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.

Learning Goals

Connect an angle to \((r\cos\theta,r\sin\theta)\).
Derive values for multiples of \(\pi/4\) and \(\pi/6\).
Use reference angles and quadrant signs.
Evaluate positive, negative, and coterminal angles exactly.
Check approximations for nonspecial angles.

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

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.

Key Takeaways

Evaluate sine and cosine at standard angles using reference angles, symmetry, and the unit circle.
Core relationship: \((\cos\theta,\sin\theta)\text{ lies on the unit circle}\)
Error check: Do not swap the cosine x-coordinate and sine y-coordinate.

Checkpoint · Topic 3.3

Derive the coordinates at \(\pi/4\).
Find sine and cosine of \(11\pi/6\).
Evaluate sine and cosine of \(-5\pi/4\).
Find the point on a radius-8 circle at \(2\pi/3\).
Approximate sine and cosine of 2 radians and justify both signs.