Unit 5 · Topic 3.2

Sine, Cosine, and Tangent

Connect directed angles, radian measure, circles, coordinates, and slope to define the three primary trigonometric functions.

Learning Goals

Draw positive and negative angles in standard position and identify coterminal angles.
Interpret radian measure as the ratio of arc length to radius.
Determine sine, cosine, and tangent from a point on any circle centered at the origin.
Use the unit circle to interpret cosine as a horizontal coordinate and sine as a vertical coordinate.
Interpret tangent as terminal-ray slope and recognize when it is undefined.

1. Put an Angle in Standard Position

An angle is in standard position when its vertex is at the origin and its initial ray lies on the positive x-axis. Rotation ends at the terminal ray.

Positive angle	Rotate counterclockwise.
Negative angle	Rotate clockwise.
One revolution	\(2\pi\) radians or \(360^\circ\).

The angle measure records directed rotation, not merely the smaller geometric angle between two rays.

2. Recognize Coterminal Angles

Angles that finish on the same terminal ray are coterminal. They differ by an integer number of complete revolutions:

\[\theta+2\pi k,\qquad k\in\mathbb Z.\]

For example, \(-\pi/3\), \(5\pi/3\), and \(11\pi/3\) are coterminal. They produce the same sine, cosine, and tangent whenever tangent is defined.

3. Understand Radian Measure

For a circle of radius \(r\), an angle subtending an arc of length \(s\) has radian measure

\[\theta=\frac{s}{r}\qquad\Longleftrightarrow\qquad s=r\theta.\]

Because the ratio compares two lengths, radians are dimensionless. On a unit circle, \(r=1\), so the numerical arc length equals the radian measure.

Rotation	Degrees	Radians
Quarter	\(90^\circ\)	\(\pi/2\)
Half	\(180^\circ\)	\(\pi\)
Full	\(360^\circ\)	\(2\pi\)

4. Connect the Angle to a Point on a Circle

Coordinates on a circle encode cosine horizontally and sine vertically.

The terminal ray intersects a circle centered at the origin at \(P=(x,y)\). The radius \(r\) is the distance from the origin to \(P\), so \(x^2+y^2=r^2\).

5. Define the Three Ratios on Any Radius

For \(P=(x,y)\) on a circle of radius \(r>0\),

\[\sin\theta=\frac{y}{r},\qquad \cos\theta=\frac{x}{r},\qquad \tan\theta=\frac{y}{x}\quad(x\ne0).\]

Changing the circle's radius scales \(x\), \(y\), and \(r\) by the same factor, so the ratios depend only on the terminal ray, not the chosen circle.

6. Simplify on the Unit Circle

When \(r=1\), the intersection point is

\[P=(\cos\theta,\sin\theta).\]

Function	Geometric meaning	Possible values
\(\cos\theta\)	Horizontal coordinate	\([-1,1]\)
\(\sin\theta\)	Vertical coordinate	\([-1,1]\)
\(\tan\theta\)	Terminal-ray slope	Any real value when defined

The identity \(\cos^2\theta+\sin^2\theta=1\) follows immediately from the unit-circle equation.

7. Interpret Tangent as Slope

Every nonvertical terminal ray through the origin has slope

\[m=\frac{y-0}{x-0}=\frac{y}{x}=\tan\theta=\frac{\sin\theta}{\cos\theta}.\]

If \(x=0\), the ray is vertical, the slope does not exist, and tangent is undefined. Sine and cosine remain defined at those angles.

8. Determine Signs from the Quadrant

Quadrant	Sign of x / cosine	Sign of y / sine	Sign of tangent
I	+	+	+
II	-	+	-
III	-	-	+
IV	+	-	-

Signs come from directed coordinates. Lengths such as \(r\) remain positive, but horizontal and vertical displacements can be negative.

9. Evaluate from a Point on a Nonunit Circle

The terminal ray passes through \(P=(-5,12)\). First find the radius:

\[r=\sqrt{(-5)^2+12^2}=\sqrt{169}=13.\]

Therefore,

\[\sin\theta=\frac{12}{13},\qquad \cos\theta=-\frac{5}{13},\qquad \tan\theta=-\frac{12}{5}.\]

The point lies in Quadrant II, which confirms positive sine, negative cosine, and negative tangent.

10. Recover a Missing Coordinate

Suppose a unit-circle point lies in Quadrant III and has x-coordinate \(-8/17\). From \(x^2+y^2=1\),

\[y^2=1-\frac{64}{289}=\frac{225}{289}.\]

Quadrant III requires \(y<0\), so \(y=-15/17\). Thus

\[\cos\theta=-\frac8{17},\qquad \sin\theta=-\frac{15}{17},\qquad \tan\theta=\frac{15}{8}.\]

11. Evaluate Angles on the Axes

Angle	Point	Cosine	Sine	Tangent
\(0\)	\((1,0)\)	1	0	0
\(\pi/2\)	\((0,1)\)	0	1	Undefined
\(\pi\)	\((-1,0)\)	-1	0	0
\(3\pi/2\)	\((0,-1)\)	0	-1	Undefined

Tangent fails exactly where the horizontal coordinate and cosine are zero.

12. Connect Circle and Right-Triangle Definitions

Dropping a perpendicular from \(P=(x,y)\) to the x-axis forms a right triangle. In Quadrant I,

\[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}.\]

The circle definitions extend these ratios to every real angle. Outside Quadrant I, signed coordinates supply the correct signs while the radius remains positive.

13. Apply Coterminal Reasoning

The angle \(-7\pi/6\) is coterminal with \(5\pi/6\) because adding \(2\pi\) gives

\[-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}.\]

Both angles terminate in Quadrant II and therefore have the same three trigonometric values. Reducing an angle to a familiar coterminal representative changes the number of revolutions, not the terminal ray.

14. Use Technology with the Correct Angle Mode

A calculator must use the same angle unit as the input. For example, \(\sin(30^\circ)=0.5\), while \(\sin(30)\) in radian mode is approximately \(-0.988\).

Identify whether the angle is in degrees or radians.
Set the calculator mode.
Estimate the sign from the quadrant.
Compute and compare the result with the geometric estimate.

Technology confirms a value; it does not replace the coordinate or slope reasoning that defines it.

15. Common Errors

Swapping the sine y-coordinate and cosine x-coordinate.
Using clockwise rotation for a positive angle.
Treating coterminal angles as different terminal rays.
Using \(y/x\) for tangent when \(x=0\).
Discarding coordinate signs because triangle side lengths are positive.
Forgetting to calculate \(r\) when the point is not on the unit circle.
Entering a radian angle while the calculator is in degree mode.

Key Takeaways

Connect right-triangle ratios and unit-circle coordinates to sine, cosine, and tangent.
Core relationship: \(\sin\theta=\frac{y}{r},\quad\cos\theta=\frac{x}{r},\quad\tan\theta=\frac{y}{x}\)
Error check: Track the quadrant signs instead of using triangle lengths alone.

Checkpoint · Topic 3.2

Find a positive and a negative angle coterminal with \(7\pi/4\).
An arc of length 15 lies on a circle of radius 6. Find its central angle in radians.
For a terminal ray through \((7,-24)\), determine sine, cosine, and tangent.
Explain geometrically why tangent is undefined at \(\theta=\pi/2\).
A calculator reports a positive value for \(\sin(4)\). Without recalculating, decide whether degree or radian mode could explain the result.