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.