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Trigonometry Syllabus

Mathematical study and application. The full Trigonometry syllabus: 8 chapters and 30 topics, with 52 flashcards written against it.

8Chapters

30Topics

52Flashcards

Trigonometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Trigonometry, not a summary of it.

Fundamentals of Trigonometry
5 topics
- Introduction to Trigonometry
- Angles and Their Measurement
- The Unit Circle
- Trigonometric Functions
- Graphs of Trigonometric Functions
Trigonometric Identities and Equations
5 topics
- Basic Trigonometric Identities
- Sum and Difference Formulas
- Double-Angle and Half-Angle Formulas
- Product-to-Sum and Sum-to-Product Formulas
- Solving Trigonometric Equations
Applications of Trigonometry
5 topics
- Law of Sines
- Law of Cosines
- Area of a Triangle
- Harmonic Motion
- Waves and Oscillations
Inverse Trigonometric Functions
3 topics
- Definition and Properties
- Graphs of Inverse Trigonometric Functions
- Applications of Inverse Trigonometric Functions
Complex Numbers and Trigonometry
4 topics
- Complex Numbers and the Complex Plane
- Polar Form of Complex Numbers
- Euler's Formula
- De Moivre's Theorem
Trigonometric Form of Complex Numbers
2 topics
- Multiplication and Division of Complex Numbers
- Powers and Roots of Complex Numbers
Analytical Trigonometry
3 topics
- Verifying Trigonometric Identities
- Sum and Product Identities
- Trigonometric Equations and Inequalities
Spherical Trigonometry
3 topics
- Spherical Law of Sines
- Spherical Law of Cosines
- Applications in Astronomy and Geography

Trigonometry flashcards

The complete Trigonometry deck: 52 questions with worked answers.

In a right triangle, how are the sine, cosine, and tangent of an acute angle $\theta$ defined in terms of the sides?
$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.
What are the reciprocal trigonometric functions cosecant, secant, and cotangent?
$\csc\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$, and $\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}$.
State the mnemonic SOH-CAH-TOA and what each part represents.
SOH: $\sin = \frac{\text{Opp}}{\text{Hyp}}$; CAH: $\cos = \frac{\text{Adj}}{\text{Hyp}}$; TOA: $\tan = \frac{\text{Opp}}{\text{Adj}}$.
How do you convert an angle from degrees to radians?
Multiply the degree measure by $\dfrac{\pi}{180^{\circ}}$. For example, $180^{\circ} = \pi$ radians.
How do you convert an angle from radians to degrees?
Multiply the radian measure by $\dfrac{180^{\circ}}{\pi}$.
What is the definition of one radian?
One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. A full circle is $2\pi$ radians.
What are the formulas for arc length $s$ and area $A$ of a circular sector with central angle $\theta$ (in radians) and radius $r$?
$s = r\theta$ and $A = \dfrac{1}{2}r^{2}\theta$.
What are coterminal angles and how do you find one?
Coterminal angles share the same terminal side. Add or subtract integer multiples of $360^{\circ}$ (or $2\pi$): $\theta + 360^{\circ}k$ for integer $k$.
On the unit circle, what do the coordinates of the point at angle $\theta$ represent?
The point is $(\cos\theta, \sin\theta)$; the $x$-coordinate is $\cos\theta$ and the $y$-coordinate is $\sin\theta$.
What is the fundamental Pythagorean identity from the unit circle?
$$\sin^{2}\theta + \cos^{2}\theta = 1$$
Give the sine and cosine of the special angle $30^{\circ}$ ($\frac{\pi}{6}$).
$\sin 30^{\circ} = \dfrac{1}{2}$ and $\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}$.
Give the sine and cosine of the special angle $45^{\circ}$ ($\frac{\pi}{4}$).
$\sin 45^{\circ} = \cos 45^{\circ} = \dfrac{\sqrt{2}}{2}$.
Give the sine and cosine of the special angle $60^{\circ}$ ($\frac{\pi}{3}$).
$\sin 60^{\circ} = \dfrac{\sqrt{3}}{2}$ and $\cos 60^{\circ} = \dfrac{1}{2}$.
What is the value of $\tan 45^{\circ}$ and why?
$\tan 45^{\circ} = 1$, because $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\sin 45^{\circ} = \cos 45^{\circ}$.
In which quadrants is each trigonometric function positive (ASTC rule)?
Quadrant I: All positive; II: Sine (and csc); III: Tangent (and cot); IV: Cosine (and sec). Mnemonic: 'All Students Take Calculus.'
State the sign and value pattern of the two quadrantal angles $0$ and $\frac{\pi}{2}$.
At $0$: $\sin 0 = 0$, $\cos 0 = 1$. At $\frac{\pi}{2}$: $\sin\frac{\pi}{2} = 1$, $\cos\frac{\pi}{2} = 0$.
What are the domain and range of $y = \sin x$?
Domain: all real numbers $(-\infty, \infty)$; Range: $[-1, 1]$.
What are the period and range of $y = \tan x$, and where are its vertical asymptotes?
Period $\pi$; range $(-\infty, \infty)$; vertical asymptotes at $x = \dfrac{\pi}{2} + \pi k$ for integer $k$.
For $y = a\sin(bx - c) + d$, what are the amplitude, period, phase shift, and vertical shift?
Amplitude $|a|$; period $\dfrac{2\pi}{|b|}$; phase shift $\dfrac{c}{b}$; vertical shift $d$.
Classify sine and cosine as even or odd functions and give the identities.
Cosine is even: $\cos(-x) = \cos x$. Sine is odd: $\sin(-x) = -\sin x$. Tangent is odd: $\tan(-x) = -\tan x$.

