Measures, Shape & Space

Advanced trigonometry

Use sine and cosine rules, radians, and trigonometric graphs in 2D and 3D applications.

Advanced trigonometry extends right-triangle work to non-right triangles, radians, and trig graphs used in periodic modeling.

Radians and arc length

Convert degrees/radians:

\[ 180^\circ=\pi\ \text{rad}. \]

Arc length and sector area:

\[ s=r\theta,\qquad A=\frac12r^2\theta \]

with \(\theta\) in radians.

Sine and cosine rules

\[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \]

\[ c^2=a^2+b^2-2ab\cos C. \]

Use sine rule for side-angle opposite pairs; cosine rule for SAS or SSS cases.

Triangle area formula

\[ \text{Area}=\frac12ab\sin C. \]

Useful when two sides and included angle are known.

Core identities

\[ \sin^2x+\cos^2x=1,\qquad \tan x=\frac{\sin x}{\cos x}. \]

Solving basic trig equations

Solve in interval (e.g., \(0^\circ\le x<360^\circ\)) using unit-circle symmetry.

Example: \(\sin x=\frac12\Rightarrow x=30^\circ,150^\circ\).

Trigonometric graphs

For \(y=a\sin(bx)+d\) (similarly cosine):

  • Amplitude \(=|a|\)
  • Period \(=\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\) in degrees mode)
  • Vertical shift \(=d\)

Exam strategy

  • Check calculator mode (degree/radian) before solving.
  • State interval when listing trig equation solutions.
  • Draw quick triangle sketch for sine/cosine rule setup.
  • Keep exact radian forms when possible.

History

Advanced trigonometric methods were developed for astronomy, navigation, and surveying. Sine and cosine rule tables were essential long before modern calculators.

Checkpoints

Convert \(150^\circ\) to radians.

Answer: \(\frac{5\pi}{6}\).

Find arc length for \(r=8\), \(\theta=\frac{\pi}{3}\).

Answer: \(s=r\theta=\frac{8\pi}{3}\).

In triangle, \(a=10,b=7,C=60^\circ\). Find \(c\).

Answer: \(c^2=10^2+7^2-2(10)(7)\cos60^\circ=79\), so \(c=\sqrt{79}\).

Solve \(2\sin x=1\) for \(0^\circ\le x<360^\circ\).

Answer: \(x=30^\circ,150^\circ\).

For \(y=3\cos(2x)-1\), state amplitude and period.

Answer: Amplitude \(3\), period \(\pi\) radians (or \(180^\circ\)).

Applications

  • Wave modeling: sound, light, and AC signals.
  • Navigation: triangulation in maps and GPS support calculations.
  • Engineering: periodic motion and oscillation analysis.

References

  • Standard high-school advanced trigonometry syllabi (IB, A-Level, DSE extension).
  • Precalculus texts on trig identities, equations, and graphs.

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