Radian measure and degree–radian conversion

Key idea

Radian measure connects angles to circles using arc length. If an angle \(\theta\) subtends an arc of length \(s\) on a circle of radius \(r\), then: \[ \theta=\frac{s}{r}. \] On the unit circle (\(r=1\)), the radian measure equals the arc length: \(\theta=s\).

To convert: \[ \text{degrees} \rightarrow \text{radians:}\quad \theta^\circ \cdot \frac{\pi}{180}, \qquad \text{radians} \rightarrow \text{degrees:}\quad \theta \cdot \frac{180}{\pi}. \] Common anchors: \(180^\circ=\pi\), \(90^\circ=\tfrac{\pi}{2}\), \(60^\circ=\tfrac{\pi}{3}\), \(45^\circ=\tfrac{\pi}{4}\), \(30^\circ=\tfrac{\pi}{6}\).

Worked example

Try it

Summary

• Radian measure: \(\theta=\dfrac{s}{r}\) (arc length divided by radius).
• Convert degrees \(\rightarrow\) radians: multiply by \(\dfrac{\pi}{180}\). Convert radians \(\rightarrow\) degrees: multiply by \(\dfrac{180}{\pi}\).

The unit circle and \((\cos\theta,\sin\theta)\)

Key idea

The unit circle is the circle \(x^2+y^2=1\) (radius \(1\)). If an angle \(\theta\) is in standard position, the point where its terminal side intersects the unit circle is: \[ (\cos\theta,\;\sin\theta). \] So:

• \(\cos\theta\) is the \(x\)-coordinate
• \(\sin\theta\) is the \(y\)-coordinate

Important quadrantal points: \[ 0:(1,0),\quad \frac{\pi}{2}:(0,1),\quad \pi:(-1,0),\quad \frac{3\pi}{2}:(0,-1),\quad 2\pi:(1,0). \]

Worked example

Try it

Summary

• On the unit circle, \((\cos\theta,\sin\theta)\) gives the point at angle \(\theta\).
• Quadrantal angles give quick exact values (like \(\sin(\pi/2)=1\), \(\cos(\pi)= -1\)).

Special angles and exact trig values

Key idea

The special angles come from special right triangles:

• \(45^\circ\)–\(45^\circ\)–\(90^\circ\): \(\sin(\pi/4)=\cos(\pi/4)=\dfrac{\sqrt{2}}{2}\)
• \(30^\circ\)–\(60^\circ\)–\(90^\circ\): \(\sin(\pi/6)=\dfrac{1}{2}\), \(\cos(\pi/6)=\dfrac{\sqrt{3}}{2}\), \(\sin(\pi/3)=\dfrac{\sqrt{3}}{2}\), \(\cos(\pi/3)=\dfrac{1}{2}\)

Then: \[ \tan\theta=\frac{\sin\theta}{\cos\theta}, \] and \(\tan\theta\) is undefined when \(\cos\theta=0\) (e.g., \(\theta=\pi/2\)).

Worked example

Try it

Summary

• Special angles give exact values for \(\sin\) and \(\cos\), then \(\tan=\dfrac{\sin}{\cos}\).
• Memorize or reconstruct values for \(\tfrac{\pi}{6},\tfrac{\pi}{4},\tfrac{\pi}{3}\) using triangles or a unit circle chart.

Reference angles and quadrant sign rules

Key idea

A reference angle is the acute angle between the terminal side of \(\theta\) and the \(x\)-axis. It helps you use a known special-angle value and then apply the correct sign by quadrant.

For an angle \(\theta\) in \([0,2\pi)\):

• Quadrant I: \(\alpha=\theta\)
• Quadrant II: \(\alpha=\pi-\theta\)
• Quadrant III: \(\alpha=\theta-\pi\)
• Quadrant IV: \(\alpha=2\pi-\theta\)

Sign rules:

• QI: \(\sin,+\;\cos,+\;\tan,+\)
• QII: \(\sin,+\;\cos,-\;\tan,-\)
• QIII: \(\sin,-\;\cos,-\;\tan,+\)
• QIV: \(\sin,-\;\cos,+\;\tan,-\)

Worked example

Try it

Summary

• Reference angle gives the “special angle” size; quadrant gives the sign.
• Always identify the quadrant before assigning \(+\) or \(-\) to \(\sin\), \(\cos\), \(\tan\).

Negative angles and even/odd identities

Key idea

The unit circle is symmetric, which creates powerful identities:

• Cosine is even: \(\cos(-\theta)=\cos(\theta)\)
• Sine is odd: \(\sin(-\theta)=-\sin(\theta)\)
• Tangent is odd: \(\tan(-\theta)=-\tan(\theta)\)

These let you convert a negative angle into a positive angle you already know.

Worked example

Try it

Summary

• \(\cos\) is even; \(\sin\) and \(\tan\) are odd.
• Use symmetry to turn negative angles into familiar positive angles.

Coterminal angles and periodic trig functions

Key idea

Angles are coterminal if they differ by a full rotation: \[ \theta \text{ and } \theta + 2k\pi \quad (k\in\mathbb{Z}). \] Trigonometric functions repeat: \[ \sin(\theta+2\pi)=\sin\theta,\quad \cos(\theta+2\pi)=\cos\theta,\quad \tan(\theta+\pi)=\tan\theta. \] This lets you simplify angles like \(5\pi/2\), \(11\pi/6\), \(9\pi/4\), and more.

Worked example

Try it

Summary

• Use coterminal angles: replace \(\theta\) with \(\theta\pm 2\pi\) to land on a familiar angle.
• Periodicity: \(\sin,\cos\) repeat every \(2\pi\); \(\tan\) repeats every \(\pi\).

Why the unit circle and radians matter

Where the unit circle shows up

• Trig graphs: \(\sin\theta\) and \(\cos\theta\) waves come directly from the unit circle coordinates.
• Calculus: derivatives and integrals of \(\sin\) and \(\cos\) assume radians for standard formulas.
• Physics: circular motion and angular speed use radians naturally (\(\omega\) in rad/s).
• Complex numbers: \(e^{i\theta}=\cos\theta+i\sin\theta\) links the unit circle to rotations.

Worked example: tangent with a reference angle

Try it

Final recap

• Radians measure angle by arc length: \(\theta=\dfrac{s}{r}\). Convert degrees \(\leftrightarrow\) radians using \(\dfrac{\pi}{180}\) and \(\dfrac{180}{\pi}\).
• Unit circle coordinates: \((\cos\theta,\sin\theta)\). Cosine is \(x\); sine is \(y\).
• Special angles (\(\tfrac{\pi}{6},\tfrac{\pi}{4},\tfrac{\pi}{3}\)) give exact values; \(\tan=\dfrac{\sin}{\cos}\).
• Reference angles + quadrant signs keep your answers correct for any angle.
• Symmetry: \(\cos(-\theta)=\cos\theta\), \(\sin(-\theta)=-\sin\theta\). Periodicity: \(\sin,\cos\) repeat every \(2\pi\); \(\tan\) repeats every \(\pi\).