Trigonometric Identities: Fundamental, Reciprocal, Quotient, Pythagorean, and Negative-Angle Identities

Trigonometric Identities

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. These identities are essential tools for simplifying expressions and solving trigonometric equations.

• Reciprocal Identities: These relate each trigonometric function to its reciprocal.

• Quotient Identities: These express tangent and cotangent in terms of sine and cosine.

• Pythagorean Identities: These are derived from the Pythagorean Theorem and relate the squares of the trigonometric functions.

• Negative-Angle (Even/Odd) Identities: These describe how trigonometric functions behave when their angle is negated.

Reciprocal and Quotient Identities

Reciprocal and quotient identities allow us to express one trigonometric function in terms of another, which is useful for simplifying expressions and solving equations.

Reciprocal Identities

• For all angles $\theta$ for which both functions are defined:

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

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

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

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

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

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

Quotient Identities

• For all angles $\theta$ for which the denominators are not zero:

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

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

Pythagorean and Negative-Angle Identities

Pythagorean identities are based on the fundamental relationship between sine and cosine, while negative-angle identities describe the symmetry properties of trigonometric functions.

Pythagorean Identities

• For all angles $\theta$ for which the function values are defined:

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Negative-Angle (Even/Odd) Identities

• $\sin(-\theta) = -\sin \theta$

• $\cos(-\theta) = \cos \theta$

• $\tan(-\theta) = -\tan \theta$

• $\csc(-\theta) = -\csc \theta$

• $\sec(-\theta) = \sec \theta$

• $\cot(-\theta) = -\cot \theta$

Note: Sine, tangent, cosecant, and cotangent are odd functions; cosine and secant are even functions.

Examples and Applications

Example 1: Fill in the Blank

• a.) If $\tan \theta = 2.6$, then $\tan(-\theta) = -2.6$

• b.) If $\tan \theta = 1.6$, then $\cot \theta = \frac{1}{1.6}$

• c.) If $\sin \theta = \frac{2}{3}$, then $-\sin(-\theta) = \frac{2}{3}$

Example 2: Find $\sin \theta$

• a.) $\cos \theta = \frac{3}{4}$, $\theta$ in QI

  • Use $\sin^2 \theta + \cos^2 \theta = 1$:

  • $\sin^2 \theta = 1 - \left(\frac{3}{4}\right)^2 = 1 - \frac{9}{16} = \frac{7}{16}$

  • Since $\theta$ is in Quadrant I, $\sin \theta > 0$:

  • $\sin \theta = \frac{\sqrt{7}}{4}$

• b.) $\sec \theta = \frac{11}{4}$, $\tan \theta < 0$

  • $\cos \theta = \frac{1}{\sec \theta} = \frac{4}{11}$

  • Use $\sin^2 \theta + \cos^2 \theta = 1$:

  • $\sin^2 \theta = 1 - \left(\frac{4}{11}\right)^2 = 1 - \frac{16}{121} = \frac{105}{121}$

  • Since $\tan \theta < 0$, $\sin \theta$ and $\cos \theta$ have opposite signs. $\cos \theta > 0$, so $\sin \theta < 0$:

  • $\sin \theta = -\frac{\sqrt{105}}{11}$

Example 3: Find the Other Five Trig Values

• a.) $\sin \theta = \frac{2}{3}$, $\theta$ in QII

  • $\cos^2 \theta = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}$

  • In Quadrant II, $\cos \theta < 0$:

  • $\cos \theta = -\frac{\sqrt{5}}{3}$

  • $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{2/3}{-\sqrt{5}/3} = -\frac{2}{\sqrt{5}}$

  • $\csc \theta = \frac{1}{\sin \theta} = \frac{3}{2}$

  • $\sec \theta = \frac{1}{\cos \theta} = -\frac{3}{\sqrt{5}}$

  • $\cot \theta = \frac{\cos \theta}{\sin \theta} = -\frac{\sqrt{5}}{2}$

• b.) $\tan \theta = -\frac{1}{4}$, $\cos \theta > 0$

  • Let $\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{1}{4}$, $\cos \theta > 0$

  • Let $\cos \theta = 4k$, $\sin \theta = -1k$ for some $k > 0$

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

  • $(-1k)^2 + (4k)^2 = 1 \implies 1k^2 + 16k^2 = 1 \implies 17k^2 = 1 \implies k = \frac{1}{\sqrt{17}}$

  • $\sin \theta = -\frac{1}{\sqrt{17}}$, $\cos \theta = \frac{4}{\sqrt{17}}$

  • $\csc \theta = -\sqrt{17}$

  • $\sec \theta = \frac{\sqrt{17}}{4}$

  • $\cot \theta = \frac{\cos \theta}{\sin \theta} = -4$

Simplifying Trigonometric Expressions Using Identities

Trigonometric identities are used to rewrite expressions in terms of sine and cosine, or to simplify powers using Pythagorean identities.

• If you have a trigonometric function, rewrite using $\sin \theta$ or $\cos \theta$:

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

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

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

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

• If you have squared trigonometric functions, use Pythagorean identities:

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

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

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

Example 4: Simplify

• a.) $\cos \theta \csc \theta$

  • $\cos \theta \csc \theta = \cos \theta \cdot \frac{1}{\sin \theta} = \frac{\cos \theta}{\sin \theta} = \cot \theta$

• b.) $\sin^2 \theta (\csc^2 \theta - 1)$

  • $\csc^2 \theta = \frac{1}{\sin^2 \theta}$

  • $\sin^2 \theta (\csc^2 \theta - 1) = \sin^2 \theta \left(\frac{1}{\sin^2 \theta} - 1\right) = 1 - \sin^2 \theta = \cos^2 \theta$

• c.) $\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta}$

  • Numerator: $\cos^2 \theta - \sin^2 \theta$

  • Denominator: $\sin \theta \cos \theta$

  • Split: $\frac{\cos^2 \theta}{\sin \theta \cos \theta} - \frac{\sin^2 \theta}{\sin \theta \cos \theta} = \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta} = \cot \theta - \tan \theta$

• d.) $\tan \theta \cos \theta \csc \theta$

  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$

  • $\tan \theta \cos \theta \csc \theta = \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \cdot \frac{1}{\sin \theta} = 1$

Summary Table: Fundamental Trigonometric Identities

Type	Identity
Reciprocal	$\sin \theta = \frac{1}{\csc \theta}$, $\cos \theta = \frac{1}{\sec \theta}$, $\tan \theta = \frac{1}{\cot \theta}$, $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$, $\tan^2 \theta + 1 = \sec^2 \theta$, $1 + \cot^2 \theta = \csc^2 \theta$
Negative-Angle	$\sin(-\theta) = -\sin \theta$, $\cos(-\theta) = \cos \theta$, $\tan(-\theta) = -\tan \theta$, $\csc(-\theta) = -\csc \theta$, $\sec(-\theta) = \sec \theta$, $\cot(-\theta) = -\cot \theta$

Additional info: The above notes include expanded explanations, step-by-step examples, and a summary table for quick reference, making them suitable for exam preparation and review of trigonometric identities.