Study Notes: Trigonometric Functions (Precalculus, Ch. 5)

Trigonometric Functions: Foundations and Applications

Introduction to Angles

Angles are fundamental in trigonometry and are measured in degrees or radians. Understanding how to represent and classify angles is essential for solving trigonometric problems.

• Angle in Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. The amount of rotation from the initial side to the terminal side determines the angle's measure.

• Types of Angles:

  • Acute Angle: $0^\circ < \theta < 90^\circ$

  • Right Angle: $\theta = 90^\circ$

  • Obtuse Angle: $90^\circ < \theta < 180^\circ$

  • Straight Angle: $\theta = 180^\circ$

• Notation: Angles are often denoted as $\angle ABC$ or $\theta$.

Radians and Degree Measure

Radians are another unit for measuring angles, commonly used in higher mathematics and science. Converting between degrees and radians is a key skill in trigonometry.

• Definition: One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

• Conversion Formulas:

  • To convert degrees to radians: $\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

  • To convert radians to degrees: $\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}$

• Full Circle: $360^\circ = 2\pi$ radians

Right Triangle Trigonometry

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. These functions are foundational for solving triangles and modeling periodic phenomena.

• Primary Trigonometric Functions:

  • Sine: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

  • Cosine: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

  • Tangent: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

• Mnemonic: SOH-CAH-TOA helps remember the definitions.

Reciprocal Trigonometric Functions

In addition to sine, cosine, and tangent, there are three reciprocal trigonometric functions: cosecant, secant, and cotangent. These are defined as follows:

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

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

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

Function	Definition	Reciprocal
sin θ	Opp/Hyp	csc θ
cos θ	Adj/Hyp	sec θ
tan θ	Opp/Adj	cot θ

Examples and Applications

• Example 1: Convert $120^\circ$ to radians.

  • $120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians

• Example 2: Find $\sin \theta$ for a right triangle with opposite side 3 and hypotenuse 5.

  • $\sin \theta = \frac{3}{5}$

• Example 3: Find $\sec \theta$ for a right triangle with adjacent side 4 and hypotenuse 5.

  • $\sec \theta = \frac{5}{4}$

Additional info: Mastery of these foundational trigonometric concepts is essential for progressing to analytic trigonometry, graphing trigonometric functions, and solving real-world problems involving periodic phenomena.