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. Mastery of these identities is essential for simplifying expressions and solving trigonometric equations.

Reciprocal Identities

• Definition: Reciprocal identities express each trigonometric function as the reciprocal of another.

• For all angles $\theta$ for which both sides 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

• Definition: Quotient identities relate tangent and cotangent to sine and cosine.

• 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

These identities are derived from the Pythagorean Theorem and the properties of even and odd functions.

Pythagorean Identities

• Definition: These identities are based on the fundamental relationship between sine and cosine on the unit circle.

• 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

• Definition: These identities describe how trigonometric functions behave when their input is negated.

• Even functions: $\cos(−\theta) = \cos \theta$, $\sec(−\theta) = \sec \theta$

• Odd functions: $\sin(−\theta) = −\sin \theta$, $\tan(−\theta) = −\tan \theta$, $\csc(−\theta) = −\csc \theta$, $\cot(−\theta) = −\cot \theta$

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 (Quadrant I): $\sin^2 \theta + \cos^2 \theta = 1$ $\sin^2 \theta = 1 - (\frac{3}{4})^2 = 1 - \frac{9}{16} = \frac{7}{16}$ $\sin \theta = \frac{\sqrt{7}}{4}$ (positive in QI)

  • (b) $\sec \theta = \frac{11}{4}$, $\tan \theta < 0$: $\cos \theta = \frac{1}{\sec \theta} = \frac{4}{11}$ $\sin^2 \theta = 1 - (\frac{4}{11})^2 = 1 - \frac{16}{121} = \frac{105}{121}$ $\sin \theta = -\frac{\sqrt{105}}{11}$ (since $\tan \theta < 0$, $\sin \theta$ is negative)

• Example 3: Find the other five trig values.

  • (a) $\sin \theta = \frac{2}{3}$, $\theta$ in QII (Quadrant II): $\cos^2 \theta = 1 - (\frac{2}{3})^2 = 1 - \frac{4}{9} = \frac{5}{9}$ $\cos \theta = -\frac{\sqrt{5}}{3}$ (negative in QII) $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{2/3}{-\sqrt{5}/3} = -\frac{2}{\sqrt{5}}$ $\csc \theta = \frac{1}{\sin \theta} = \frac{3}{2}$ $\sec \theta = \frac{1}{\cos \theta} = -\frac{3}{\sqrt{5}}$ $\cot \theta = \frac{\cos \theta}{\sin \theta} = -\frac{\sqrt{5}}{2}$

  • (b) $\tan \theta = -\frac{1}{4}$, $\cos \theta > 0$: Let $\sin \theta = -1$, $\cos \theta = 4$ (proportional values). Normalize: $r = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}$ $\sin \theta = -\frac{1}{\sqrt{17}}$, $\cos \theta = \frac{4}{\sqrt{17}}$ $\csc \theta = -\sqrt{17}$ $\sec \theta = \frac{\sqrt{17}}{4}$ $\cot \theta = -4$

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

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

• (c) $\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} = \frac{(\cos \theta + \sin \theta)(\cos \theta - \sin \theta)}{\sin \theta \cos \theta}$ (can be further simplified depending on context)

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

Additional info: When simplifying, always express all functions in terms of sine and cosine first, then apply identities as needed. This approach is especially useful for verifying identities and solving trigonometric equations.