Comprehensive Trigonometry Study Notes: Functions, Triangles, Radian Measure, and Applications

Trigonometric Functions and Right Triangles

Definitions and Basic Relationships

Trigonometric functions relate the angles of a triangle to the lengths of its sides. In right triangles, these functions are fundamental for solving unknown sides or angles.

• Sine (sin): $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine (cos): $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent (tan): $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Pythagorean Theorem: $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse)

Example: For a right triangle with sides $a = 6$, $b = 8$, $c = 10$:

• $\sin(\theta) = \frac{6}{10} = 0.6$

• $\cos(\theta) = \frac{8}{10} = 0.8$

• $\tan(\theta) = \frac{6}{8} = 0.75$

Radian Measure and the Unit Circle

Understanding Radians

Angles can be measured in degrees or radians. Radians are based on the radius of a circle and are essential for advanced trigonometric calculations.

• Conversion: $1\text{ radian} = \frac{180}{\pi}$ degrees

• Unit Circle: A circle with radius 1, used to define trigonometric functions for all angles.

Example: $30^\circ = \frac{\pi}{6}$ radians

Arc Length and Sector Area

The arc length and area of a sector in a circle can be calculated using radian measure.

• Arc Length: $s = r\theta$ (where $\theta$ is in radians)

• Sector Area: $A = \frac{1}{2} r^2 \theta$

Example: For $r = 8$, $\theta = 2$ radians: $s = 8 \times 2 = 16$ units

Graphs of Trigonometric Functions

Graphing Sine and Secant Functions

Trigonometric functions can be graphed to show their periodic nature. The sine and secant functions are commonly graphed over one period.

• Sine Function: $y = \sin(x - \pi)$ shifts the standard sine curve horizontally.

• Secant Function: $y = \sec(x)$ is the reciprocal of the cosine function and has vertical asymptotes where $\cos(x) = 0$.

Example: The graph of $y = \sin(x - \pi)$ is a sine wave shifted right by $\pi$ units.

Trigonometric Identities and Equations

Solving Trigonometric Equations

Trigonometric equations can be solved for unknown angles or values using algebraic manipulation and identities.

• Example Equation: $2\sin(x) - 2 = 0$

• Solution: $\sin(x) = 1 \implies x = \frac{\pi}{2}$

• Secant Equation: $\sec(x) = 1$; since $\sec(x) = \frac{1}{\cos(x)}$, $\cos(x) = 1 \implies x = 0$

Solving Triangles: Law of Sines

Law of Sines

The Law of Sines relates the sides and angles of any triangle, not just right triangles.

• $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Example: Given $A = 20^\circ$, $B = 107^\circ$, $a = 15$:

• Find $C = 180^\circ - 20^\circ - 107^\circ = 53^\circ$

• Use Law of Sines to solve for other sides.

Complex Numbers and Polar Form

Converting Between Forms

Complex numbers can be expressed in rectangular form $(a + bi)$ or polar form $(r\text{cis}\theta)$.

• Rectangular to Polar: $r = \sqrt{a^2 + b^2}$, $\theta = \arctan\left(\frac{b}{a}\right)$

• Polar to Rectangular: $a = r\cos\theta$, $b = r\sin\theta$

Example: $5\text{cis}(120^\circ) = 5\cos(120^\circ) + 5i\sin(120^\circ)$

Applications: Circular Motion and Bicycles

Relating Angular and Linear Speed

Problems involving wheels or circular motion use the relationship between angular speed, radius, and linear speed.

• Linear Speed: $v = r\omega$ (where $\omega$ is angular speed in radians per unit time)

• Distance per Revolution: $d = 2\pi r$

Example: If a bicycle wheel has a radius of $32$cm and rotates $180$ times, total distance is $180 \times 2\pi \times 32$ cm.

Summary Table: Key Trigonometric Relationships

Function	Definition	Example Value
sin(θ)	Opposite / Hypotenuse	$\sin(30^\circ) = 0.5$
cos(θ)	Adjacent / Hypotenuse	$\cos(60^\circ) = 0.5$
tan(θ)	Opposite / Adjacent	$\tan(45^\circ) = 1$
Arc Length	$s = r\theta$	$r = 5$, $\theta = 2$; $s = 10$
Law of Sines	$\frac{a}{\sin A} = \frac{b}{\sin B}$	$a = 8$, $A = 30^\circ$, $b = 16$, $B = 60^\circ$

Additional info: Some context and examples have been inferred to clarify fragmented notes and ensure completeness for exam preparation.