Trigonometry Comprehensive Cheat Sheet

1. Basic Trigonometric Ratios

1.1 Sine, Cosine, and Tangent

In a right triangle, the basic trigonometric ratios are defined as:

Sine (sin): \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
Cosine (cos): \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
Tangent (tan): \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

Example:

In a right triangle, if the angle \( \theta \) is 30°, the hypotenuse is 10, and the side opposite to \( \theta \) is 5, find the sine, cosine, and tangent of \( \theta \).

\(\sin 30° = \frac{5}{10} = 0.5\)
\(\cos 30° = \frac{\sqrt{3} \times 5}{10} = \frac{\sqrt{3}}{2}\)
\(\tan 30° = \frac{5}{\sqrt{3} \times 5} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)

Result: \( \sin 30° = 0.5 \), \( \cos 30° = \frac{\sqrt{3}}{2} \), \( \tan 30° = \frac{\sqrt{3}}{3} \).

2. Trigonometric Identities

2.1 Pythagorean Identity

The Pythagorean Identity is a fundamental relation among the trigonometric functions:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Example:

If \( \sin \theta = \frac{3}{5} \), find \( \cos \theta \).

Use the identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
Substitute \( \sin \theta \): \( \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \)
Simplify: \( \frac{9}{25} + \cos^2 \theta = 1 \)
Subtract \( \frac{9}{25} \) from both sides: \( \cos^2 \theta = \frac{16}{25} \)
Take the square root: \( \cos \theta = \pm \frac{4}{5} \)

Result: \( \cos \theta = \pm \frac{4}{5} \).

2.2 Sum and Difference Formulas

These formulas are used to find the sine, cosine, and tangent of the sum or difference of two angles:

Sine: \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
Cosine: \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
Tangent: \( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)

Example:

Find \( \sin(75°) \) using the sum formula.

Express 75° as the sum of 45° and 30°: \( \sin(75°) = \sin(45° + 30°) \)
Use the sum formula: \( \sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30° \)
Substitute known values: \( \sin 45° = \frac{\sqrt{2}}{2}, \cos 30° = \frac{\sqrt{3}}{2}, \cos 45° = \frac{\sqrt{2}}{2}, \sin 30° = \frac{1}{2} \)
Calculate: \( \sin(75°) = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Result: \( \sin(75°) = \frac{\sqrt{6} + \sqrt{2}}{4} \).

2.3 Double-Angle Formulas

These formulas are used to express trigonometric functions of double angles in terms of single angles:

Sine: \( \sin(2\theta) = 2 \sin \theta \cos \theta \)
Cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \)
Tangent: \( \tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

Example:

Find \( \cos(2\theta) \) if \( \sin \theta = \frac{3}{5} \).

Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to find \( \cos \theta \).
Substitute \( \sin \theta = \frac{3}{5} \): \( \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} \).
So, \( \cos \theta = \pm \frac{4}{5} \).
Use the double-angle formula: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \).

Result: \( \cos(2\theta) = \frac{7}{25} \).

3. Inverse Trigonometric Functions

3.1 Definitions and Properties

Inverse trigonometric functions are used to find the angle that corresponds to a given trigonometric ratio:

Inverse Sine: \( \sin^{-1}(x) = \theta \), where \( \sin(\theta) = x \) and \( -1 \leq x \leq 1 \).
Inverse Cosine: \( \cos^{-1}(x) = \theta \), where \( \cos(\theta) = x \) and \( -1 \leq x \leq 1 \).
Inverse Tangent: \( \tan^{-1}(x) = \theta \), where \( \tan(\theta) = x \) and \( x \) can be any real number.

Example:

Find \( \theta \) if \( \sin^{-1}\left(\frac{1}{2}\right) = \theta \).

Find the angle \( \theta \) such that \( \sin(\theta) = \frac{1}{2} \).
From the unit circle, \( \theta = 30° \) or \( \theta = \frac{\pi}{6} \).
Result: \( \theta = 30° \) or \( \theta = \frac{\pi}{6} \).

4. Trigonometric Equations

4.1 Solving Basic Trigonometric Equations

To solve trigonometric equations, use algebraic techniques and trigonometric identities:

Isolate the trigonometric function.
Use inverse trigonometric functions to find the angle.
Check for all possible solutions within the given interval.

