Fundamental Trigonometric Identities and Their Applications

Fundamental Trigonometric Identities

Reciprocal Identities

Reciprocal identities express each trigonometric function as the reciprocal of another. These are essential for simplifying expressions and solving equations in trigonometry.

• Cotangent: $\cot \theta = \frac{1}{\tan \theta}$

• Secant: $\sec \theta = \frac{1}{\cos \theta}$

• Cosecant: $\csc \theta = \frac{1}{\sin \theta}$

Quotient Identities

Quotient identities relate the tangent and cotangent functions to sine and cosine. They are useful for rewriting expressions and solving trigonometric equations.

• Tangent: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• Cotangent: $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities

Pythagorean identities are derived from the Pythagorean Theorem and relate the squares of sine, cosine, tangent, secant, cotangent, and cosecant.

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Even-Odd Identities

Even-odd identities describe how trigonometric functions behave when their angle is negated. These are important for understanding symmetry and periodicity.

• $\sin(-\theta) = -\sin \theta$

• $\cos(-\theta) = \cos \theta$

• $\tan(-\theta) = -\tan \theta$

• $\csc(-\theta) = -\csc \theta$

• $\sec(-\theta) = \sec \theta$

• $\cot(-\theta) = -\cot \theta$

Note: In trigonometric identities, $\theta$ can represent an angle in degrees, a real number, or a variable.

Applications of Fundamental Identities

Finding Trigonometric Function Values Given One Value and the Quadrant

Given the value of one trigonometric function and the quadrant in which the angle lies, you can determine the values of other trigonometric functions using identities and the signs of functions in each quadrant.

• Example: If $\cos \theta = \frac{5}{8}$ and $\theta$ is in quadrant IV, find $\sin \theta$, $\tan \theta$, and $\sec(-\theta)$.

• Solution Outline:

  • Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$.

  • Determine the sign of $\sin \theta$ in quadrant IV (negative).

  • Calculate $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

  • Use the even-odd identity for $\sec(-\theta)$: $\sec(-\theta) = \sec \theta$.

Expressing One Trigonometric Function in Terms of Another

Trigonometric identities allow you to rewrite one function in terms of another, which is useful for simplifying expressions and solving equations.

• Example 1: Write $\cos x$ in terms of $\tan x$.

  • Use $\tan x = \frac{\sin x}{\cos x}$ and $\sin^2 x + \cos^2 x = 1$ to solve for $\cos x$.

• Example 2: Write $\tan \theta$ in terms of $\cos \theta$.

  • Use $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin^2 \theta + \cos^2 \theta = 1$ to express $\sin \theta$ in terms of $\cos \theta$.

Rewriting Expressions in Terms of Sine and Cosine

Many trigonometric expressions can be rewritten using only sine and cosine, which often simplifies calculations and proofs.

• Example 1: Write $\tan \theta + \cot \theta$ in terms of $\sin \theta$ and $\cos \theta$, and simplify so that no quotients appear.

  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$

  • Combine: $\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}$

  • Since $\sin^2 \theta + \cos^2 \theta = 1$, the expression simplifies to $\frac{1}{\sin \theta \cos \theta}$

• Example 2: Write $\frac{1 + \tan^2 \theta}{1 - \sec^2 \theta}$ in terms of $\sin \theta$ and $\cos \theta$, and simplify so that no quotients appear.

  • Use $\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}$ and $\sec^2 \theta = \frac{1}{\cos^2 \theta}$

  • Substitute and simplify using Pythagorean identities.

Summary Table: Fundamental Trigonometric Identities

Identity Type	Formula	Example
Reciprocal	$\cot \theta = \frac{1}{\tan \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\csc \theta = \frac{1}{\sin \theta}$	$\sec 60^\circ = \frac{1}{\cos 60^\circ} = 2$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$	$\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = 1$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$ $\tan^2 \theta + 1 = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$	$\sin^2 30^\circ + \cos^2 30^\circ = 1$
Even-Odd	$\sin(-\theta) = -\sin \theta$ $\cos(-\theta) = \cos \theta$ $\tan(-\theta) = -\tan \theta$	$\sin(-90^\circ) = -\sin 90^\circ = -1$

Additional info: The examples and solution outlines are inferred based on standard trigonometry curriculum and the context of the slides.