Trigonometric Identities

Trigonometric identities are foundational concepts in trigonometry that provide relationships between the angles and sides of triangles. They simplify complex trigonometric expressions and are essential for solving equations and proving other mathematical concepts. In this article, we will delve into key trigonometric identities, including the Pythagorean identities, angle sum identities, and double angle identities. We'll go through their proofs and provide examples to enhance your understanding.

1. Pythagorean Identities

The Pythagorean identities arise from the Pythagorean theorem, where the square of the hypotenuse equals the sum of the squares of the other two sides. In trigonometry, these identities relate the sine and cosine functions:

  1. First Identity: \[ \sin^2(x) + \cos^2(x) = 1 \]

  2. Second Identity: \[ 1 + \tan^2(x) = \sec^2(x) \]

  3. Third Identity: \[ 1 + \cot^2(x) = \csc^2(x) \]

Proof of the First Pythagorean Identity

To understand the first identity, consider a right triangle where \( x \) is one of the angles. According to the definitions of sine and cosine in a right triangle, we have:

  • \(\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}\)
  • \(\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

If we let the hypotenuse be \( r \), the opposite side be \( a \), and the adjacent side be \( b \), we can apply the Pythagorean theorem:

\[ r^2 = a^2 + b^2 \]

Dividing all terms by \( r^2 \):

\[ 1 = \left(\frac{a}{r}\right)^2 + \left(\frac{b}{r}\right)^2 \]

This leads to:

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

Thus, the first Pythagorean identity is proven.

Examples

  • Use the identity \( \sin^2(30^\circ) + \cos^2(30^\circ) = 1 \).

    \[ \sin(30^\circ) = \frac{1}{2} \quad \text{and} \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

    \[ \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1 \]

As expected, the identity holds true.

2. Angle Sum Identities

The angle sum identities express the sine and cosine of a sum of angles in terms of the sine and cosine of the individual angles:

  1. Sine of a Sum: \[ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \]

  2. Cosine of a Sum: \[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]

  3. Tangent of a Sum: \[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)} \quad (if \quad 1 - \tan(a)\tan(b) \neq 0) \]

Proof of the Sine of a Sum Identity

We derive the sine of a sum identity using the unit circle. Let \((x_1, y_1)\) represent the coordinates for angle \( a \) and \((x_2, y_2)\) for angle \( b \):

  • \( x_1 = \cos(a), y_1 = \sin(a) \)
  • \( x_2 = \cos(b), y_2 = \sin(b) \)

We can now express \( \sin(a + b) \) in terms of these coordinates as follows:

  • Rotate the point \((\cos(b), \sin(b))\) by an angle \( a \):

\[ \begin{pmatrix} \cos(a) & -\sin(a) \ \sin(a) & \cos(a) \end{pmatrix} \begin{pmatrix} \cos(b) \ \sin(b) \end{pmatrix} \]

After performing matrix multiplication and simplifying, we find:

\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]

Examples

  • Compute \( \sin(30^\circ + 45^\circ) \) using the angle sum identity:

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

Substituting the values:

\[ = \left(\frac{1}{2} \cdot \frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \]

3. Double Angle Identities

Double angle identities express trigonometric functions of double angles in terms of single angles:

  1. Sine Double Angle: \[ \sin(2x) = 2\sin(x)\cos(x) \]

  2. Cosine Double Angle: \[ \cos(2x) = \cos^2(x) - \sin^2(x) \] This can also be expressed as: \[ \cos(2x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x) \]

  3. Tangent Double Angle: \[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \quad (if \quad 1 - \tan^2(x) \neq 0) \]

Proof of the Sine Double Angle Identity

The sine double angle identity can be derived directly from the angle sum identity:

\[ \sin(2x) = \sin(x + x) = \sin(x) \cos(x) + \cos(x) \sin(x) = 2\sin(x)\cos(x) \]

Examples

  • Calculate \( \sin(2 \times 30^\circ) \):

\[ \sin(60^\circ) = 2\sin(30^\circ)\cos(30^\circ) = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

  • Verify \( \cos(60^\circ) \) using the cosine double angle identity:

\[ \cos(2 \times 30^\circ) = \cos^2(30^\circ) - \sin^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \]

Conclusion

Trigonometric identities are powerful tools in Pre-Calculus that enable us to simplify and solve problems involving angles and triangulations. Familiarizing yourself with these identities, including the Pythagorean, angle sum, and double angle identities, will strengthen your understanding of trigonometric functions. With practice and application, these identities can transform complex problems into manageable solutions, making them essential knowledge in your mathematical toolkit.