Glossary
Cosine Difference Identity
The identity `cos(a - b) = cos(a)cos(b) + sin(a)sin(b)`, used to expand the cosine of the difference of two angles.
Example:
To prove that cos(π/2 - x) equals sin(x), you would apply the Cosine Difference Identity.
Cosine Double-Angle Identities
A set of three identities for `cos(2a)`: `cos²(a) - sin²(a)`, `1 - 2sin²(a)`, and `2cos²(a) - 1`. The choice depends on the problem's context.
Example:
To simplify cos(2x) + sin²(x), choosing the form 1 - 2sin²(x) for the Cosine Double-Angle Identities quickly leads to 1 - sin²(x) or cos²(x).
Cosine Sum Identity
The identity `cos(a + b) = cos(a)cos(b) - sin(a)sin(b)`, used to expand the cosine of a sum of two angles.
Example:
If you need to simplify cos(x + π/2), the Cosine Sum Identity reveals it's equivalent to -sin(x).
Cotangent and Cosecant Identity
A derived Pythagorean identity stating `1 + cot²(x) = csc²(x)`. It relates the cotangent and cosecant functions.
Example:
To prove that csc²(x) - cot²(x) = 1, you can directly apply the Cotangent and Cosecant Identity.
Difference Identities
Formulas used to find the trigonometric values of the difference between two angles, such as `sin(a - b)` or `cos(a - b)`.
Example:
Calculating sin(15°) can be achieved using Difference Identities by writing 15° as 45° - 30°.
Double-Angle Identities
Special cases of sum identities where the two angles are equal, used to express trigonometric functions of `2a` in terms of functions of `a`.
Example:
When solving equations involving sin(2x) or cos(2x), you'll frequently use Double-Angle Identities to rewrite them in terms of sin(x) and cos(x).
Pythagorean Identity
The most fundamental trigonometric identity, stating that for any angle x, `sin²(x) + cos²(x) = 1`. It is derived from the Pythagorean theorem on the unit circle.
Example:
If you know sin(x) = 3/5, you can quickly find cos(x) using the Pythagorean Identity: (3/5)² + cos²(x) = 1, leading to cos(x) = 4/5 (in the first quadrant).
Sine Difference Identity
The identity `sin(a - b) = sin(a)cos(b) - cos(a)sin(b)`, used to expand the sine of the difference of two angles.
Example:
The expression sin(90° - x) can be simplified to cos(x) using the Sine Difference Identity.
Sine Double-Angle Identity
The identity `sin(2a) = 2sin(a)cos(a)`, which expresses the sine of a double angle.
Example:
If sin(x) = 0.6 and cos(x) = 0.8, then sin(2x) can be found using the Sine Double-Angle Identity as 2 * 0.6 * 0.8 = 0.96.
Sine Sum Identity
The identity `sin(a + b) = sin(a)cos(b) + cos(a)sin(b)`, used to expand the sine of a sum of two angles.
Example:
Finding sin(π/3 + π/6) can be done using the Sine Sum Identity, which simplifies to sin(π/2) = 1.
Sum Identities
Formulas used to find the trigonometric values of the sum of two angles, such as `sin(a + b)` or `cos(a + b)`.
Example:
To calculate the exact value of cos(75°), you can use Sum Identities by expressing 75° as 45° + 30°.
Tangent and Secant Identity
A derived Pythagorean identity stating `1 + tan²(x) = sec²(x)`. It relates the tangent and secant functions.
Example:
When simplifying sec²(x) - tan²(x), you can use the Tangent and Secant Identity to immediately see the expression equals 1.
Trigonometric Identities
Equations involving trigonometric functions that are true for all values of the variables for which the functions are defined.
Example:
Understanding trigonometric identities allows you to simplify complex expressions like sin(x)cos(y) + cos(x)sin(y) into a simpler form, sin(x+y).