Trigonometry Course

The six trig functions relate angles to side lengths in a right triangle:

• Sine \(\sin\theta=\tfrac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine \(\cos\theta=\tfrac{\text{adjacent}}{\text{hypotenuse}}\)
• Tangent \(\tan\theta=\tfrac{\text{opposite}}{\text{adjacent}}\)
• Cosecant \(\csc\theta=\tfrac{1}{\sin\theta}\)
• Secant \(\sec\theta=\tfrac{1}{\cos\theta}\)
• Cotangent \(\cot\theta=\tfrac{1}{\tan\theta}\)

Quick Examples

Example: In a right triangle with opposite 3 and hypotenuse 5: \(\sin\theta=3/5=0.6\).

Example: With adjacent 4 and hypotenuse 5: \(\cos\theta=4/5=0.8\).

Radians vs Degrees

Radians are the natural angle unit: \(\pi\,\text{rad}=180^\circ\). Conversion:

• Degrees → Radians: \(\theta_\text{rad}=\theta_\text{deg}\cdot\tfrac{\pi}{180}\)
• Radians → Degrees: \(\theta_\text{deg}=\theta_\text{rad}\cdot\tfrac{180}{\pi}\)

Enter angles in radians

Pythagorean

\(\sin^2\theta+\cos^2\theta=1\)

Angle Sum/Difference

\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)

\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)

Double/Half Angle

\(\sin 2A=2\sin A\cos A\), \(\cos 2A=\cos^2A-\sin^2A\)

\(\sin^2\tfrac A2=\tfrac{1-\cos A}{2}\), \(\cos^2\tfrac A2=\tfrac{1+\cos A}{2}\)

Product-to-Sum

\(\sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]\) (etc.)

Exact Values (Special Angles)

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	\(\sqrt{3}/2\)	\(\sqrt{3}/3\)
45°	\(\sqrt{2}/2\)	\(\sqrt{2}/2\)	1
60°	\(\sqrt{3}/2\)	1/2	\(\sqrt{3}\)
90°	1	0	—

Reference Angles & Signs

Use the reference angle (distance from nearest \(x\)-axis) and quadrant signs to find values quickly.

Quadrant	sin	cos	tan
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Law of Sines

\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\), where \(R\) is circumradius.

Law of Cosines

\(c^2=a^2+b^2-2ab\cos C\) (and cyclic permutations).

Area Formulas

\(K=\tfrac12 ab\sin C\) and Heron: \(K=\sqrt{s(s-a)(s-b)(s-c)}\) with \(s=\tfrac{a+b+c}{2}\).

Cofunction Rules

Complementary angles add to \(90^\circ\) (or \(\tfrac{\pi}{2}\) rad). Use cofunctions to convert between pairs:

• \(\sin(90^\circ-\theta)=\cos\theta\), \(\cos(90^\circ-\theta)=\sin\theta\)
• \(\tan(90^\circ-\theta)=\cot\theta\)
• \(\csc(90^\circ-\theta)=\sec\theta\), \(\sec(90^\circ-\theta)=\csc\theta\)

Exam tip: Turn everything into sine/cosine when simplifying.

Reciprocal & Quotient

• \(\csc\theta=\tfrac{1}{\sin\theta}\), \(\sec\theta=\tfrac{1}{\cos\theta}\), \(\cot\theta=\tfrac{1}{\tan\theta}\)
• \(\tan\theta=\tfrac{\sin\theta}{\cos\theta}\), \(\cot\theta=\tfrac{\cos\theta}{\sin\theta}\)

Use radians for clean formulas:

• Arc length: \(s=r\,\theta\)
• Sector area: \(A=\tfrac12 r^2\,\theta\)

45-45-90

Legs equal; if hypotenuse is 1 then each leg is \(\tfrac{\sqrt2}{2}\). Hence \(\sin45^\circ=\cos45^\circ=\tfrac{\sqrt2}{2}\).

30-60-90

Side ratios \(1:\sqrt3:2\). So \(\sin30^\circ=\tfrac12\), \(\cos30^\circ=\tfrac{\sqrt3}{2}\), \(\tan30^\circ=\tfrac{\sqrt3}{3}\).

Proof Sketches

\(\sin^2\theta+\cos^2\theta=1\) via unit circle: \(x^2+y^2=1\Rightarrow(\cos\theta)^2+(\sin\theta)^2=1\).

\(\sin(\alpha+\beta)\) from projections or \(e^{i\theta}\) (complex numbers) gives the standard sum formulas.

We restrict domains so each inverse is a function:

• \(\arcsin x\) range \([-\tfrac{\pi}{2},\tfrac{\pi}{2}]\)
• \(\arccos x\) range \([0,\pi]\)
• \(\arctan x\) range \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\)

Technique: If \(y=\arcsin x\), then \(\sin y=x\) with \(y\) in its principal range.

Use radians for clean formulas:

• Arc length: \(s = r\,\theta\)
• Sector area: \(A = \tfrac12 r^2\,\theta\)

Graph Notes

• Period of \(\sin\) and \(\cos\): \(\tfrac{2\pi}{|b|}\) for \(y=a\sin(bx+c)\)
• Vertical asymptotes for \(\tan\) at \(x=\tfrac{\pi}{2}+k\pi\)
• Amplitude is \(|a|\) for \(\sin\) and \(\cos\)