Comprehensive Study Guide: College Mathematics II – Trigonometry (MAT 124)

Trigonometric Functions

Angles and Radian Measure

Understanding angles and their measurement is fundamental in trigonometry. Angles can be measured in degrees or radians, and converting between these units is a key skill.

• Angle: Formed by two rays with a common endpoint (vertex).

• Degree Measure: One full rotation is 360 degrees.

• Radian Measure: One full rotation is $2\pi$ radians.

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

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

Example: Convert $90^\circ$ to radians: $90^\circ = \frac{\pi}{2}$ radians.

Right Triangle Trigonometry

Trigonometric functions can be defined using the ratios of sides in a right triangle.

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

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

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

• Cosecant: $\csc\theta = \frac{1}{\sin\theta}$

• Secant: $\sec\theta = \frac{1}{\cos\theta}$

• Cotangent: $\cot\theta = \frac{1}{\tan\theta}$

Example: In a triangle with sides 3 (opposite), 4 (adjacent), 5 (hypotenuse): $\sin\theta = \frac{3}{5}$, $\cos\theta = \frac{4}{5}$, $\tan\theta = \frac{3}{4}$.

Trigonometric Functions of Any Angle

Trigonometric functions can be extended to any angle using the unit circle.

• Definition: For an angle $\theta$, the coordinates $(x, y)$ on the unit circle give $\cos\theta = x$, $\sin\theta = y$.

• Periodic Functions: Trigonometric functions repeat their values in regular intervals.

• Domain and Range: For example, $\sin\theta$ and $\cos\theta$ have domain $(-\infty, \infty)$ and range $[-1, 1]$.

Example: $\sin(360^\circ) = \sin(0^\circ) = 0$.

Graphs of Trigonometric Functions

Graphing trigonometric functions helps visualize their periodic nature and transformations.

• Parent Functions: Sine and cosine graphs have amplitude 1 and period $2\pi$.

• Transformations: Shifts, stretches, and reflections can alter amplitude, period, and phase.

• General Form: $y = A\sin(Bx + C) + D$

Example: $y = 2\sin(x)$ has amplitude 2.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles given function values.

• Notation: $\sin^{-1}(x)$, $\cos^{-1}(x)$, $\tan^{-1}(x)$

• Domain and Range: For $\sin^{-1}(x)$, domain $[-1, 1]$, range $[-\frac{\pi}{2}, \frac{\pi}{2}]$

Example: $\sin^{-1}(0.5) = \frac{\pi}{6}$

Analytic Trigonometry

Trigonometric Identities

Identities are equations true for all values in the domain. They are essential for simplifying expressions and solving equations.

• Fundamental Identities:

  • $\sin^2\theta + \cos^2\theta = 1$

  • $1 + \tan^2\theta = \sec^2\theta$

  • $1 + \cot^2\theta = \csc^2\theta$

• Cofunction Identities: $\sin(90^\circ - \theta) = \cos\theta$

• Even-Odd Identities: $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$

Example: Verify $\sin^2\theta + \cos^2\theta = 1$ for $\theta = 30^\circ$.

Sum and Difference Formulas

These formulas allow calculation of trigonometric values for sums or differences of angles.

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

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

• $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Example: $\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

Double-Angle, Power-Reducing, and Half-Angle Formulas

These formulas simplify expressions and solve equations involving multiple angles.

• Double-Angle: $\sin(2\theta) = 2\sin\theta\cos\theta$

• $\cos(2\theta) = \cos^2\theta - \sin^2\theta$

• Power-Reducing: $\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$

• Half-Angle: $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$

Example: $\cos(2\theta)$ for $\theta = 45^\circ$ is $\cos(90^\circ) = 0$.

Product-to-Sum and Sum-to-Product Formulas

These formulas convert products of trigonometric functions into sums or vice versa.

• $\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

• $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$

• $\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$

Example: $\sin 30^\circ \sin 60^\circ = \frac{1}{2}[\cos(-30^\circ) - \cos(90^\circ)]$

Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given equation.

• General Solution: $\sin\theta = a \implies \theta = \sin^{-1}(a) + 2\pi n$ or $\theta = \pi - \sin^{-1}(a) + 2\pi n$

• Quadratic Form: Equations like $a\sin^2\theta + b\sin\theta + c = 0$ can be solved by substitution.

• Factoring: Used to separate different functions.

Example: Solve $\sin\theta = 0.5$ for $0 \leq \theta < 2\pi$.

Additional Topics in Trigonometry

The Law of Sines

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

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

• Ambiguous Case: When given two sides and a non-included angle (SSA), there may be zero, one, or two possible triangles.

Example: Given $a = 7$, $A = 30^\circ$, $b = 10$, find $B$.

The Law of Cosines

The Law of Cosines is used for triangles when the Law of Sines cannot be applied, such as when two sides and the included angle are known.

• $c^2 = a^2 + b^2 - 2ab\cos C$

• Can be used to find unknown sides or angles.

Example: Given $a = 5$, $b = 7$, $C = 60^\circ$, find $c$.

Vectors

Vectors are quantities with both magnitude and direction. They are used in physics and engineering to represent forces, velocities, and more.

• Vector Addition: $\vec{u} + \vec{v}$

• Scalar Multiplication: $k\vec{v}$

• Dot Product: $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2$

• Rectangular Coordinates: $\vec{v} = v_1\mathbf{i} + v_2\mathbf{j}$

• Magnitude: $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$

• Unit Vector: $\vec{u} = \frac{\vec{v}}{|\vec{v}|}$

Example: $\vec{v} = 3\mathbf{i} + 4\mathbf{j}$, $|\vec{v}| = 5$

Summary Table: Trigonometric Functions and Properties

Function	Definition (Right Triangle)	Domain	Range	Period
sin	$\frac{\text{opposite}}{\text{hypotenuse}}$	$(-\infty, \infty)$	$[-1, 1]$	$2\pi$
cos	$\frac{\text{adjacent}}{\text{hypotenuse}}$	$(-\infty, \infty)$	$[-1, 1]$	$2\pi$
tan	$\frac{\text{opposite}}{\text{adjacent}}$	$(-\infty, \infty)$ except $\theta \neq \frac{\pi}{2} + n\pi$	$(-\infty, \infty)$	$\pi$
csc	$\frac{1}{\sin\theta}$	$(-\infty, \infty)$ except $\theta \neq n\pi$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$
sec	$\frac{1}{\cos\theta}$	$(-\infty, \infty)$ except $\theta \neq \frac{\pi}{2} + n\pi$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$
cot	$\frac{1}{\tan\theta}$	$(-\infty, \infty)$ except $\theta \neq n\pi$	$(-\infty, \infty)$	$\pi$

Applications and Problem Solving

Applied Problems

Trigonometry is used to solve real-world problems involving triangles, vectors, and periodic phenomena.

• Uniform Circular Motion: Describes objects moving in a circle at constant speed.

• Simple Harmonic Motion: Models oscillations, such as springs and pendulums.

• Bearings: Used in navigation and surveying.

Example: Find the height of a building using angle of elevation and distance.

Additional info:

• Polar coordinates and graphs of polar equations are also covered, but not detailed in the syllabus text.

• Students are expected to use calculators for computation, but not those with CAS capabilities.

• Practice tests are required before module tests to ensure readiness.