reflecting a point
\[ (x,y) \]across the \(x\)-axis gives
\[ (x,-y). \]Therefore cosine stays the same while sine changes sign.
This leads to
\[ \boxed{\cos(-\theta)=\cos\theta} \]and
\[ \boxed{\sin(-\theta)=-\sin\theta}. \]Tangent under Negative Angles
Since
\[ \tan\theta=\frac{\sin\theta}{\cos\theta}, \]we have
\[ \tan(-\theta) = \frac{-\sin\theta}{\cos\theta}. \]Therefore,
\[ \boxed{\tan(-\theta)=-\tan\theta}. \]Even and Odd Trigonometric Behavior
Cosine satisfies
\[ \cos(-\theta)=\cos\theta, \]so cosine is an even function.
Sine and tangent satisfy
\[ \sin(-\theta)=-\sin\theta \]and
\[ \tan(-\theta)=-\tan\theta, \]so sine and tangent are odd functions.
The unit circle makes these algebraic symmetries visible geometrically.
Horizontal and Vertical Projection
The point
\[ (\cos\theta,\sin\theta) \]can be understood as the endpoint of a unit-length vector pointing in direction \(\theta\).
Its horizontal component is
\[ \cos\theta, \]and its vertical component is
\[ \sin\theta. \]This connects the unit circle directly to vector components.
Scaling beyond the Unit Circle
If a vector has length \(r\) instead of \(1\), its endpoint is
\[ \boxed{ (r\cos\theta,\ r\sin\theta) }. \]The unit-circle coordinates therefore serve as direction ratios that can be scaled to any magnitude.
Connection to Circular Motion
Imagine a point moving around the unit circle.
As the angle changes, the point's coordinates change:
\[ x=\cos\theta, \qquad y=\sin\theta. \]The horizontal and vertical positions repeatedly increase, decrease, and reverse direction.
This circular motion is the geometric source of sine and cosine waves.
Why the Unit Circle Matters for Periodicity
Right-triangle trigonometry describes fixed acute angles.
The unit circle adds motion: an angle can rotate continuously around the origin.
Because the circle returns to the same positions again and again, trigonometric values repeat.
This makes the unit circle the bridge from triangle ratios to periodic functions.
Common Error: Reversing the Coordinates
On the unit circle,
\[ \boxed{(\cos\theta,\sin\theta)} \]is the correct coordinate order.
The \(x\)-coordinate is cosine, and the \(y\)-coordinate is sine.
Writing
\[ (\sin\theta,\cos\theta) \]reverses the roles.
Common Error: Ignoring Coordinate Signs
The coordinate magnitudes may match familiar special-angle values, but the quadrant determines the signs.
For example,
\[ \left( -\frac12, \frac{\sqrt3}{2} \right) \]has negative cosine and positive sine.
Exact magnitudes alone are not enough.
Common Error: Treating Tangent as the \(y\)-Coordinate
Sine is the \(y\)-coordinate:
\[ \sin\theta=y. \]Tangent is the ratio
\[ \boxed{\tan\theta=\frac{y}{x}}. \]Tangent is not simply one coordinate of the point.
Common Error: Forgetting Undefined Values
At
\[ (0,1), \]sine and cosine are perfectly defined:
\[ \sin\theta=1, \qquad \cos\theta=0. \]But tangent requires division by the \(x\)-coordinate:
\[ \tan\theta=\frac10, \]so tangent is undefined.
Common Error: Assuming Angles Stop at \(360^\circ\)
An angle may represent any amount of rotation.
Values such as
\[ 450^\circ, \qquad 720^\circ, \qquad -90^\circ \]are all meaningful.
The unit circle naturally handles positive, negative, and multi-revolution angles.
Common Error: Confusing the Circle Equation with a Trigonometric Ratio
The equation
\[ x^2+y^2=1 \]describes the geometric circle.
The relationships
\[ x=\cos\theta \]and
\[ y=\sin\theta \]connect the angle to a point on that circle.
These ideas work together but play different roles.
A Reliable Procedure for Reading the Unit Circle
- Place the angle in standard position.
- Identify the terminal point on the unit circle.
- Read the \(x\)-coordinate as cosine.
- Read the \(y\)-coordinate as sine.
- Compute tangent as \(y/x\) when \(x\neq0\).
- Use reciprocals for secant, cosecant, and cotangent.
- Check the quadrant signs.
- Check undefined cases before dividing.
A Reliable Procedure for Finding a Missing Coordinate
- Use the unit-circle equation \(x^2+y^2=1\).
- Substitute the known coordinate.
- Solve for the square of the missing coordinate.
- Take the square root.
- Use the quadrant to choose the correct sign.
- Verify the completed point in \(x^2+y^2=1\).
A Reliable Procedure for Large or Negative Angles
- Begin from the positive \(x\)-axis.
- Rotate counterclockwise for positive angles and clockwise for negative angles.
- Add or subtract full \(360^\circ\) revolutions when convenient.
- Locate the resulting terminal side.
- Use the terminal point to determine the trigonometric values.
Practice 1: Read a Unit-Circle Point
Suppose
\[ P=\left(\frac35,\frac45\right) \]lies on the terminal side of \(\theta\).
Find \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
Practice 2: Find Reciprocal Functions
For
\[ P=\left(-\frac5{13},\frac{12}{13}\right), \]find \(\csc\theta\), \(\sec\theta\), and \(\cot\theta\).
Practice 3: Verify a Unit-Circle Point
Determine whether
\[ \left(\frac8{17},\frac{15}{17}\right) \]lies on the unit circle.
Practice 4: Find a Missing Coordinate
A point on the unit circle lies in Quadrant II and has
\[ x=-\frac{12}{13}. \]Find \(y\).
Practice 5: Axis Values
Find
\[ \sin180^\circ, \qquad \cos180^\circ, \qquad \tan180^\circ. \]Practice 6: Undefined Value
Explain why
\[ \tan90^\circ \]is undefined using unit-circle coordinates.
Practice 7: Exact Point
Give the exact unit-circle point corresponding to
\[ 45^\circ. \]Practice 8: Coterminal Angles
Give one positive and one negative angle coterminal with
\[ 60^\circ. \]Practice 9: Repeated Rotation
Find
\[ \sin450^\circ \]and
\[ \cos450^\circ. \]Practice 10: Identify the Error
A student says that the point
\[ \left(-\frac{\sqrt3}{2},\frac12\right) \]has
\[ \sin\theta=-\frac{\sqrt3}{2} \]because sine is always the first coordinate.
Explain the error.
Practice Answers
Practice 1
Since
\[ (\cos\theta,\sin\theta) = \left(\frac35,\frac45\right), \]we have
\[ \boxed{\cos\theta=\frac35}, \] \[ \boxed{\sin\theta=\frac45}, \]and
\[ \boxed{\tan\theta=\frac{4/5}{3/5}=\frac43}. \]Practice 2
From
\[ P=\left(-\frac5{13},\frac{12}{13}\right), \]we have
\[ \cos\theta=-\frac5{13} \]and
\[ \sin\theta=\frac{12}{13}. \]Therefore,
\[ \boxed{\csc\theta=\frac{13}{12}}, \] \[ \boxed{\sec\theta=-\frac{13}{5}}, \]and
\[ \boxed{\cot\theta=-\frac5{12}}. \]Practice 3
\[ \left(\frac8{17}\right)^2+ \left(\frac{15}{17}\right)^2 = \frac{64}{289}+\frac{225}{289} = \frac{289}{289} = 1. \]Therefore,
\[ \boxed{ \left(\frac8{17},\frac{15}{17}\right) \text{ lies on the unit circle} }. \]Practice 4
\[ \left(-\frac{12}{13}\right)^2+y^2=1. \] \[ \frac{144}{169}+y^2=1. \] \[ y^2=\frac{25}{169}. \]Quadrant II has positive \(y\), so
\[ \boxed{y=\frac5{13}}. \]Practice 5
At \(180^\circ\), the terminal point is
\[ (-1,0). \]Therefore,
\[ \boxed{\sin180^\circ=0}, \] \[ \boxed{\cos180^\circ=-1}, \]and
\[ \boxed{\tan180^\circ=0}. \]Practice 6
At \(90^\circ\), the unit-circle point is
\[ (0,1). \]Since
\[ \tan\theta=\frac{y}{x}, \]we would have
\[ \tan90^\circ=\frac10, \]which is undefined.
Practice 7
\[ \boxed{ \left( \frac{\sqrt2}{2}, \frac{\sqrt2}{2} \right) }. \]Practice 8
Add \(360^\circ\):
\[ 60^\circ+360^\circ=420^\circ. \]Subtract \(360^\circ\):
\[ 60^\circ-360^\circ=-300^\circ. \]Therefore, one possible pair is
\[ \boxed{420^\circ\text{ and }-300^\circ}. \]Practice 9
Since
\[ 450^\circ=360^\circ+90^\circ, \]the angle is coterminal with \(90^\circ\).
Therefore,
\[ \boxed{\sin450^\circ=1} \]and
\[ \boxed{\cos450^\circ=0}. \]Practice 10
Sine is the second coordinate, not the first.
For
\[ \left(-\frac{\sqrt3}{2},\frac12\right), \]we have
\[ \boxed{\cos\theta=-\frac{\sqrt3}{2}} \]and
\[ \boxed{\sin\theta=\frac12}. \]What to Remember
The unit circle is
\[ \boxed{x^2+y^2=1}. \]For the terminal point
\[ P=(x,y), \]the primary relationships are
\[ \boxed{ \cos\theta=x, \qquad \sin\theta=y, \qquad \tan\theta=\frac{y}{x} }. \]- The \(x\)-coordinate is cosine.
- The \(y\)-coordinate is sine.
- Tangent is \(y/x\) when \(x\neq0\).
- Signs follow directly from the quadrant.
- The reciprocal functions come from reciprocating the coordinate ratios.
- The Pythagorean identity comes from the circle equation.
- Angles may be positive, negative, or larger than one revolution.
- Coterminal angles end at the same point.
- Repeated position produces trigonometric periodicity.
- The unit circle connects triangle trigonometry to coordinates, vectors, rotation, and waves.
Where We Go Next
The unit circle shows how trigonometric values extend around all four quadrants, but finding exact values efficiently requires a systematic way to compare a general angle with a familiar acute angle.
The next lesson, Reference Angles and Quadrants, develops that method and uses quadrant signs to determine exact trigonometric values throughout the circle.