10.4 The Unit Circle, Radian Measure & Trigonometric Functions

10.4 The Unit Circle, Radian Measure and Trigonometric Functions

Competency 011 asks you to use the unit circle in the coordinate plane to explore properties of trigonometric functions, and Competency 006 lists trigonometric functions among the nonlinear relations you must analyze. Section 7.3 defined sine, cosine and tangent for acute angles in right triangles. This section extends them to every angle with the coordinate plane and adds the Law of Sines and Law of Cosines, which appear on the official Definitions and Formulas page.

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Angles in Standard Position and Radian Measure

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. Counterclockwise rotation is positive and clockwise rotation is negative. Coterminal angles share a terminal side. For example, $60^\circ$, $420^\circ$ and $-300^\circ$ are coterminal.

A radian is the angle that subtends an arc equal in length to the radius. For any circle,

$$\theta \text{ (in radians)} = \frac{s}{r} \qquad\text{so}\qquad s = r\theta$$

A full turn subtends the whole circumference, $2\pi r$, so $360^\circ = 2\pi$ radians and $180^\circ = \pi$ radians.

• Degrees to radians: multiply by $\frac{\pi}{180}$. For example, $135^\circ = \frac{3\pi}{4}$.
• Radians to degrees: multiply by $\frac{180}{\pi}$. For example, $\frac{5\pi}{6} = 150^\circ$.
• Arc length: a central angle of $\frac{3\pi}{4}$ in a circle of radius 8 cm cuts off $s = 8 \cdot \frac{3\pi}{4} = 6\pi \approx 18.85$ cm.
• Sector area in radians: $A = \frac{1}{2}r^2\theta$.

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The Unit Circle Definitions

On the circle $x^2 + y^2 = 1$, let the terminal side of angle $\theta$ meet the circle at $(x, y)$. Then

$$\cos\theta = x, \qquad \sin\theta = y, \qquad \tan\theta = \frac{y}{x}\ (x \ne 0)$$

For an acute angle, drop a perpendicular to the $x$-axis. You get a right triangle with hypotenuse 1, so these definitions match SOH-CAH-TOA from section 7.3.

Because every point satisfies $x^2 + y^2 = 1$, the Pythagorean identity follows at once:

$$\sin^2\theta + \cos^2\theta = 1$$

Special angles in Quadrant I

These coordinates come straight from the 45°-45°-90° and 30°-60°-90° triangles scaled to hypotenuse 1.

Other quadrants: reference angles and signs

The reference angle is the acute angle between the terminal side and the $x$-axis. Find the values for the reference angle, then attach signs by quadrant:

• Quadrant I: all positive.
• Quadrant II: sine positive (cosine and tangent negative).
• Quadrant III: tangent positive (sine and cosine negative).
• Quadrant IV: cosine positive (sine and tangent negative).

Examples:

• $\sin 150^\circ = +\sin 30^\circ = \frac12$ (Quadrant II).
• $\cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt2}{2}$ (Quadrant III).
• $\tan 300^\circ = -\tan 60^\circ = -\sqrt3$ (Quadrant IV).
• The point at $210^\circ$ is $\left(-\frac{\sqrt3}{2}, -\frac12\right)$.

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Graphs of the Trigonometric Functions

Unwrap the unit circle. Plot the angle $\theta$ on the horizontal axis and the $y$-coordinate (sine) or $x$-coordinate (cosine) on the vertical axis, and you get periodic waves:

• $y = \sin x$: period $2\pi$, range $[-1, 1]$, zeros at multiples of $\pi$, maximum at $\frac{\pi}{2}$.
• $y = \cos x$: the same shape shifted left by $\frac{\pi}{2}$; it starts at its maximum, $\cos 0 = 1$.
• $y = \tan x$: period $\pi$, with vertical asymptotes where $\cos x = 0$ (at $x = \pm\frac{\pi}{2}, \dots$).

For $y = a\sin\big(b(x - h)\big) + d$ (the same transformation rule as section 6.4):

• Amplitude $= \lvert a \rvert$.
• Period $= \dfrac{2\pi}{\lvert b \rvert}$.
• Midline $y = d$, and phase shift $h$.

Modeling example. A Ferris wheel 40 m in diameter has its center 22 m above the ground and completes a turn every 8 minutes. A rider starts at the bottom. Height as a function of time is $H(t) = 22 - 20\cos\left(\frac{2\pi}{8}t\right)$. The amplitude is 20 (the radius), the midline is 22, and the period is 8. At $t = 4$ minutes, $H = 22 - 20\cos\pi = 42$ m, the top of the wheel.

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Solving Any Triangle: Law of Sines and Law of Cosines

Both laws are printed on the Definitions and Formulas page. You still need to know when each one applies.

Law of Sines: $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$. Use it when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA, the ambiguous case from section 8.2).

Example: In a triangle with $A = 40^\circ$, $B = 65^\circ$ and $a = 10$,

$$b = \frac{10\sin 65^\circ}{\sin 40^\circ} \approx \frac{10(0.9063)}{0.6428} \approx 14.10$$

Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$. Use it when you know two sides and the included angle (SAS), or all three sides (SSS).

Example: Two sides of 5 and 8 enclose a $60^\circ$ angle. Then

$$c^2 = 25 + 64 - 2(5)(8)(0.5) = 49, \qquad c = 7$$

When $C = 90^\circ$, $\cos C = 0$ and the Law of Cosines reduces to the Pythagorean theorem. This connection is worth pointing out to students.

Area of any triangle: $\text{Area} = \frac12 ab\sin C$. For the 5–8–60° triangle the area is $\frac12(5)(8)\frac{\sqrt3}{2} = 10\sqrt3 \approx 17.3$.

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Calculator and Teaching Notes

• Scientific calculators like the TI-30XS include SIN, COS, TAN and their inverses, plus a degree/radian mode setting. A wrong mode is the most common reason for a "wrong" trigonometric answer: $\sin 30$ in radian mode is about $-0.988$, not $0.5$.
• Middle school students meet the ideas behind this topic informally, through similar right triangles (the ratios stay constant) and circle measurement ($C = 2\pi r$ and $\pi$ as a ratio). Radians grow naturally from the question "how many radius-lengths fit along the arc?"
• A common misconception is that $\sin(2x) = 2\sin x$. Test $x = 90^\circ$: $\sin 180^\circ = 0$, but $2\sin 90^\circ = 2$.