Example:

Solve \( 2\sin \theta - 1 = 0 \) for \( 0 \leq \theta \leq 360° \).

Isolate the sine function: \( \sin \theta = \frac{1}{2} \).
Find the angles where \( \sin \theta = \frac{1}{2} \): \( \theta = 30°, 150° \).
Result: \( \theta = 30°, 150° \).

4.2 Solving Trigonometric Equations with Identities

Some trigonometric equations require the use of identities to simplify and solve:

Use Pythagorean identities, double-angle formulas, or sum and difference formulas to simplify the equation.
Solve for the variable as in basic trigonometric equations.

Example:

Solve \( \cos 2\theta = \sin \theta \) for \( 0 \leq \theta \leq 360° \).

Use the double-angle identity: \( \cos 2\theta = 1 - 2\sin^2 \theta \).
Set up the equation: \( 1 - 2\sin^2 \theta = \sin \theta \).
Rearrange into a quadratic equation: \( 2\sin^2 \theta + \sin \theta - 1 = 0 \).
Factor the quadratic: \( (2\sin \theta - 1)(\sin \theta + 1) = 0 \).
Solve for \( \sin \theta = \frac{1}{2} \) and \( \sin \theta = -1 \).
Result: \( \theta = 30°, 150°, 270° \).

5. Law of Sines and Cosines

5.1 Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its angles:

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

Example:

Find the length of side \( b \) in a triangle where \( A = 30° \), \( B = 45° \), and \( a = 10 \).

Use the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
Substitute the known values: \( \frac{10}{\sin 30°} = \frac{b}{\sin 45°} \).
Solve for \( b \): \( b = 10 \times \frac{\sin 45°}{\sin 30°} = 10 \times \frac{\frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2} \).

Result: \( b = 10\sqrt{2} \).

5.2 Law of Cosines

The Law of Cosines is used to find the sides or angles of a triangle when you know two sides and the included angle, or all three sides:

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

Example:

Find the angle \( C \) in a triangle where \( a = 7 \), \( b = 10 \), and \( c = 12 \).

Use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cos C \).
Substitute the known values: \( 12^2 = 7^2 + 10^2 - 2(7)(10) \cos C \).
Solve for \( \cos C \): \( 144 = 49 + 100 - 140 \cos C \), \( 144 = 149 - 140 \cos C \), \( 140 \cos C = 5 \), \( \cos C = \frac{5}{140} = \frac{1}{28} \).
Find \( C \): \( C = \cos^{-1}\left(\frac{1}{28}\right) \approx 87.9° \).

Result: \( C \approx 87.9° \).

6. Trigonometric Applications

6.1 Solving Right Triangles

To solve a right triangle, use the trigonometric ratios (sine, cosine, tangent) along with the Pythagorean Theorem to find the unknown sides and angles.

Example:

Solve a right triangle with hypotenuse 13 and one leg 5.

Use the Pythagorean Theorem to find the other leg: \( a^2 + 5^2 = 13^2 \), \( a^2 + 25 = 169 \), \( a^2 = 144 \), \( a = 12 \).
Find the angles using trigonometric ratios: \( \sin \theta = \frac{5}{13} \), \( \theta = \sin^{-1}\left(\frac{5}{13}\right) \approx 22.6° \).
Result: The other leg is 12, and the angles are \( \theta \approx 22.6° \) and \( 90° - 22.6° = 67.4° \).

6.2 Area of a Triangle Using Trigonometry

The area of a triangle can be found using the formula:

\[ \text{Area} = \frac{1}{2}ab \sin C \]

where \( a \) and \( b \) are sides, and \( C \) is the included angle.

Example:

Find the area of a triangle with sides \( a = 7 \), \( b = 10 \), and included angle \( C = 60° \).

Use the area formula: \( \text{Area} = \frac{1}{2} \times 7 \times 10 \times \sin 60° \).
Substitute the known values: \( \text{Area} = \frac{1}{2} \times 7 \times 10 \times \frac{\sqrt{3}}{2} \).
Simplify: \( \text{Area} = 35 \times \frac{\sqrt{3}}{2} \).

Result: \( \text{Area} = 35\sqrt{3} \) square units.