Formulas & Identities

Trigonometric identities are essential tools in simplifying and solving trigonometric equations. These identities are foundational for advanced concepts in calculus, physics, and engineering. Some of the most important identities include the Pythagorean Identity, the Sum and Difference Formulas, and the Double Angle Formulas, which express trigonometric functions of one angle in terms of functions of another.

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental identities and stems from the Pythagorean theorem. It states that: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

For example, if \( \sin(\theta) = \frac{3}{5} \), then: \( \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25} \). Therefore, \( \cos(\theta) = \frac{4}{5} \).

Sum and Difference Formulas

The Sum and Difference Formulas allow us to calculate the sine and cosine of the sum or difference of two angles. This is useful for breaking down complex angles into simpler expressions.

• For Sine: \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)
• For Cosine: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)

Example: If \( A = 30^\circ \) and \( B = 45^\circ \):
\( \sin(30^\circ + 45^\circ) = \sin(30^\circ)\cos(45^\circ) + \cos(30^\circ)\sin(45^\circ) \).
Using values: \( \sin(30^\circ) = \frac{1}{2}, \cos(45^\circ) = \frac{\sqrt{2}}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}, \sin(45^\circ) = \frac{\sqrt{2}}{2} \), we get:
\( \sin(75^\circ) = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \).
Result: \( \sin(75^\circ) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \).

Formulas & Identities

Trigonometric identities are essential tools in simplifying and solving trigonometric equations. Some of the most important identities include the Pythagorean Identity, Sum and Difference Formulas, and Double Angle Formulas.

Pythagorean Identity

Formula: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

Example: If \( \sin(\theta) = 0.6 \), find \( \cos(\theta) \):

\[ \cos^2(\theta) = 1 - \sin^2(\theta) \]

\[ \cos^2(\theta) = 1 - (0.6)^2 = 1 - 0.36 = 0.64 \]

\[ \cos(\theta) = \sqrt{0.64} = 0.8 \]

Sum and Difference Formulas

• Sine Sum: \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)
• Sine Difference: \( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \)
• Cosine Sum: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)
• Cosine Difference: \( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \)

Example: Calculate \( \sin(45^\circ + 30^\circ) \):

\[ \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) \]

Using values: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = 0.5 \)

\[ \sin(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot 0.5 = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Inverse Trigonometric Functions

Inverse trigonometric functions allow you to find the angle that corresponds to a given trigonometric value. They include:

• \( \sin^{-1}(x) \) or arcsin(x)
• \( \cos^{-1}(x) \) or arccos(x)
• \( \tan^{-1}(x) \) or arctan(x)

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles:

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

This law is useful for calculating the length of a side or an angle in any triangle.

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin. It is fundamental in trigonometry as it helps to define trigonometric functions for all angles.

Key angles on the unit circle include:

• 0°, 90°, 180°, and 270°
• Corresponding radians: 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \)

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry and is derived from the Pythagorean theorem.

Formula:

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

This identity shows that for any angle \( \theta \), the square of the sine of the angle plus the square of the cosine of the angle always equals one.

Sum and Difference Formulas

The Sum and Difference Formulas help find the sine or cosine of the sum or difference of two angles.

Formulas:

• Sine: \( \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \)
• Cosine: \( \cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) \)

These formulas are especially useful in solving trigonometric equations or in simplifying expressions involving multiple angles.

Double Angle Formulas

The Double Angle Formulas allow us to find the trigonometric values of double angles (2A) in terms of single angles (A).

Formulas:

• Sine: \( \sin(2A) = 2\sin(A)\cos(A) \)
• Cosine: \( \cos(2A) = \cos^2(A) - \sin^2(A) \)

These formulas are particularly useful for solving equations and proving identities in trigonometry.

Trigonometric Functions

Trigonometric Functions