Show the other 32 Trigonometry flashcards

What is the amplitude and period of $y = 3\cos(2x)$?
Amplitude $= 3$; period $= \dfrac{2\pi}{2} = \pi$.
State the two Pythagorean identities derived by dividing $\sin^{2}\theta + \cos^{2}\theta = 1$.
$1 + \tan^{2}\theta = \sec^{2}\theta$ and $1 + \cot^{2}\theta = \csc^{2}\theta$.
State the cofunction identities relating sine and cosine.
$\sin\!\left(\dfrac{\pi}{2} - \theta\right) = \cos\theta$ and $\cos\!\left(\dfrac{\pi}{2} - \theta\right) = \sin\theta$.
What is the quotient identity for tangent and cotangent?
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ and $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$.
State the sine sum and difference formulas.
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$.
State the cosine sum and difference formulas.
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$.
State the tangent sum and difference formulas.
$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$.
Using a difference formula, find $\cos 15^{\circ}$.
$\cos 15^{\circ} = \cos(45^{\circ} - 30^{\circ}) = \dfrac{\sqrt{6} + \sqrt{2}}{4}$.
State the double-angle formula for sine.
$\sin 2\theta = 2\sin\theta\cos\theta$.
State the three forms of the double-angle formula for cosine.
$\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta$.
State the double-angle formula for tangent.
$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^{2}\theta}$.
State the half-angle formulas for sine and cosine.
$\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}$ and $\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}$.
State the half-angle formula for tangent (a non-radical form).
$\tan\dfrac{\theta}{2} = \dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}$.
State the power-reducing formulas for $\sin^{2}\theta$ and $\cos^{2}\theta$.
$\sin^{2}\theta = \dfrac{1 - \cos 2\theta}{2}$ and $\cos^{2}\theta = \dfrac{1 + \cos 2\theta}{2}$.
State the product-to-sum formula for $\sin A\cos B$.
$\sin A\cos B = \dfrac{1}{2}\left[\sin(A+B) + \sin(A-B)\right]$.
State the product-to-sum formula for $\cos A\cos B$.
$\cos A\cos B = \dfrac{1}{2}\left[\cos(A-B) + \cos(A+B)\right]$.
State the product-to-sum formula for $\sin A\sin B$.
$\sin A\sin B = \dfrac{1}{2}\left[\cos(A-B) - \cos(A+B)\right]$.
State the sum-to-product formula for $\sin A + \sin B$.
$\sin A + \sin B = 2\sin\!\left(\dfrac{A+B}{2}\right)\cos\!\left(\dfrac{A-B}{2}\right)$.
State the sum-to-product formula for $\cos A + \cos B$.
$\cos A + \cos B = 2\cos\!\left(\dfrac{A+B}{2}\right)\cos\!\left(\dfrac{A-B}{2}\right)$.
How do you write the general solution of $\sin x = 0$?
$x = \pi k$ for any integer $k$ (i.e., all integer multiples of $\pi$).
Give the general solution of $\cos x = 0$.
$x = \dfrac{\pi}{2} + \pi k$ for any integer $k$.
What are all solutions of $\sin x = \dfrac{1}{2}$ on $[0, 2\pi)$?
$x = \dfrac{\pi}{6}$ and $x = \dfrac{5\pi}{6}$.
State the Law of Sines.
$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$ where each side is opposite its named angle.
State the Law of Cosines for side $c$.
$$c^{2} = a^{2} + b^{2} - 2ab\cos C$$
Which triangle cases (given information) call for the Law of Sines versus the Law of Cosines?
Law of Sines: ASA, AAS, and the ambiguous SSA case. Law of Cosines: SAS and SSS.
What is the 'ambiguous case' in solving triangles?
The SSA case, where two sides and a non-included angle are given, can yield zero, one, or two possible triangles.
Give two formulas for the area of a triangle: one using two sides and the included angle, and one using base and height.
$\text{Area} = \dfrac{1}{2}ab\sin C$ and $\text{Area} = \dfrac{1}{2}bh$ (base $\times$ height).
State Heron's formula for the area of a triangle with sides $a, b, c$.
$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \dfrac{a+b+c}{2}$ is the semiperimeter.
Write the general equation of simple harmonic motion and identify amplitude and period.
$y = A\sin(\omega t + \phi)$ where $A$ is the amplitude, $\omega$ is the angular frequency, and the period is $T = \dfrac{2\pi}{\omega}$.
For a wave $y = A\sin(\omega t)$, how are frequency $f$, angular frequency $\omega$, and period $T$ related?
$\omega = 2\pi f$ and $f = \dfrac{1}{T}$, so $\omega = \dfrac{2\pi}{T}$.
What are the domains and ranges of the inverse functions $\arcsin x$ and $\arctan x$?
$\arcsin x$: domain $[-1,1]$, range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. $\arctan x$: domain $(-\infty,\infty)$, range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
What is the range of $y = \arccos x$, and how does its graph differ from $\arcsin x$?
Range of $\arccos x$ is $[0, \pi]$; it is a decreasing function on $[-1,1]$, whereas $\arcsin x$ is increasing on $[-1,1]$ with range $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

Trigonometry syllabus FAQ

What is in the Trigonometry syllabus?

The Trigonometry syllabus is split into 8 chapters: Fundamentals of Trigonometry, Trigonometric Identities and Equations, Applications of Trigonometry, Inverse Trigonometric Functions, Complex Numbers and Trigonometry, Trigonometric Form of Complex Numbers, Analytical Trigonometry and Spherical Trigonometry. Together they contain 30 topics.

How many chapters and topics are in the Trigonometry syllabus?

8 chapters and 30 topics. The largest chapter is Fundamentals of Trigonometry, with 5 topics.

Are there Trigonometry flashcards?

Yes — 52 question-and-answer flashcards, printed in full on this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.

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