Study guide: Trigonometry Essentials

📋 Course Outline

1. Unit Circle and Arc Lengths
2. Angles and Radian Measures
3. Coterminal Angles
4. Principal Measures
5. Sine and Cosine on the Circle
6. Basic Trigonometric Equations
7. Trigonometric Inequalities
8. Symmetries and Shift Formulas
9. Equations with Sine and Cosine
10. Addition and Double-Angle Formulas

📖 1. Unit Circle and Arc Lengths

🔑 Key Concepts & Definitions

• Unit circle: a circle centered at the origin with radius 1, whose circumference is $2\pi$

★ Must-know

• On the unit circle, a complete positive turn has arc length $2\pi$, a half-turn has length $\pi$, and a quarter-turn has length $\frac{\pi}{2}$.

Further detail

• The positive direction on the trigonometric circle is counterclockwise, while the negative direction is clockwise.

💡 Memory Hook

Picture a unit circle marked from 0 around one complete turn to 2π.

📖 2. Angles and Radian Measures

★ Must-know

📌 A radian measures an arc length on the unit circle, whereas a degree measures an angle relative to a full turn of 360 degrees.

📐 Formula — The conversion relation between radians and degrees is $\pi\text{ radians}=180^\circ$.

Further detail

• The commonly memorized correspondences are $\frac{\pi}{6}=30^\circ$, $\frac{\pi}{4}=45^\circ$, $\frac{\pi}{3}=60^\circ$, and $\frac{\pi}{2}=90^\circ$.

💡 Memory Hook

Radians measure arc length, whereas degrees measure angular opening.

📖 3. Coterminal Angles

📝 Essential Points

• All angles coterminal with an angle $x$ are given by $x+2k\pi$, where $k\in\mathbb{Z}$.

📌 Two angles represent the same point on the unit circle exactly when their difference is an integer multiple of $2\pi$.

💡 Memory Hook

Start at x, then add or subtract complete turns 2πk.

📖 4. Principal Measures

🔑 Key Concepts & Definitions

• Principal measure: the unique coterminal angle belonging to the interval $[-\pi,\pi]$

📝 Essential Points

• To find a principal measure, subtract or add complete turns $2\pi$ until the angle lies in $[-\pi,\pi]$.

💡 Memory Hook

Reduce by complete turns until the angle lies between −π and π.

📖 5. Sine and Cosine on the Circle

🔑 Key Concepts & Definitions

• Cosine: the horizontal, or abscissa, coordinate of its point on the unit circle
• Sine: the vertical, or ordinate, coordinate of its point on the unit circle

📝 Essential Points

📌 For every real number $x$, the cosine and sine satisfy $-1\leq\cos x\leq1$ and $-1\leq\sin x\leq1$.

📐 Formula — For every real number $x$, the fundamental identity is $\cos^2x+\sin^2x=1$.

💡 Memory Hook

Cosine is the horizontal coordinate, whereas sine is the vertical coordinate.

📖 6. Basic Trigonometric Equations

★ Must-know

• 🔄 The solving method is:
  1. draw the unit circle
  2. mark the required sine or cosine value
  3. identify the corresponding angles
  4. retain the angles in the stated interval

Further detail

• The solutions of $\cos x=-\frac12$ on $[-\pi,\pi]$ are $x=\frac{2\pi}{3}$ and $x=-\frac{2\pi}{3}$.

📌 When solving an equation such as $\cos x=-1$, the only point on the unit circle is the angle $\pi$ modulo complete turns.

💡 Memory Hook

Sketch the circle, place the target value, then select solutions in the interval.

📖 7. Trigonometric Inequalities

📝 Essential Points

• 🔄 The inequality method is: draw the unit circle, mark the boundary angles, select the arc satisfying the inequality, include or exclude endpoints according to the sign

📌 Dividing a trigonometric inequality by a negative number reverses its inequality sign.

💡 Memory Hook

Sketch, mark the boundary values, keep the correct arcs, then respect endpoints.

📖 8. Symmetries and Shift Formulas

★ Must-know

📐 Formula — Cosine is even, so $\cos(-x)=\cos x$, while sine is odd, so $\sin(-x)=-\sin x$.

Further detail

📐 Formula — The half-turn identities are $\cos(x+\pi)=-\cos x$ and $\sin(x+\pi)=-\sin x$.

📐 Formula — The quarter-turn identities are $\cos\left(x+\frac{\pi}{2}\right)=-\sin x$ and $\sin\left(x+\frac{\pi}{2}\right)=\cos x$.

💡 Memory Hook

Cosine is even, while sine changes sign under x → −x.

📖 9. Equations with Sine and Cosine

★ Must-know

📌 For an equation $\cos A=\cos B$, all solutions satisfy either $A=B+2k\pi$ or $A=-B+2k\pi$, with $k\in\mathbb{Z}$.

• To solve an equation containing both sine and cosine, first rewrite one function using a quarter-turn identity so that both sides use the same trigonometric function.

Further detail

📐 Formula — The fundamental identity allows one function to be recovered from the other through $\sin^2x=1-\cos^2x$ or $\cos^2x=1-\sin^2x$.

💡 Memory Hook

Transform unlike functions into the same function, then use the standard equation patterns.

📖 10. Addition and Double-Angle Formulas

★ Must-know

📐 Formula — The addition formulas are $\cos(a+b)=\cos a\cos b-\sin a\sin b$ and $\sin(a+b)=\sin a\cos b+\cos a\sin b$.

📐 Formula — The double-angle formulas are $\cos(2a)=2\cos^2a-1$ and $\sin(2a)=2\sin a\cos a$.

Further detail

📐 Formula — The subtraction formulas are $\cos(a-b)=\cos a\cos b+\sin a\sin b$ and $\sin(a-b)=\sin a\cos b-\cos a\sin b$.

• To compute an exact value such as $\cos\left(\frac{\pi}{12}\right)$, express $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$ and apply the cosine subtraction formula.

💡 Memory Hook

An angle decomposition enables addition formulas, which produce double-angle identities and exact values.

📊 Synthesis Tables

Sine and cosine on the unit circle