Introduction to Trigonometry

Learning Objectives

• Understand the derivation and application of trigonometric ratios in right triangles, including identities such as $\sin^2 A + \cos^2 A = 1$, $1 + \tan^2 A = \sec^2 A$, and $1 + \cot^2 A = \csc^2 A$.
• Calculate the trigonometric ratios for specific angles such as $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ using geometric constructions and the Pythagorean theorem.
• Express trigonometric ratios in terms of other ratios using identities, such as expressing $\cos A$, $\tan A$, and $\sec A$ in terms of $\sin A$.
• Solve problems involving heights and distances using trigonometric ratios, including real-world applications like finding the height of a building or the distance across a river.
• Prove trigonometric equations and identities using algebraic manipulation and known identities, such as proving $\sec A (1 - \sin A)(\sec A + \tan A) = 1$.

Chapter Notes

Trigonometric Ratios and Identities

• Trigonometric Ratios in Right Triangle:
  • Sine ($\sin A$): $\frac{\text{side opposite to angle } A}{\text{hypotenuse}}$
  • Cosine ($\cos A$): $\frac{\text{side adjacent to angle } A}{\text{hypotenuse}}$
  • Tangent ($\tan A$): $\frac{\text{side opposite to angle } A}{\text{side adjacent to angle } A}$
  • Cosecant ($\csc A$): $\frac{1}{\sin A}$
  • Secant ($\sec A$): $\frac{1}{\cos A}$
  • Cotangent ($\cot A$): $\frac{1}{\tan A}$
• Trigonometric Identities:
  • $\sin^2 A + \cos^2 A = 1$
  • $1 + \tan^2 A = \sec^2 A$
  • $1 + \cot^2 A = \csc^2 A$

Trigonometric Ratios of Specific Angles

• For $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$:
  • $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, $\tan 0^\circ = 0$
  • $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$
  • $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\cos 45^\circ = \frac{1}{\sqrt{2}}$, $\tan 45^\circ = 1$
  • $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$
  • $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ$ is not defined

Trigonometric Ratios in Terms of Each Other

• Expressions:
  • If $\sin A$ is known, $\cos A = \sqrt{1 - \sin^2 A}$
  • $\tan A = \frac{\sin A}{\cos A}$
  • $\sec A = \frac{1}{\cos A}$

Applications of Trigonometric Ratios

• Solving Real-World Problems:
  • Use trigonometric ratios to find unknown heights and distances.
  • Example: Calculating the height of a building using the angle of elevation and distance from the building.

Trigonometric Equations and Proofs

• Example Proof:
  • Prove $\sec A (1 - \sin A)(\sec A + \tan A) = 1$
  • Solution:
    • LHS: $\sec A (1 - \sin A)(\sec A + \tan A)$
    • $= \frac{1}{\cos A} (1 - \sin A) \left(\frac{1}{\cos A} + \frac{\sin A}{\cos A}\right)$
    • $= \frac{1}{\cos A} (1 - \sin A) \frac{1 + \sin A}{\cos A}$
    • $= \frac{1 - \sin^2 A}{\cos^2 A}$
    • $= \frac{\cos^2 A}{\cos^2 A} = 1$