Unit-circle coordinate definitions
On the unit circle, a terminal point has a horizontal and a vertical coordinate. These give cosine and sine respectively. Tangent is the vertical coordinate divided by the horizontal coordinate when the latter is non-zero
\[\cos\theta=x,\quad\sin\theta=y,\quad\tan\theta=\frac{y}{x}\quad\left(x\ne0\right)\]
- Read the horizontal coordinate The x-coordinate of a terminal point gives the cosine of its angle \(\displaystyle P=\left(x,y\right)\quad\Rightarrow\quad\cos\theta=x\)
- Read the vertical coordinate The y-coordinate of a terminal point gives the sine of its angle \(\displaystyle P=\left(x,y\right)\quad\Rightarrow\quad\sin\theta=y\)
- Form tangent as a quotient When the horizontal coordinate is non-zero, divide the vertical coordinate by the horizontal coordinate \(\displaystyle \tan\theta=\frac{y}{x}\quad\left(x\ne0\right)\)
- Check domains and ranges Sine and cosine accept every real input and have outputs from -1 to 1, while tangent excludes inputs where x is zero and has all real outputs \(\displaystyle \operatorname{Dom}(\tan)=\mathbb{R}\setminus\left\{\frac{(2k+1)\pi}{2}:k\in\mathbb{Z}\right\},\quad\operatorname{Ran}(\tan)=\mathbb{R}\)
Worked example
For the unit-circle point shown, determine the exact values of cosine, sine and tangent
\[P=\left(-\frac{33}{65},\frac{56}{65}\right)\]
- \(\displaystyle P=\left(x,y\right)=\left(-\frac{33}{65},\frac{56}{65}\right),\quad x=-\frac{33}{65},\quad y=\frac{56}{65}\)Identify the horizontal and vertical coordinates
- \(\displaystyle \cos\theta=x=-\frac{33}{65}\)Cosine is the horizontal coordinate
- \(\displaystyle \sin\theta=y=\frac{56}{65}\)Sine is the vertical coordinate
- \(\displaystyle \tan\theta=\frac{y}{x}=\frac{\frac{56}{65}}{-\frac{33}{65}}=-\frac{56}{33}\)Divide the vertical coordinate by the horizontal coordinate; the common denominator cancels
Common mistakes
Swapping x and y
✕\(\displaystyle P=\left(-\frac{4}{5},\frac{3}{5}\right):\quad\sin\theta=-\frac{4}{5}\)✓\(\displaystyle P=\left(-\frac{4}{5},\frac{3}{5}\right):\quad\sin\theta=\frac{3}{5}\)
Sine is read from the vertical y-coordinate, not the horizontal x-coordinateReversing the tangent quotient
✕\(\displaystyle P=\left(-\frac{4}{5},\frac{3}{5}\right):\quad\tan\theta=\frac{x}{y}=\frac{-\frac{4}{5}}{\frac{3}{5}}=-\frac{4}{3}\)✓\(\displaystyle P=\left(-\frac{4}{5},\frac{3}{5}\right):\quad\tan\theta=\frac{y}{x}=\frac{\frac{3}{5}}{-\frac{4}{5}}=-\frac{3}{4}\)
Tangent is the vertical coordinate divided by the horizontal coordinate, so reversing the quotient changes the valueIgnoring the vertical boundary
✕\(\displaystyle P=\left(0,-1\right):\quad\tan\theta=\frac{-1}{0}=0\)✓\(\displaystyle P=\left(0,-1\right):\quad x=0\Rightarrow\tan\theta\text{ is undefined}\)
Tangent is the vertical coordinate divided by the horizontal coordinate; when the horizontal coordinate is zero, the quotient is undefinedDifficulty levels
introUse an axis or rational terminal point to read sine or cosine from its coordinates
standardUse a rational or standard-angle point to determine exact coordinates and all three circular-function values
challengeEvaluate a sine, cosine or tangent value in radians to 1 or 2 decimal places and classify a tangent boundary or a function domain and range
examChoose the correct coordinate definition, tangent boundary statement, domain and range, or range statement
Questions about this topic
Why is cosine the x-coordinate on the unit circle?+
A unit-circle terminal point has cosine as its horizontal coordinate and sine as its vertical coordinate
Why is tangent undefined when x is zero?+
Tangent is the vertical coordinate divided by the horizontal coordinate; when the horizontal coordinate is zero, the quotient is undefined
What are the domains and ranges of the three basic circular functions?+
Sine and cosine accept all real inputs and have outputs from -1 to 1. Tangent has all real outputs but excludes angles that are odd multiples of a quarter-turn from its inputs
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations
In this chapter
Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems