Trigonometric Functions of Any Angle, Radian Measure, and Angular Velocity

Chapter 8: Trigonometric Functions of Any Angle

8.1 Signs of Trigonometric Functions

This section explores how the signs of trigonometric functions depend on the quadrant in which the terminal side of an angle lies. The CAST rule is a mnemonic to remember which trigonometric functions are positive in each quadrant.

• CAST Rule: The CAST rule divides the coordinate plane into four quadrants, indicating where each trigonometric function is positive.

Quadrant	Functions Positive
I (0° to 90°)	All (sin, cos, tan, csc, sec, cot)
II (90° to 180°)	sin, csc
III (180° to 270°)	tan, cot
IV (270° to 360°)	cos, sec

• Reciprocal Functions: The reciprocal of a positive trigonometric function is also positive in the same quadrant.

Examples:

• sin 350°: 350° is in Quadrant IV, where only cos and sec are positive. Therefore, sin 350° is negative.

• sec 175°: 175° is in Quadrant II, where only sin and csc are positive. Therefore, sec 175° is negative.

• tan θ < 0 and sec θ < 0: tan is negative in Quadrants II and IV; sec is negative in Quadrants II and III. Therefore, θ is in Quadrant II.

• cos θ = 0.3415: cos is positive in Quadrants I and IV.

8.2 Reference Angles

The reference angle is the acute angle formed between the terminal side of θ and the x-axis. It is always positive and less than 90° (or π/2 radians).

• Reference Angle Notation: θref

• Formulas for Reference Angles (in degrees):

Quadrant	Reference Angle Formula
I	θref = θ
II	θref = 180° – θ
III	θref = θ – 180°
IV	θref = 360° – θ

• Examples:

• θ = 217° (Quadrant III): θref = 217° – 180° = 37°

• θ = 115° (Quadrant II): θref = 180° – 115° = 65°

• θ = 55° (Quadrant I): θref = 55°

• θ = 330° (Quadrant IV): θref = 360° – 330° = 30°

8.2 Trigonometric Functions of Any Angle

To solve trigonometric equations for θ in the interval 0° ≤ θ < 360°, use the reference angle and CAST rule to determine all possible solutions.

• Procedure:

  1. Identify the quadrants where the function is positive or negative (using CAST).

  2. Find the reference angle θref using the inverse trigonometric function.

  3. Determine the actual angles θ in the specified quadrants.

• Examples:

• Given cos θ = 0.2985, θref = cos-1(0.2985) = 72.6°

• Possible θ: θ1 = 72.6° (Quadrant I), θ2 = 360° – 72.6° = 287.4° (Quadrant IV)

• Given tan θ = –1.830, θref = tan-1(1.830) = 61.3°

• Possible θ: θ1 = 180° – 61.3° = 118.7° (Quadrant II), θ2 = 360° – 61.3° = 298.7° (Quadrant IV)

8.3 Radians: Evaluating Trigonometric Functions

Trigonometric functions can be evaluated for angles in degrees or radians. Always ensure your calculator is in the correct mode.

• Six Trigonometric Functions: sin θ, cos θ, tan θ, csc θ, sec θ, cot θ

• Examples:

• sin 167° ≈ 0.225

• sec 219° ≈ –1.287

• tan 3π = 0

• csc 4.12 ≈ –1.205

• sin (4π/5) ≈ –0.7071

• cot 1.376 ≈ 0.197

8.4 Applications of Radian Measure

Arc Length

The length of an arc (s) on a circle of radius r, subtended by a central angle θ (in radians), is given by:

$s = \theta r$

• Derivation: For a full circle, s = circumference = 2πr, θ = 2π radians. Therefore, s = θr.

• Example: Find the arc length for r = 3.00 cm, θ = π/6:

• $s = \frac{\pi}{6} \times 3.00 = 1.57 \text{ cm}$

• Example: Given s = 1.010 m, θ = 136°, find r:

• Convert θ to radians: $\theta = 136^\circ \times \frac{\pi}{180^\circ} = 2.374$ radians

• $r = \frac{s}{\theta} = \frac{1.010}{2.374} = 0.425 \text{ m}$

Area of a Sector

The area (A) of a sector of a circle with radius r and central angle θ (in radians) is:

$A = \frac{1}{2} r^2 \theta$

• Derivation: The area of a full circle is $\pi r^2$. For a sector, $\frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2} r^2 \theta$.

• Example: r = 5.25 cm, θ = 218°:

• Convert θ to radians: $\theta = 218^\circ \times \frac{\pi}{180^\circ} = 3.8048$ radians

• $A = \frac{1}{2} (5.25)^2 (3.8048) = 52.4 \text{ cm}^2$

• Example: r = 110 m, θ = 75°:

• Convert θ to radians: $\theta = 75^\circ \times \frac{\pi}{180^\circ} = 1.309$ radians

• $A = \frac{1}{2} (110)^2 (1.309) = 7919 \text{ m}^2$

Angular Velocity

Angular velocity (ω) is the rate of change of the central angle θ with respect to time t. It is measured in radians per unit time.

$\omega = \frac{\theta}{t}$

• Units: radians/second (rad/s), radians/minute (rad/min), revolutions/minute (rev/min)

• Conversion: 1 revolution = 2π radians

• Examples:

• 13.5 rev/min to rad/s: $13.5 \times \frac{2\pi}{60} = 1.41 \text{ rad/s}$

• 2.49 rad/s to rev/min: $2.49 \times \frac{60}{2\pi} = 23.8 \text{ rev/min}$

Relationship Between Linear and Angular Velocity

For an object moving in a circle of radius r with angular velocity ω, the linear (tangential) velocity v is:

$v = r \omega$

• Derivation: The arc length s = rθ, so $v = \frac{s}{t} = r \frac{\theta}{t} = r \omega$

• Units: v in m/s, ω in rad/s, r in meters

• Examples:

• Propeller blade: r = 0.225 m, ω = 130 rad/s, $v = 0.225 \times 130 = 29.3 \text{ m/s}$

• Pulley: s = 7.5 m, r = 0.19 m, t = 15 s

• θ = s/r = 39.474 radians, ω = θ/t = 2.632 rad/s

• Convert to RPM: $2.632 \times \frac{60}{2\pi} = 25.13 \text{ rev/min}$

Formula Recap

• Arc Length: $s = \theta r$

• Sector Area: $A = \frac{1}{2} r^2 \theta$

• Linear Velocity: $v = r \omega$

• Angular Velocity: $\omega = \frac{\theta}{t}$

Summary Table: Key Formulas

Quantity	Formula	Units
Arc Length (s)	$s = \theta r$	length (e.g., m, cm)
Sector Area (A)	$A = \frac{1}{2} r^2 \theta$	area (e.g., m2, cm2)
Angular Velocity (ω)	$\omega = \frac{\theta}{t}$	rad/s, rad/min
Linear Velocity (v)	$v = r \omega$	m/s, cm/s

Additional info:

• When solving trigonometric equations, always consider the interval for θ (degrees or radians) and the sign of the function to determine the correct quadrants.

• For all formulas involving θ, ensure θ is in radians unless otherwise specified.

• When converting between revolutions and radians, use 1 revolution = 2π radians.