Equivalent Representations of Trigonometric Functions

Equivalent Representations of Trigonometric Functions

This note explores ways to rewrite trigonometric expressions using trigonometric identities and properties derived from the unit circle.

Trigonometric Identities

Trigonometric identities allow manipulation and rewriting of trigonometric expressions in equivalent forms.

• Reciprocal Identities:

  • $sin(x) = \frac{1}{csc(x)}$

  • $csc(x) = \frac{1}{sin(x)}$

  • $cos(x) = \frac{1}{sec(x)}$

  • $sec(x) = \frac{1}{cos(x)}$

  • $tan(x) = \frac{1}{cot(x)}$

  • $cot(x) = \frac{1}{tan(x)}$

• Quotient Identities:

  • $tan(x) = \frac{sin(x)}{cos(x)}$

  • $cot(x) = \frac{cos(x)}{sin(x)}$

Example 1: Rewrite an expression involving $tan(x)$ and $csc(x)$ as an expression involving $sin(x)$.

Example 2: Rewrite an expression involving $sec(x)$ as an expression involving $cos(x)$.

Unit Circle and Pythagorean Identity

For the unit circle, any point $(x, y)$ can be expressed as $(cos(\theta), sin(\theta))$, where:

• $x = cos(\theta)$

• $y = sin(\theta)$

Applying the Pythagorean Theorem to the right triangle formed by the point on the unit circle gives the Pythagorean identity:

• $sin^2(\theta) + cos^2(\theta) = 1$

This identity holds true for any angle $\theta$.

Manipulating the Pythagorean Identity

The Pythagorean identity can be manipulated to derive additional identities:

1. Dividing each term by $cos^2(\theta)$:

   • $\frac{sin^2(\theta)}{cos^2(\theta)} + \frac{cos^2(\theta)}{cos^2(\theta)} = \frac{1}{cos^2(\theta)}$

   • $tan^2(\theta) + 1 = sec^2(\theta)$

2. Dividing each term by $sin^2(\theta)$:

   • $\frac{sin^2(\theta)}{sin^2(\theta)} + \frac{cos^2(\theta)}{sin^2(\theta)} = \frac{1}{sin^2(\theta)}$

   • $1 + cot^2(\theta) = csc^2(\theta)$

• Pythagorean Identities:

  • $sin^2(\theta) + cos^2(\theta) = 1$

  • $1 + tan^2(\theta) = sec^2(\theta)$

  • $1 + cot^2(\theta) = csc^2(\theta)$

Right Triangle in the Unit Circle

Inscribing a right triangle in the unit circle allows expressing the height (h) using the Pythagorean Theorem:

• $x^2 + h^2 = 1$

• $h = \sqrt{1 - x^2}$

This leads to identities such as:

• $cos(arccos(x)) = x$

• $sin(arcsin(x)) = x$

Therefore:

• $sin(arccos(x)) = \sqrt{1 - x^2}$

• $cos(arcsin(x)) = \sqrt{1 - x^2}$

• Inverse Trigonometric Identities:

  • $arccos(x) + arcsin(x) = \frac{\pi}{2}$

  • $arcsin(x) + arccos(x) = \frac{\pi}{2}$

Example 3: Rewrite $f(x) = tan(x) \cdot csc(x)$ as a fraction involving powers of $cos(x)$.

Sum and Difference Identities

• Sum Identities:

  • $sin(\alpha + \beta) = sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta)$

  • $cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$

• Difference Identities:

  • $sin(\alpha - \beta) = sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta)$

  • $cos(\alpha - \beta) = cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)$

Double Angle Identities

• $sin(2\theta) = 2sin(\theta)cos(\theta)$

• $cos(2\theta) = cos^2(\theta) - sin^2(\theta) = 1 - 2sin^2(\theta) = 2cos^2(\theta) - 1$

Example 4: Simplify $2sin(\frac{\pi}{14})cos(\frac{\pi}{14})$.

Using the identity $sin(2\theta) = 2sin(\theta)cos(\theta)$, we have $sin(2 * \frac{\pi}{14}) = sin(\frac{\pi}{7})$.

Example 5: Given $k(x) = 4cos^2(x)$, find an equivalent form.

Example 6: Simplify $cos(\frac{\pi}{8})cos(\frac{\pi}{16}) - sin(\frac{\pi}{8})sin(\frac{\pi}{16})$.

Using the identity $cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$, we have $cos(\frac{\pi}{8} + \frac{\pi}{16}) = cos(\frac{3\pi}{16})$.

Example 7: Point P (5, 12) on a circle centered at the origin. Find $sin(2\theta)$.

Example 8: Two circles centered at the origin with angles $\alpha$ and $\beta$. Point P (-5, 11) is on angle $\alpha$ and Point Q (2, 5) is on angle $\beta$.

a) Find $cos(2\alpha)$.
b) Find $sin(\alpha + \beta)$.
c) Find $cos(\alpha - \beta)$.