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MathChapter 15: Trigonometric Functions

Notes

The Radian Measure of an Angle Notes

Interpret radians by arc length, convert exactly between degrees and radians, draw signed standard-position angles, and use coterminal and periodic relationships.

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Degrees divide a turn into \(360\) parts. Radians measure a central angle by comparing its intercepted arc length with the circle radius, which makes radians natural for trigonometric functions.

What one radian means

One radianThe central angle intercepts an arc whose length equals the circle radius.
One radianThe central angle intercepts an arc whose length equals the circle radius.1 radrarc length rcounterclockwise positive rotationxyO
One radian
The central angle that intercepts an arc whose length equals the circle radius: \(s=r\), so \(\theta=s/r=1\) radian.
A full revolution
\[2\pi\text{ radians}=360^\circ,\qquad \pi\text{ radians}=180^\circ,\qquad 1\text{ radian}\approx57.3^\circ\]

Convert degrees and radians exactly

Benchmark angle conversions
DegreesRadians
\(0^\circ\)\(0\)
\(30^\circ\)\(\pi/6\)
\(45^\circ\)\(\pi/4\)
\(60^\circ\)\(\pi/3\)
\(90^\circ\)\(\pi/2\)
\(180^\circ\)\(\pi\)
\(270^\circ\)\(3\pi/2\)
\(360^\circ\)\(2\pi\)

Choose the conversion factor that cancels the old unit

  1. Degrees to radians

    Multiply by \(\pi\text{ rad}/180^\circ\), then reduce.

  2. Radians to degrees

    Multiply by \(180^\circ/\pi\text{ rad}\), then cancel \(\pi\).

  3. Keep exact form

    Leave answers as reduced multiples of \(\pi\) unless an approximation is requested.

Worked example

Convert in both directions

Convert \(225^\circ\) to radians and \(7\pi/12\) radians to degrees.

  1. \(225^\circ(\pi/180^\circ)=5\pi/4\).

  2. \((7\pi/12)(180^\circ/\pi)=105^\circ\).

\(225^\circ=5\pi/4\) and \(7\pi/12=105^\circ\).

Standard position and signed rotation

A positive angle in standard positionThe initial side lies on the positive x-axis and the terminal side rotates counterclockwise 135 degrees.
A positive angle in standard positionThe initial side lies on the positive x-axis and the terminal side rotates counterclockwise 135 degrees.135°positive: counterclockwisexyO
A negative angle in standard positionThe terminal side is reached by a clockwise rotation of 150 degrees.
A negative angle in standard positionThe terminal side is reached by a clockwise rotation of 150 degrees.−150°negative: clockwisexyO

How signed angles rotate

Initial side

For standard position, it begins on the positive \(x\)-axis with vertex at the origin.

Positive angle

Rotate counterclockwise from the initial side.

Negative angle

Rotate clockwise from the initial side.

Coterminal angles share a terminal side

Coterminal anglesSixty degrees and negative 300 degrees finish on the same terminal side.
Coterminal anglesSixty degrees and negative 300 degrees finish on the same terminal side.−300°60°same terminal side; rotations differ by 360°xyO
All coterminal angles
\[\theta+360^\circ k\quad\text{or}\quad\theta+2\pi k,\qquad k\in\mathbb Z\]
Worked example

Find one positive and one negative coterminal angle

Start with \(-75^\circ\).

  1. Add \(360^\circ\): \(-75^\circ+360^\circ=285^\circ\), a positive coterminal angle.

  2. Subtract \(360^\circ\): \(-75^\circ-360^\circ=-435^\circ\), another negative coterminal angle.

Examples are \(285^\circ\) and \(-435^\circ\).

Periodicity repeats function values

Periods of the primary trigonometric functions
FunctionDegree periodRadian periodIdentity
Sine\(360^\circ\)\(2\pi\)\(\sin(\theta+2\pi)=\sin\theta\)
Cosine\(360^\circ\)\(2\pi\)\(\cos(\theta+2\pi)=\cos\theta\)
Tangent\(180^\circ\)\(\pi\)\(\tan(\theta+\pi)=\tan\theta\)
Mini check

Use the correct period

Which expression always equals \(\tan\theta\)?

  1. \(\tan(\theta+\pi)\)
  2. \(\tan(\theta+\pi/2)\)
  3. \(\tan(\theta+2\pi/3)\)
  4. \(\tan(\theta+3\pi/2)\)
Show answer and explanation

Answer: \(\tan(\theta+\pi)\)

Tangent's fundamental period is \(\pi\).

Key takeaways

Radian-measure checklist

  • One radian intercepts an arc equal in length to the radius.
  • Use \(\pi\text{ rad}=180^\circ\) as the exact conversion relationship.
  • Counterclockwise angles are positive; clockwise angles are negative.
  • Coterminal angles differ by whole turns: \(360^\circ k\) or \(2\pi k\).
  • Sine and cosine have period \(2\pi\); tangent has period \(\pi\).
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.