Unit 5 · Topic 3.3
Sine and Cosine Function Values
Derive exact unit-circle coordinates from special triangles, then use reference angles and symmetry to evaluate sine and cosine throughout the circle.
1. Coordinates Encode Function Values
If a terminal ray intersects a radius-\(r\) circle at \(P=(x,y)\), then
\[x=r\cos\theta,\qquad y=r\sin\theta.\]
Equivalently, \(\cos\theta=x/r\) and \(\sin\theta=y/r\). Radius changes the coordinate scale, not the function values.
2. Read the Unit-Circle Point
When \(r=1\), the terminal point is
\[P=(\cos\theta,\sin\theta),\qquad \cos^2\theta+\sin^2\theta=1.\]
Cosine is the first, horizontal coordinate. Sine is the second, vertical coordinate.
3. Derive the \(\pi/4\) Values
A \(45^\circ\!\!-45^\circ\!\!-90^\circ\) triangle with legs 1 has hypotenuse \(\sqrt2\). Scale the hypotenuse to 1:
\[\frac1{\sqrt2}=\frac{\sqrt2}{2}.\]
\[\left(\cos\frac\pi4,\sin\frac\pi4\right)=\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right).\]
4. Derive the \(\pi/6\) and \(\pi/3\) Values
Bisect an equilateral triangle of side 2 to obtain side lengths \(1,\sqrt3,2\), then scale by \(1/2\):
\[\left(\cos\frac\pi6,\sin\frac\pi6\right)=\left(\frac{\sqrt3}{2},\frac12\right)\]
\[\left(\cos\frac\pi3,\sin\frac\pi3\right)=\left(\frac12,\frac{\sqrt3}{2}\right).\]
The coordinates swap because \(\pi/6\) and \(\pi/3\) are complementary.
5. See Coordinates on the Circle
Sine and Cosine Function Values example graph The point at angle \(\frac{\pi}{6}\) has vertical coordinate \(\frac{1}{2}\). The point at angle \(\frac{\pi}{6}\) has vertical coordinate \(\frac{1}{2}\). Moving to another quadrant preserves special-triangle magnitudes while changing coordinate signs.
6. Build the First-Quadrant Table
Angle \(0\) \(\pi/6\) \(\pi/4\) \(\pi/3\) \(\pi/2\) Cosine 1 \(\sqrt3/2\) \(\sqrt2/2\) \(1/2\) 0 Sine 0 \(1/2\) \(\sqrt2/2\) \(\sqrt3/2\) 1
From 0 to \(\pi/2\), cosine decreases while sine increases. This helps detect swapped values.
7. Find the Reference Angle
The reference angle \(\alpha\) is the acute angle between the terminal ray and the x-axis.
Quadrant Reference angle I \(\alpha=\theta\) II \(\alpha=\pi-\theta\) III \(\alpha=\theta-\pi\) IV \(\alpha=2\pi-\theta\)
8. Apply Quadrant Signs
Quadrant Cosine / x Sine / y I + + II - + III - - IV + -
The reference angle supplies magnitudes; the original quadrant supplies signs.
9. Evaluate a Quadrant II Angle
For \(\theta=5\pi/6\), \(\alpha=\pi-5\pi/6=\pi/6\). Quadrant II has negative x and positive y:
\[\cos\frac{5\pi}{6}=-\frac{\sqrt3}{2},\qquad \sin\frac{5\pi}{6}=\frac12.\]
10. Evaluate a Quadrant IV Angle
For \(7\pi/4\), the reference angle is \(\pi/4\), and Quadrant IV gives
\[\cos\frac{7\pi}{4}=\frac{\sqrt2}{2},\qquad \sin\frac{7\pi}{4}=-\frac{\sqrt2}{2}.\]
11. Reduce Large and Negative Angles
\[\frac{17\pi}{6}-2\pi=\frac{5\pi}{6}.\]
Thus \(17\pi/6\) shares values with \(5\pi/6\). Also, \(-\pi/3\) is coterminal with \(5\pi/3\):
\[\cos\left(-\frac\pi3\right)=\frac12,\qquad \sin\left(-\frac\pi3\right)=-\frac{\sqrt3}{2}.\]
12. Find Coordinates on a Nonunit Circle
For radius 10 and \(\theta=4\pi/3\), the unit-circle point is \((-1/2,-\sqrt3/2)\). Scale by 10:
\[P=(10\cos\theta,10\sin\theta)=(-5,-5\sqrt3).\]
Check: \((-5)^2+(-5\sqrt3)^2=100=10^2\).
13. Use Symmetry as a Check
\[\cos(-\theta)=\cos\theta,\qquad \sin(-\theta)=-\sin\theta\]
\[\cos(\pi-\theta)=-\cos\theta,\qquad \sin(\pi-\theta)=\sin\theta.\]
These identities summarize reflection of unit-circle coordinates across the axes.
14. Approximate a Nonspecial Angle
In radian mode,
\[\cos(1.2)\approx0.362,\qquad \sin(1.2)\approx0.932.\]
Both are positive because \(1.2\) lies in Quadrant I, and \(0.362^2+0.932^2\approx1\).
Confirm angle mode. Predict signs. Calculate. Check the unit-circle identity.
15. Common Errors
Reading \((\sin\theta,\cos\theta)\) instead of \((\cos\theta,\sin\theta)\). Swapping the \(\pi/6\) and \(\pi/3\) coordinates. Ignoring quadrant signs after finding a reference angle. Reducing by \(\pi\) instead of a full \(2\pi\) revolution. Forgetting to multiply coordinates by \(r\) on a nonunit circle. Rounding when an exact radical is requested.