Dening Trigonometric Functions: Sine And Cosine
Understand how sine and cosine are defined on the unit circle for Queensland Year 11 Mathematical Methods (QCAA). Cosine gives the horizontal coordinate of the point at a given angle, and sine the vertical one.
You will learn to read quadrant signs with the ASTC rule, use reference angles for exact values, and see why both ratios stay between minus one and one — groundwork for graphing and solving equations in the QCAA course.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), the functions cosine and sine are defined on the unit circle: for the point \(P\) reached by rotating through an angle \(t\), the cosine is the \(x\)-coordinate and the sine is the \(y\)-coordinate. This page shows how to read \(\cos t\) and \(\sin t\) from the circle, the values at the quadrantal angles, the signs by quadrant (ASTC), and exact values using a reference angle.
Start at \((1,0)\) and rotate anticlockwise through an angle \(t\) around the unit circle (radius \(1\), centre the origin). You arrive at a point \(P\). By definition \(\cos t\) is the \(x\)-coordinate of \(P\) and \(\sin t\) is its \(y\)-coordinate, so \(P=(\cos t,\ \sin t)\).
Because \(P\) sits on a circle of radius \(1\), each coordinate lies between \(-1\) and \(1\); hence \(-1\le\cos t\le 1\) and \(-1\le\sin t\le 1\). The signs of the coordinates give the signs of cosine and sine in each quadrant — summarised by ASTC (All, Sine, Tangent, Cosine positive in quadrants 1 to 4).
For an angle that is not in the first quadrant, use the reference angle (the acute angle to the \(x\)-axis) to get the size of the value, then attach the correct sign from the quadrant.
The unit-circle definitions and the resulting bounds:
The quadrantal values (the point sits on an axis):
How to find \(\cos t\) or \(\sin t\)
- Locate the point: place the angle \(t\) on the unit circle and note its quadrant (or that it lands on an axis).
- Reference angle: find the acute angle to the \(x\)-axis, and read the size of the coordinate (for example \(\dfrac{\sqrt3}{2}\) at a reference of \(\dfrac{\pi}{3}\)).
- Attach the sign: use ASTC to give cosine (the \(x\)-sign) and sine (the \(y\)-sign) the correct sign for that quadrant.
By definition, cosine is the \(x\)-coordinate of \(P\):
| \(\cos t\) | \(=\) | \(x\text{-coordinate}\) |
| \(=\) | \(-\dfrac{3}{5}\) |
Sine is the \(y\)-coordinate of \(P\):
| \(\sin t\) | \(=\) | \(y\text{-coordinate}\) |
| \(=\) | \(\dfrac{4}{5}\) |
The signs \((-,\,+)\) confirm \(P\) is in the second quadrant, and \(\left(-\dfrac{3}{5}\right)^2+\left(\dfrac{4}{5}\right)^2=1\) as required.
\(\cos t=-\dfrac{3}{5}\) and \(\sin t=\dfrac{4}{5}\).
At angle \(\pi\) the point is \((-1,\,0)\); cosine is the \(x\)-coordinate:
| \(\cos\pi\) | \(=\) | \(x\text{-coordinate of }(-1,0)\) |
| \(=\) | \(-1\) |
At angle \(\dfrac{3\pi}{2}\) the point is \((0,\,-1)\); sine is the \(y\)-coordinate:
| \(\sin\dfrac{3\pi}{2}\) | \(=\) | \(y\text{-coordinate of }(0,-1)\) |
| \(=\) | \(-1\) |
\(\cos\pi=-1\) and \(\sin\dfrac{3\pi}{2}=-1\).
\(\dfrac{2\pi}{3}\) is in the second quadrant, where sine is positive. Reference angle:
| \(\text{ref}\) | \(=\) | \(\pi-\dfrac{2\pi}{3}\) |
| \(=\) | \(\dfrac{\pi}{3}\) |
So \(\sin\dfrac{2\pi}{3}=+\sin\dfrac{\pi}{3}\):
| \(\sin\dfrac{2\pi}{3}\) | \(=\) | \(\sin\dfrac{\pi}{3}\) |
| \(=\) | \(\dfrac{\sqrt3}{2}\) |
\(\sin\dfrac{2\pi}{3}=\dfrac{\sqrt3}{2}\).
\(\dfrac{5\pi}{4}\) is in the third quadrant, where cosine is negative. Reference angle:
| \(\text{ref}\) | \(=\) | \(\dfrac{5\pi}{4}-\pi\) |
| \(=\) | \(\dfrac{\pi}{4}\) |
Attach the third-quadrant sign to \(\cos\dfrac{\pi}{4}\):
| \(\cos\dfrac{5\pi}{4}\) | \(=\) | \(-\cos\dfrac{\pi}{4}\) |
| \(=\) | \(-\dfrac{\sqrt2}{2}\) |
\(\cos\dfrac{5\pi}{4}=-\dfrac{\sqrt2}{2}\).
Common pitfalls
Frequently asked questions
How are sine and cosine defined on the unit circle?
For the point \(P\) reached by rotating through angle \(t\), \(\cos t\) is the \(x\)-coordinate of \(P\) and \(\sin t\) is the \(y\)-coordinate, so \(P=(\cos t,\ \sin t)\).
Which is the x-coordinate, sine or cosine?
Cosine is the \(x\)-coordinate (the horizontal run) and sine is the \(y\)-coordinate (the vertical rise).
In which quadrants are sine and cosine positive?
Cosine is positive where \(x>0\) (quadrants 1 and 4); sine is positive where \(y>0\) (quadrants 1 and 2). This is the ASTC rule.
What are the largest and smallest values of sine and cosine?
Both lie between \(-1\) and \(1\), because each is a coordinate of a point on the unit circle: \(-1\le\cos t\le 1\) and \(-1\le\sin t\le 1\).
How do you use a reference angle for an exact value?
Find the acute angle to the \(x\)-axis to get the size of the value, then attach the sign for the quadrant. For example \(\sin\dfrac{2\pi}{3}=+\sin\dfrac{\pi}{3}=\dfrac{\sqrt3}{2}\).