Precalculus Study Guide: Trigonometry, Vectors, and Systems of Equations

Trigonometric Functions and Angles

Angle Measure, Coterminal Angles, Arc Length, and Circular Motion

Understanding angles and their measurement is fundamental in trigonometry. Angles can be measured in degrees or radians, and knowing how to convert between these units is essential. Coterminal angles share the same terminal side, and arc length relates the distance traveled along a circle to the angle subtended.

• Degree and Radian Measure: Degrees and radians are two units for measuring angles. The conversion formula is $\text{radians} = \text{degrees} \times \frac{\pi}{180}$.

• Coterminal Angles: Angles that differ by integer multiples of $360^{\circ}$ or $2\pi$ radians.

• Arc Length: The length of an arc is given by $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

• Circular Motion: Angular speed is the rate at which an angle changes, often measured in radians per unit time.

• Example: Convert $150^{\circ}$ to radians: $150 \times \frac{\pi}{180} = \frac{5\pi}{6}$.

• Example: Find the arc length for a circle of radius $10$and angle$\frac{3\pi}{5}$radians:$s = 10 \times \frac{3\pi}{5} = 6\pi$.

Right-Triangle Trigonometry

Right-triangle trigonometry uses ratios of side lengths to define trigonometric functions. The mnemonic SOH CAH TOA helps remember these relationships.

• SOH CAH TOA: Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Reciprocal Functions: Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$ Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$ Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$

• Cofunctions: Sine and cosine, tangent and cotangent, secant and cosecant are cofunctions: $\sin(90^{\circ} - \theta) = \cos(\theta)$, etc.

• Example: If $\text{opp} = 3$, $\text{hyp} = 5$, then $\sin(\theta) = \frac{3}{5}$.

• Example: If $\sec(\theta) = \frac{13}{5}$ and $\theta$ is acute, $\cos(\theta) = \frac{5}{13}$, so $\sin(\theta) = \frac{12}{13}$.

Trigonometric Functions of Any Angle; The Unit Circle

Trigonometric functions can be defined for any angle using the unit circle. The signs of these functions depend on the quadrant, and reference angles help evaluate values.

• Coordinate Definitions: For a point $(x, y)$ on a circle of radius $r$, $\sin(\theta) = \frac{y}{r}$, $\cos(\theta) = \frac{x}{r}$, $\tan(\theta) = \frac{y}{x}$.

• Reference Angles: The acute angle formed with the x-axis, used to find exact values.

• Quadrant Signs (ASTC): Quadrant I: All positive Quadrant II: Sine positive Quadrant III: Tangent positive Quadrant IV: Cosine positive

• Example: $\cos\left(\frac{5\pi}{6}\right)$: Reference angle $\frac{\pi}{6}$ in Quadrant II, so $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.

• Example: Terminal side passes through $(-8, 15)$: $r = \sqrt{(-8)^2 + 15^2} = 17$, $\sin(\theta) = \frac{15}{17}$, $\cos(\theta) = -\frac{8}{17}$.

Graphs and Inverse Trigonometric Functions

Graphing Sine and Cosine

The graphs of sine and cosine functions are periodic and can be transformed by changing amplitude, period, phase shift, and vertical shift.

• Amplitude: The height from the midline to the peak, $|A|$ in $y = A \sin(Bx + C) + D$.

• Period: The length of one cycle, $\frac{2\pi}{|B|}$.

• Phase Shift: Horizontal shift, $-\frac{C}{B}$.

• Vertical Shift: The midline, $y = D$.

• Example: $y = 3 \sin(4x)$: Period $= \frac{2\pi}{4} = \frac{\pi}{2}$.

• Example: $y = -2 \cos(\pi x - \frac{\pi}{2}) + 1$: Amplitude $2$, period $2$, phase shift $\frac{1}{2}$ right, midline $y = 1$.

Graphs of Tangent, Secant, Cosecant, and Cotangent

These functions have unique graphs with vertical asymptotes and different periods. Secant and cosecant use sine and cosine as guide curves.

• Tangent: Period $\pi$, vertical asymptotes at $x = \frac{\pi}{2} + \pi k$.

• Secant and Cosecant: Use sine and cosine guide curves; vertical asymptotes where the denominator is zero.

• Example: $y = \tan(x)$: First positive vertical asymptote at $x = \frac{\pi}{2}$.

• Example: $y = \tan(2x)$: Period $= \frac{\pi}{2}$, asymptotes at $x = \frac{\pi}{4} + \frac{\pi}{2}k$.

Inverse Trigonometric Functions and Compositions

Inverse trigonometric functions return the angle whose trigonometric function equals a given value. Principal-value intervals are important for correct evaluation.

• Principal-Value Intervals: $\arcsin(x)$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$ $\arccos(x)$: $[0, \pi]$ $\arctan(x)$: $(-\frac{\pi}{2}, \frac{\pi}{2})$

• Compositions: Expressions like $\cos(\arcsin(x))$ use right-triangle relationships.

• Example: $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$.

• Example: $\cos(\arctan(\frac{3}{4}))$: Use a triangle with sides $3$and $4$, hypotenuse $5$, so$\cos(\theta) = \frac{4}{5}$.

Trigonometric Identities, Formulas, and Equations

Trigonometric Identities

Identities are equations true for all values in the domain. They are used to simplify and verify expressions.

• Reciprocal Identities: $\sin(\theta) = \frac{1}{\csc(\theta)}$, $\cos(\theta) = \frac{1}{\sec(\theta)}$, $\tan(\theta) = \frac{1}{\cot(\theta)}$

• Quotient Identities: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

• Pythagorean Identities: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, $1 + \cot^2(\theta) = \csc^2(\theta)$

• Example: Simplify $\tan(x) \cos(x)$: $\frac{\sin x}{\cos x} \cos x = \sin(x)$.

• Example: Verify $\frac{1}{1-\sin x} + \frac{1}{1+\sin x} = 2 \sec^2 x$ using common denominators and $1 - \sin^2 x = \cos^2 x$.

Sum and Difference Formulas

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

• Formulas:

  • $\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

  • $\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

  • $\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

• Example: $\sin(15^{\circ})$ using $45^{\circ} - 30^{\circ}$: $\frac{\sqrt{6} - \sqrt{2}}{4}$.

• Example: $\cos(75^{\circ})$ using $45^{\circ} + 30^{\circ}$: $\frac{\sqrt{6} - \sqrt{2}}{4}$.

Double-Angle and Half-Angle Formulas

These formulas are used to simplify expressions and solve equations involving multiples or fractions of angles.

• Double-Angle Formulas:

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

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

  • $\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

• Half-Angle Formulas:

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

  • $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

• Example: Simplify $\cos^2(15^{\circ}) - \sin^2(15^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}$.

• Example: If $\cos(\theta) = \frac{3}{5}$ and $\theta$ is in Quadrant IV, $\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 3/5}{2}} = \frac{\sqrt{5}}{5}$ (Quadrant II, positive sign).

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the function, factoring, or using identities. Solutions may be restricted to specific intervals.

• Isolate the Function: Rearranging to solve for the trigonometric function.

• Factoring and Quadratic Form: Some equations can be factored or written as quadratics.

• Use Identities: Apply identities to simplify and solve.

• Example: Solve $2 \cos(x) - 1 = 0$ on $[0, 2\pi)$: $\cos(x) = \frac{1}{2}$, $x = \frac{\pi}{3}, \frac{5\pi}{3}$.

• Example: Solve $\sin(2x) = \cos(x)$ on $[0, 2\pi)$: Use $2 \sin x \cos x = \cos x$, factor, $x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{3\pi}{2}$.

Vectors and Systems of Linear Equations

Vector Components, Operations, Direction, and Unit Vectors

Vectors are quantities with magnitude and direction. They can be written in component form or using unit vectors $\mathbf{i}$ and $\mathbf{j}$.

• Component Form: $\langle a, b \rangle$ or $a\mathbf{i} + b\mathbf{j}$.

• Vector from Two Points: $\overrightarrow{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle$.

• Magnitude: $||\mathbf{u}|| = \sqrt{a^2 + b^2}$.

• Direction Angle: $\theta = \arctan\left(\frac{b}{a}\right)$.

• Unit Vector: $\mathbf{u}_{\text{unit}} = \frac{\mathbf{u}}{||\mathbf{u}||}$.

• Example: $\mathbf{u} = \langle 3, -4 \rangle$, $||\mathbf{u}|| = \sqrt{3^2 + (-4)^2} = 5$.

• Example: Vector from $P(2, -1)$ to $Q(-3, 3)$: $\overrightarrow{PQ} = \langle -3 - 2, 3 - (-1) \rangle = \langle -5, 4 \rangle$.

Dot Product, Angles, and Orthogonality

The dot product is a scalar that measures the extent to which two vectors point in the same direction. It is used to find angles and test orthogonality.

• Dot Product: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$.

• Angle Between Vectors: $\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cos \theta$.

• Orthogonality: Vectors are orthogonal if their dot product is zero.

• Example: $\mathbf{u} = \langle 1, 3 \rangle$, $\mathbf{v} = \langle -2, 4 \rangle$, $\mathbf{u} \cdot \mathbf{v} = 1 \times (-2) + 3 \times 4 = 10$.

• Example: $\mathbf{a} = \langle 4, 6 \rangle$, $\mathbf{b} = \langle -3, 2 \rangle$, $\mathbf{a} \cdot \mathbf{b} = 4 \times (-3) + 6 \times 2 = -12 + 12 = 0$ (orthogonal).

Systems of Linear Equations in Two Variables

Systems of equations can be solved by graphing, substitution, or elimination. The solution may be unique, nonexistent, or infinite.

• Graphing: Plot both equations and find the intersection.

• Substitution: Solve one equation for a variable and substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Types of Solutions: One solution: lines intersect No solution: lines are parallel Infinitely many solutions: lines coincide

• Example: $x + y = 5$, $x - y = 1$: Solution $(3, 2)$.

• Example: $2x + y = 7$, $4x + 2y = 14$: Second equation is a multiple of the first, so infinitely many solutions.

Summary Table: Trigonometric Identities

Identity Type	Formula
Reciprocal	$\sin(\theta) = \frac{1}{\csc(\theta)}$, $\cos(\theta) = \frac{1}{\sec(\theta)}$, $\tan(\theta) = \frac{1}{\cot(\theta)}$
Quotient	$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$
Pythagorean	$\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, $1 + \cot^2(\theta) = \csc^2(\theta)$
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-Angle	$\sin(2\theta) = 2 \sin \theta \cos \theta$, $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$
Half-Angle	$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$, $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

Summary Table: Vector Operations

Operation	Formula
Magnitude	$||\mathbf{u}|| = \sqrt{a^2 + b^2}$
Direction Angle	$\theta = \arctan\left(\frac{b}{a}\right)$
Dot Product	$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$
Unit Vector	$\mathbf{u}_{\text{unit}} = \frac{\mathbf{u}}{||\mathbf{u}||}$

Summary Table: Methods for Solving Systems of Equations

Method	Description
Graphing	Plot both equations and find intersection point
Substitution	Solve one equation for a variable and substitute
Elimination	Add/subtract equations to eliminate a variable