Trigonometry - a MaS Tutor.

Definitions:

\sin θ = \frac{opposite}{hypotenuse}

\cos θ = \frac{adjacent}{hypotenuse}

\tan θ = \frac{opposite}{adjacent}

Some say "S^OH C^AH T^OA"

Trigonometry introduction:: Practice Questions; Textbook Exercise
Missing sides:: Practice Questions; Textbook Exercise
Missing angles:: Practice Questions; Textbook Exercise
Trigonometry in 3D Video: Practice Questions; Textbook Exercise
Area of a triangle Video: Proof; Practice Questions; Textbook Exercise

Trigonometric Graphs

Warm-up

Sine graph Video
Cosine graph Video
Tangent graph Video: Practice Questions; Textbook Exercise
Transformations of trigonometric graphs Video: Practice Questions

Inverse Trigonometric Functions

y = sin x

then

arcsin y = x

$arcsin x$ can be written as ${sin}^{-1} x$ , and this is how it usually appears on a calculator.

Note: ${sin}^{-1} x \neq \frac{1}{sin x}$ , as $\frac{1}{sin x} = cosec x$

There are similar inverse functions for cosine and tangent:

Inverse Trig Function	Domain	Range
$y = arcsin x$	$-1 \leq x \leq 1$	$-90 ° \leq y \leq 90 °$ or $-π / 2 \leq y \leq π / 2$
$y = arccos x$	$-1 \leq x \leq 1$	$0 ° \leq y \leq 180 °$ or $0 \leq y \leq π$
$y = arctan x$	$x \in ℝ$	$-90 ° \leq y \leq 90 °$ or $-π / 2 \leq y \leq π / 2$

Graphs of Inverse Trig Functions:

Three graphs of the inverse trig functions side-by-side

Questions:

Year 2 Edexcel Pure Text Exercise 6E page 160 [ MS (archive version)]

Inverse Trigonometric Functions Practice Problems
(Care of www.geeksforgeeks.org)

Inverse Trig Functions
(Care of mmerevise.co.uk)

Questions
(Care of www.analyzemath.com)

... Search for More ...
(Care of duckduckgo.com)

Trig Values to Remember

You need to learn the exact values of the trigonometric ratios for certain angles measured in Radians (and degrees).

Angle	degrees / °	0	30	45	60	90
Angle	Radians	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
sine		0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
cosine		1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
tangent		0	$\frac{\sqrt{3}}{3}$	1	$\sqrt{3}$	[undefined]

Trigonometry: Exact Values Video: Practice Questions; Textbook Exercise

You can use these rules to find sin, cos or tan of any positive or negative angles measured in Radians using the corresponding acute angle from the table above.

$sin (π - θ) = sin θ$
$sin (π + θ) = - sin θ$
$sin (2 π - θ) = - sin θ$
$cos (π - θ) = - cos θ$
$cos (π + θ) = - cos θ$
$cos (2 π - θ) = cos θ$
$tan (π - θ) = - tan θ$
$tan (π + θ) = tan θ$
$tan (2 π - θ) = - tan θ$

(Note: similar relationships can be created for angles in degrees by replacing $π$ with $180°$ in the above.)

Small Angle Approximations:

When $θ$ is small and measured in Radians:

$sin θ \approx θ$

$cos θ \approx 1 - \frac{θ^{2}}{2}$

$tan θ \approx θ$

These approximations are truncations of the associated Maclaurin series.

Questions:

Small Angle Approximation (Care of www.mathsgenie.co.uk)

Small Angle Approximation (Care of www.jethwamaths.com)

Small Angle Approximation (Care of alevelmathsrevision.com)

Sine and Cosine Rule

Standard Labeling of a Triangle

The Law of Sines
(AKA the sine rule)

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

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

Sine rule - missing sides video;
Sine rule - missing angles video;
Sine rule - ambiguous case video;: Practice Questions;; Textbook Questions.; (Care of corbettmaths.com)

al-Kashi's Theorem
(AKA the cosine rule)

c^{2} = a^{2} + b^{2} - 2 a b \cos C

a^{2} = b^{2} + c^{2} - 2 b c \cos A

b^{2} = a^{2} + c^{2} - 2 a c \cos B

Cosine rule - missing sides video;
Cosine rule - missing angles video.: Sine & Cosine Rule Practice Questions.; Textbook Questions.; (Care of corbettmaths.com)

Sine and Cosine Rule
(Care of revisionmaths.com)

Cosine Rule
(Care of www.mathsisfun.com)

Cosine Rule
Sine Rule
(Care of gcse-math.co.uk)

Trigonometric Identities

Trigonometric Equations

Trigonometry Cheat Sheets and Worksheets
(Aimed at A level Edexcel)
(Care of www.physicsandmathstutor.com)

Pythag and Trig

a^{2} + b^{2} = c^{2}

Divide throughout by $c^{2}$

\frac{a^{2}}{c^{2}} + \frac{b^{2}}{c^{2}} = \frac{c^{2}}{c^{2}}

{(\frac{a}{c})}^{2} + {(\frac{b}{c})}^{2} = 1

{(sin θ)}^{2} + {(cos θ)}^{2} = 1

{sin}^{2} θ + {cos}^{2} θ = 1

Taking

{sin}^{2} θ + {cos}^{2} θ = 1

and dividing by ${cos}^{2} θ$

\frac{{sin}^{2} θ}{{cos}^{2} θ} + \frac{{cos}^{2} θ}{{cos}^{2} θ} = \frac{1}{{cos}^{2} θ}

{tan}^{2} θ + 1 = {sec}^{2} θ

Taking

{sin}^{2} θ + {cos}^{2} θ = 1

and dividing by ${sin}^{2} θ$

\frac{{sin}^{2} θ}{{sin}^{2} θ} + \frac{{cos}^{2} θ}{{sin}^{2} θ} = \frac{1}{{sin}^{2} θ}

1 + {cot}^{2} θ = {cosec}^{2} θ

Trigonometric Identities

tan x = \frac{oppposite}{adjacent} = \frac{hypotenuse \times sin x}{hypotenuse \times cos x} = \frac{sin x}{cos x}

Reciprocal Trigonometric Functions

By definition:

cosec x = \frac{1}{sin x}

sec x = \frac{1}{cos x}

cot x = \frac{1}{tan x} = \frac{cos x}{sin x} = \frac{cosec x}{sec x}

Trigonometric version of Pythagorus

\sin^{2} θ + \cos^{2} θ = 1

... dividing above by $sin θ$

1 + {cot}^{2} θ = {cosec}^{2} θ

... dividing the first by $cos θ$

\tan^{2} θ + 1 = \sec^{2} θ

Addition Formulae

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

when $B = A$ then ...

\sin 2 A = 2 \sin A \cos A

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

when $B = A$ then ...

\cos 2 A = \cos^{2} A - \sin^{2} A

= 2 \cos^{2} A - 1

= 1 - 2 \sin^{2} A

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

when $B = A$ then ...

\tan 2 A = \frac{2 \tan A}{1 - \tan^{2} A}

Definitions:

Some say "S O H C A H T O A"

Trigonometric Graphs

Warm-up

Inverse Trigonometric Functions

Graphs of Inverse Trig Functions:

Questions:

Year 2 Edexcel Pure Text Exercise 6E page 160 [ MS (archive version)]

Inverse Trigonometric Functions Practice Problems (Care of www.geeksforgeeks.org)

Inverse Trig Functions (Care of mmerevise.co.uk)

Questions (Care of www.analyzemath.com)

... Search for More ... (Care of duckduckgo.com)

Trig Values to Remember

Small Angle Approximations:

sinθ≈θ

cosθ≈1-θ22

tanθ≈θ

Questions:

Small Angle Approximation (Care of www.mathsgenie.co.uk)

Small Angle Approximation (Care of www.jethwamaths.com)

Small Angle Approximation (Care of alevelmathsrevision.com)

Sine and Cosine Rule

Standard Labeling of a Triangle

The Law of Sines(AKA the sine rule)

al-Kashi's Theorem(AKA the cosine rule)

More Guidance and Questions:

Sine and Cosine Rule (Care of revisionmaths.com)

Cosine Rule (Care of www.mathsisfun.com)

Cosine Rule Sine Rule (Care of gcse-math.co.uk)

Further Reading:

Trigonometric Identities

Trigonometric Equations

Trigonometry Cheat Sheets and Worksheets (Aimed at A level Edexcel) (Care of www.physicsandmathstutor.com)

Pythag and Trig

Trigonometric Identities

Reciprocal Trigonometric Functions

Addition Formulae

Questions:

Questions (Care of www.revisely.co.uk)

Questions (Care of www.examsolutions.net)

Questions (Care of revisionmaths.com)

Some say "S^OH C^AH T^OA"

Inverse Trigonometric Functions Practice Problems
(Care of www.geeksforgeeks.org)

Inverse Trig Functions
(Care of mmerevise.co.uk)

Questions
(Care of www.analyzemath.com)

... Search for More ...
(Care of duckduckgo.com)

$sin θ \approx θ$

$cos θ \approx 1 - \frac{θ^{2}}{2}$

$tan θ \approx θ$

The Law of Sines
(AKA the sine rule)

al-Kashi's Theorem
(AKA the cosine rule)

Sine and Cosine Rule
(Care of revisionmaths.com)

Cosine Rule
(Care of www.mathsisfun.com)

Cosine Rule
Sine Rule
(Care of gcse-math.co.uk)

Trigonometry Cheat Sheets and Worksheets
(Aimed at A level Edexcel)
(Care of www.physicsandmathstutor.com)

Questions
(Care of www.revisely.co.uk)

Questions
(Care of www.examsolutions.net)

Questions
(Care of revisionmaths.com)